W Arendt

Isabelle Chalendar
email: isabelle.chalendar@u-pem.fr

ESSENTIALLY COERCIVE FORMS AND ASYMPTOTICALLY COMPACT SEMIGROUPS

Keywords: 2010 Mathematics Subject Classification. 47A07, 47B44, 47A12, 47A10, 47D03 sesquilinear coercive form, essentially coercive forms, elliptic forms, numerical range, essential spectrum, selfadjoint operators, asymptotically compact semigroups

Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e. that dist(S(t), K(H)) → 0 as t → ∞, where K(H) denotes the space of all compact operators on the underlying Hilbert space.

Introduction

Form methods are most efficient for proving well-posedness results for parabolic equations. We refer to the monography of Kato [START_REF] Kato | Perturbation theory for linear operators[END_REF] and Ouhabaz [START_REF] Ouhabaz | Analysis of heat equations on domains[END_REF] for example.

The scenario is the following. Given are Hilbert spaces H, V such that V ↪ d H (i.e. V is densely and continuously embedded into H) and a continuous sesquilinear form a ∶ V × V → C. Then there is a unique operator A on H whose graph is given by

G(A) = {(u, f ) ∶ u ∈ V, a(u, v) = ⟨f, v⟩ H for all v ∈ V }.
We call A the operator associated with a and write A ∼ a. The following is a classical result. Assume that a is positive-coercive, i.e.

Re a(u, u) ≥ α u 2 V for all u ∈ V and some α > 0. Then -A generates a holomorphic C 0 -semigroup S which is uniformly exponentially stable, i.e.

S(t)

≤ M e -δt for all t > 0 and for some δ > 0, M ≥ 0.

The purpose of this paper is to relax the condition of positive coerciveness as well as similar notions and to replace them by a weaker topological notion, which we call "essential versions". Definition 1.1. A continuous sesquilinear form a on V × V is called essentially positive-coercive if

u n ⇀ 0 in V and lim sup n→∞ Re a(u n , u n ) ≤ 0 implies u n V → 0 as n → ∞.
Here u n ⇀ 0 means that u n converges weakly to 0 in V . Then we show the following.

Theorem 1.2. Assume that a is essentially positive-coercive. Then -A generates a holomorphic C 0 -semigroup S which is asymptotically compact.

Here we call S asymptotially compact (sometimes called quasicompact in the literature) if there exist δ > 0 and M ≥ 0 such that S(t) ess ≤ M e -δt for all t ≥ 0, where, for T ∈ L(H), T ess is the distance of T to the space of all compact operators on the Hilbert space H, denoted by K(H). We also show that each asymptotically compact quasi-contractive holomorphic semigroup is obtained via a form in this way if we allow to pass to an equivalent scalar product on H. Theorem 1.2 seems of interest for two reasons. A specific topological property of the form a is responsible for the fact that the semigroup behaves like a finite dimensional system at infinity. But also the pure generation property is surprising since no range condition on the operator is needed. Much more generally, we investigate the numerical range W (a) ∶= {a(u, u) ∶ u ∈ V, u H = 1} of the form a, which is a convex set.

Note that the domain D(A) of A is dense in V for the topology of V . As a consequence, the numerical range of A defined by

W (A) ∶= {⟨Au, u⟩ H ∶ u ∈ D(A), u H = 1}
is dense in W (a). We call the form a coercive if a(u, u) ≥ α u 2 V for all u ∈ V and some α > 0. Note that the real part in the definition of positive-coerciveness is replaced by the absolute value.

We call a essentially coercive if u n ⇀ 0 in V and a(u n , u n ) → 0 implies u n V → 0.

Then we show in Section 2 that σ(A) ⊂ W (a) = W (A), whenever a is essentially coercive.

This spectral inclusion is remarkable since it fails in general for unbounded operators. It implies generation results. For example we show the following: if a is essentially coercive and accretive, then -A is maccretive. In other words, again, the range condition comes automatically; i.e. it is a consequence of essential coerciveness. We investigate in particular the case where a is symmetric. Letting a λ (u, v) ∶= a(u, v) -λ⟨u, v⟩ H , we prove the following.

Theorem 1.3. Assume that a is symmetric and that there exists λ ∈ C such that a λ is essentially coercive. Then A is selfadjoint and semibounded. Moreover, if λ ∈ R, then σ ess (A) ⊂ (-∞, λ) or σ ess (A) ⊂ (λ, ∞).

We also show that each semibounded selfadjoint operator can be obtained in this way by a unique essentially coercive form.

We should say some words about preceding results. The notion of essential coerciveness has been introduced in [START_REF] Arendt | Galerkin approximation in Banach and Hilbert spaces[END_REF], where it is shown that this property is equivalent to the convergence of arbitrary Galerkin approximations defined by the form. So the motivation and applications (to finite elements) were completely different in this preceding paper. A predecessor with investigations on semigroups is the paper [START_REF] Arendt | The Dirichlet-to-Neumann operator via hidden compactness[END_REF] where compactly elliptic forms play a role for investigating the Dirichlet-to-Neumann operator. One of our result shows that compact ellipticity is actually equivalent to essential positive coerciveness. We also mention that ter Elst, Sauter and Vogt [START_REF] Ter Elst | A generalisation of the form method for accretive forms and operators[END_REF] established form methods which lead to contractive semigroups which are not necessarily holomorphic. However, the use of topological conditions such as essential coerciveness (Definion 1.1) seems to be new.

The paper is organized as follows:

Contents

1

Numerical range and essential coercivity

Let V be a Hilbert space over

K = R or C. By V ′ ∶= {f ∶ V → K ∶ f is linear and continuous} we denote the anti-dual of V (where f (v) denotes the complex conjugate of f (v)). Thus, if K = R, then V ′ is just the dual space of V .
The notion of essential coerciveness has been introduced in [START_REF] Arendt | Galerkin approximation in Banach and Hilbert spaces[END_REF] to study Galerkin approximation. For our purposes, the following Fredholmproperty plays a crucial role. (2.1)

for u ∈ V, a(u, v) = 0 for all v ∈ V implies u = 0.
Then for all f ∈ V ′ , there exists a unique u ∈ V such that a(u, v) = ⟨f, v⟩ for all v ∈ V .

Here we use the bracket (without index) to indicate the evaluation; i.e., ⟨f, v⟩ = f (v) for all f ∈ V ′ , v ∈ V . The bracket ⟨u, v⟩ V for u, v ∈ V denotes the scalar product in V . Now let H be a second Hilbert space over K with scalar product ⟨u, v⟩ H for u, v ∈ H and let j ∈ L(V, H) a linear mapping with dense range. Let a ∶ V × V → K be a continuous sesquilinear form satisfying (2.2) if u ∈ ker j and a(u, v) = 0 for all v ∈ ker j then u = 0.

Then there exists a unique operator A on H whose graph is given by

G(A) = {(x, f ) ∈ H×H ∶ ∃u ∈ V, j(u) = x and a(u, v) = ⟨f, j(v)⟩ H , v ∈ V }
We call A the operator on H associated with (a, j) and write A ∼ (a, j). Property (2.2) makes sure that G(A) is the graph of a (single-valued) operator, but this property is also used in the proof of Theorem 2.3 below. Throughout this article an operator is by definition singlevalued.

The setting above has been introduced in [START_REF] Arendt | Sectorial forms and degenerate differential operators[END_REF], generalizing the common case where V ↪ d H (i.e. V embeds continuously in H with dense range) for which j is the identity mapping. See also [START_REF] Arendt | Spectral theory, mathematical system theory, evolution equations, differential and difference equations[END_REF] for an introduction and [START_REF] Sauter | Degenerate elliptic operators with boundary conditions via form methods[END_REF][START_REF] Arendt | The Dirichlet-to-Neumann operator via hidden compactness[END_REF] for more information. In general A is not a closed operator. Nevertheless, if a is essentially coercive, then we can prove quite strong spectral properties.

By W (a, j) ∶= {a(u, u) ∶ u ∈ V ; j(u) H = 1} we denote the numerical range of (a, j). For λ ∈ K, we let

a λ (u, v) = a(u, v) -λ⟨j(u), j(v)⟩ H . Thus a λ ∶ V × V → K is a continuous sesquilinear form.
We first prove a lemma.

Lemma 2.2. Let λ ∈ K W (a, j), where W (a, j) denotes the closure of W (a, j). If a is essentially coercive then so is a λ .

Proof.

Let u n ⇀ 0 in V and assume that a λ (u n , u n ) → 0 as n → ∞. Then j(u n ) H → 0. Indeed, if not, there exist a subsequence (u n k ) k and δ > 0 such that j(u n k ) H ≥ δ (k ∈ N). Let w k = u n k j(u n k ) H . Then a λ (w k , w k ) = 1 j(u n k ) 2 H a λ (u n k , u n k ) → 0 as n → ∞.
But a λ (w k , w k ) = a(w k , w k )λ and a(w k , w k ) ∈ W (a, j). This contradicts the fact that λ ∈ W (a, j).

It follows that a

(u n , u n ) = a λ (u n , u n ) + λ j(u n ) 2 H → 0 as n → ∞. Now since a is essentially coercive, we get lim n→∞ u n V = 0.
We denote by ρ(A) the resolvent set of A and for µ ∈ ρ(A) by R(µ, A) = (µ Id -A) -1 the resolvent at µ. Moreover σ(A) ∶= K ρ(A) is the spectrum of A. The following is the main result of this section.

Theorem 2.3. Assume that there exists λ ∈ K such that a λ is essentially coercive. Then

σ(A) ⊂ W (a, j) and R(µ, A) ≤ 1 dist(µ, W (a, j))
for all µ ∈ W (a, j).

Proof. a) Assume that λ = 0. Let µ ∈ K W (a, j). Then a µ is essentially coercive by Lemma 2.2. We show that a µ satisfies the uniqueness condition (2.1). Let u ∈ V such that

a µ (u, v) = a(u, v) -µ⟨j(u), j(v)⟩ H = 0 for all v ∈ V.
If j(u) ≠ 0, then a(w, w)-µ = 0 where w = u j(u) H . Thus µ ∈ W (a, j), in contradiction with the assumption. Thus j(u) = 0 and it follows from (2.2) that u = 0.

Let f ∈ H. By Proposition 2.1 there exists a unique u ∈ V such that

a µ (u, v) = ⟨-f, j(v)⟩ H for all v ∈ V.
It follows that

a(u, v) = ⟨-f + µj(u), j(v)⟩ H for all v ∈ V. Thus x ∶= j(u) ∈ D(A) and Ax = -f + µx. Moreover, if f ≠ 0, a(w, w) -µ = ⟨-f, j(w) j(u) H ⟩ H , with w ∶= u j(u) H . It follows that dist(µ, W (a, j)) ≤ ⟨f, j(w)⟩ H 1 j(u) H ≤ f H j(u) H .
Hence

x H = j(u) H ≤ 1 dist(µ, W (a, j)) f H . b) Let λ ∈ K {0} such that a λ is essentially coercive. Let µ ∈ K W (a, j).
Observe that W (a λ , j) = W (a, j)λ.

Since µ ∈ W (a λ , j) it follows that µλ ∈ W (a λ , j). Note also that (a λ , j) ∼ Aλ Id. It follows from a) that (µλ) Id -(Aλ Id) = µ Id -A is invertible and

(µ Id -A) -1 = ((µ -λ) Id -(A -λ Id)) -1 ≤ 1 dist(µ -λ, W (a λ , j)) = 1 dist(µ, W (a, j))
.

We mention that Theorem 2.3 fails in general if the form is not essentially coercive. Here is an example which is well-known (see [START_REF] Adler | A semigroup approach to the numerical range of operators on Banach spaces[END_REF]Ex. 5.3] for more information).

Example 2.4. Let H = L 2 (0, ∞), V = H 1 0 (0, ∞), j the identity and a(u, v) = ∫ ∞ 0 u ′ v. Then W (a, j) ⊂ iR as can easily be seen by integration by parts. Here the associated operator A on H is given by D(A) = V , Af = -f ′ . Thus -A is the generator of the right shift semigroup and its spectrum is σ(A) = {λ ∈ C ∶ Re (λ) ≥ 0}.

Remark 2.5. Given a closed operator A on H we always might choose V = D(A) with the graph norm u 2 A = u 2 H + Au 2 H and define a(u, v) = ⟨Au, v⟩ H . If a is essentially coercive, then

σ(A) ⊂ W (a) = {⟨Au, u⟩ H ∶ u ∈ D(A) u H = 1}.
But this a is rarely essentially coercive. In fact, if the injection V ↪ H is compact, then a is essentially coercive if and only if dim V < ∞.

Proof. Let u n ⇀ 0 in V . Then Au n ⇀ 0 in H by the definition of the norm in V . Moreover, u n → 0 in H since the embbeding is compact. It follows that ⟨Au n , u n ⟩ H → 0. Since a is essentially coercive we conclude that u n V → 0. In other words, we have shown that each weakly convergent sequence in V is norm convergent and thus dim V < ∞.

But in this situation, it may well happen that

σ(A) ⊂ {⟨Au, u⟩ H ∶ u ∈ D(A), u H = 1}
(for example if A is selfadjoint with compact resolvent). Thus the essential coercivity is not a necessary condition in Theorem 2.3.

Next we want to use Theorem 2.3 to prove several generation theorems. The first concerns selfadjoint operators, the second contraction semigroups and the third holomorphic semigroups. The point is that in the usual versions of the Lumer-Philipps Theorem, Theorem 2.3 allows us to replace the range condition by essential coercivity.

3. Selfadjoint, m-accretive and m-sectorial operators 3.1. Selfadjoint operators. Let H be a Hilbert space over C. An operator A on H is called symmetric if ⟨Ax, y⟩ H = ⟨x, Ay⟩ H for all x, y ∈ D(A).

An operator A is selfadjoint if it is densely defined and A = A * . By a well-known criterion, A is selfadjoint if and only if it is symmetric and ±i Id -A ∶ D(A) → H are surjective. Now we consider a Hilbert space V and a mapping j ∈ L(V, H) with dense range. Let a ∶ V × V → C be a continuous sesquilinear form satisfying (2.2). Let A ∼ (a, j) and a λ (u, v) ∶= a(u, v) -λ⟨j(u), j(v)⟩ H (λ ∈ C) as in Section 2.

Theorem 3.1. Assume that a is symmetric. If a λ is essentially coercive for some λ ∈ C, then A is selfadjoint.

Proof. It follows from the definition that A is symmetric. By Theorem 2.3, σ(A) ⊂ W (a, j) ⊂ R. This implies that A is selfadjoint.

We will show in Section 8 that the operator A in Theorem 3.1 is semibounded if we assume in addition that j is injective. In that case, each semibounded selfadjoint operator is obtained in that way.

We give an example to illustrate how Theorem 3.1 can be used. It is convenient (even though surprising) that only one complex number, for instance λ = i, suffices to check the essential coercivity.

Example 3.2. Let H = L 2 (R d ), d ≥ 3, m ∈ L d (R d ) real-valued, V = H 1 (R d ) and j ∶ V → H the identity. Then a(u, v) =
∇u∇v + muv defines a continuous, symmetric sesquilinear form on V × V . In fact, by Sobolev-embedding,

H 1 (R d ) ⊂ L 2d (d-2) (R d ). Since m ∈ L d (R d ), it follows that (3.1) mL 2d (d-2) (R d ) ⊂ L 2 (R d ).
We show that

a i (u, v) ∶= a(u, v) -i⟨u, v⟩ H is essentially coercive. In fact, consider a sequence (u n ) n such that u n ⇀ 0 in H 1 (R d ) and a i (u n , u n ) → 0. Then u n 2 L 2 = -Im a i (u n , u n ) → 0 as n → ∞. Since u n ⇀ 0 in H 1 (R d ), by (3.1), mu n ⇀ 0 in L 2 (R d ). Moreover, since u n → 0 in L 2 (R d ), this implies that ⟨mu n , u n ⟩ H = m u n 2 → 0 as n → ∞.
Consequently

∇u n 2 = Re a i (u n , u n ) - m u n 2 → 0 as n → ∞.
We have shown that u n H 1 → 0 and thus a i is essentially coercive. It follows from Theorem 3.1 that the operator A associated with (a, j) on H is selfadjoint. Using (3.1) one sees that

D(A) = H 2 (R d ) and Au = ∆u + mu for all u ∈ D(A).
We next present an example where j is not injective. In fact, we consider the Dirichlet-to-Neumann operator associated to the problem

-∆u + mu = 0,
where m is a measurable function. This was the prototype example in [START_REF] Arendt | The Dirichlet-to-Neumann operator via hidden compactness[END_REF], where m had been chosen bounded. Here we allow more general m to which Theorem 3.1 can be applied conveniently.

Example 3.3. (Dirichlet-to-Neumann operator) Let Ω ⊂ R d be a bounded, open Lipschitz domain, d ≥ 3, V = H 1 (Ω), m ∈ L d (Ω). Then a(u, v) = Ω ∇u∇v + Ω muv defines a continuous, symmetric form on V . We claim that a is essen- tially coercive. Proof. Let u n → 0 in H 1 (Ω) such that a(u n , u n ) → 0. Since H 1 (Ω) ↪ L 2d (d-2) , there exists c ≥ 0 such that u n L 2d (d-2) ≤ c for all n ∈ N.
Since the embedding

H 1 (Ω) ↪ L 2 (Ω) is compact, u n → 0 in L 2 (Ω) as n → ∞. Thus Ω m u n 2 ≤ u n L 2 (Ω) Ω m 2 u n 2 1 2 ≤ u n L 2 (Ω) Ω m 2 d 2 2 d 1 2 Ω u n 2 d d-2 d-2 d 1 2 ≤ u n L 2 (Ω) m L d (Ω) c → 0 as n → ∞. Hence ∫ Ω ∇u n 2 → 0 as n → ∞. We have shown that u n H 1 (Ω) → 0 as n → ∞. Now let H = L 2 (∂Ω) and let j = tr ∶ H 1 (Ω) → L 2 (∂Ω)
be the trace operator. Denote by A the operator associated with (a, j) on L 2 (∂Ω).Then A is selfadjoint by Theorem 3.1.

We claim that

G(A) = {(g, h) ∈ L 2 (∂Ω) × L 2 (∂Ω) ∶ ∃u ∈ H 1 (Ω), -∆u + mu = 0, tr u = g, ∂ ν u = h}. Proof. ⊂ Let (g, h) ∈ G(A). By definition there exists u ∈ H 1 (Ω) such that tr u = g and a(u, v) = ∂Ω htr(v) for all v ∈ H 1 (Ω). Choosing v ∈ D(Ω), one sees that -∆u + mu = 0. Thus Ω ∆uv + Ω ∇u∇v = a(u, v) = ∂Ω htr(v)
for all v ∈ H 1 (Ω). This means by definition that ∂ ν u = h in the weak sense.

⊃ The proof is similar.

The characterization of selfadjoint operators which are associated with an essentially coercive form will be given in Section 6 (see Corollary 6.6 and 6.5).

m-accretive operators.

Let K = R or C and let H be a Hilbert space. An operator A on H is called accretive if Re ⟨Au, u⟩ H ≥ 0 for all u ∈ D(A). The operator A is called m-accretive if it is accretive and Id +A ∶ D(A) → H is surjective. By the Lumer-Philipps Theorem A is m-accretive if and only if -A generates contractive C 0 -semigroup on H. Let V be a Hilbert space over K and j ∈ L(V, H) with dense range. Let a ∶ V × V → K be a continuous sesquilinear form satisfying (2.2). We say that a is accretive if Re a(u, u) ≥ 0 for all u ∈ V.

Denote by A the operator on H associated with (a, j). Theorem 3.4. Assume that a is accretive and a λ is essentially coercive for some λ ∈ K. Then A is m-accretive.

Proof. It follows from the definition of the operator associated with (a, j) that A is accretive. By Theorem 2.3,

σ(A) ⊂ W (a, j) ⊂ {λ ∈ C ∶ Re (λ) ≥ 0}. Thus -1 ∈ σ(A).
3.3. m-sectorial operators. Let V, H be complex Hilbert spaces and let j ∈ L(V, H) have dense range. Let a ∶ V × V → C be a continuous sesquilinear form satisfying (2.2). We say that a is j-sectorial if there exist θ ∈ (0, π 2 ) and w ∈ R such that (3.2) a(u, u) + w j(u) 2 H ∈ Σ θ for all u ∈ V, where Σ θ ∶= {re iα ∶ r > 0, α < θ}. Recall that an operator A on H is called m-sectorial if there exist 0 < θ < π 2 and ω ≥ 0 such that the numerical range W (A) of A is contained in Σ θ +ω and such that A+ω+1 is surjective. This can be reformulated by saying that -A generates a holomorphic C 0 -semigroup which is quasi-contractive on a sector. We refer to [START_REF] Kato | Perturbation theory for linear operators[END_REF] for this and further information.

Theorem 3.5. Assume that a) a is j-sectorial; b) a λ is essentially coercive for some λ ∈ C. Then the operator A associated with (a, j) on H is m-sectorial. Proof. Replacing a(u, v) by a(u, v) + w⟨j(u), j(v)⟩ H for u, v ∈ V , we may assume that w = 0 in (3.2). Let θ ′ = π 2 -θ.
Then Re e ±iθ ′ a(u, u) ≥ 0 for all u ∈ V . It follows from Theorem 3.4 that -e ±iθ ′ A is the generator of a contractive C 0 -semigroup. This implies that -A generates a holomorphic C 0 -semigroup which is contractive on Σ θ ′ , see, e.g., [START_REF] Arendt | Spectral theory, mathematical system theory, evolution equations, differential and difference equations[END_REF]Theorem 3.2].

Structure theorems for essential coerciveness

Throughout this section V is a complex, separable Hilbert space of infinite dimension and a ∶ V × V → C is a continuous sesquilinear form.

Recall that a is called coercive if there exists α > 0 such that

a(u, u) ≥ α u 2 V for all u ∈ V. We call a real-coercive if there exists α > 0 such that Re (a(u, u)) ≥ α u 2
V for all u ∈ V and positive-coercive if there exists α > 0 such that Re (a(u, u)) ≥ α u 2

V for all u ∈ V. Recall that the numerical range of a (with respect to V )

W (a, V ) ∶= {a(u, u) ∶ u V = 1}
is convex, see [START_REF] Gustafson | The Toeplitz-Hausdorff theorem for linear operators[END_REF]. This implies the following.

u n ⇀ 0 in V and Re a(u n , u n ) → 0 implies u n V → 0. b) The form a is called essentially positive-coercive if u n ⇀ 0 in V and lim sup n→∞ Re a(u n , u n ) ≤ 0 implies u n V → 0.
It is obvious that ess. positive-coercive ⇒ ess. real-coercive ⇒ ess. coercive, where ess. is the abbreviation of essentially.

For the following proof of the essential analogue of Proposition 4.1 a) we use a result from [START_REF] Arendt | Galerkin approximation in Banach and Hilbert spaces[END_REF].

Proposition 4.3. The form a is essentially coercive if and only if there exists θ ∈ R such that e iθ a is essentially positive-coercive.

Proof. Assume that a is essentially coercive. By [START_REF] Arendt | Galerkin approximation in Banach and Hilbert spaces[END_REF]Theorem 4.4], there exists a compact operator K ∶ V → V ′ such that the form b given by b(u, v) = a(u, v) + ⟨Ku, v⟩ is coercive. Thus, by Proposition 4.1, there exist θ ∈ R and α > 0 such that Re e iθ (a(u, u) + ⟨Ku, u⟩) ≥ α u 2 V for all u ∈ V. This implies that e iθ a is essentially positive-coercive. In fact, consider

u n ⇀ 0 in V such that lim sup n→∞ Re (e iθ a(u n , u n )) ≤ 0. Since K is compact, Ku n V ′ → 0. Consequently, ⟨Ku n , u n ⟩ V → 0 as n → ∞. We conclude by α lim sup n→∞ u n 2 V ≤ lim sup n→∞ Re (e iθ a(u n , u n )) ≤ 0.
We will see that also Proposition 4.1 b) has an essential analogue. Before that we give the following characterization of essential positive coerciveness which shows that this property is indeed the same as positive coerciveness after compact perturbation. Recall that dim V = ∞.

Theorem 4.4. The following assertions are equivalent:

(i) the form a is essentially positive-coercive;

(ii) there exist a finite dimensional subspace

V 1 of V and α > 0 such that Re a(u, u) ≥ α u 2 V for all u ∈ V ⊥ 1 ; (iii) there exist a finite rank operator K ∶ V → V ′ and α > 0 such that Re (a(u, u) + ⟨Ku, u⟩) ≥ α u 2 V for all u ∈ V ; (iv) there exist a compact operator K ∶ V → V ′ and α > 0 such that Re (a(u, u) + ⟨Ku, u⟩) ≥ α u 2 V for all u ∈ V. (v) there exist a Hilbert space Y , a compact operator K ∶ V → Y and α > 0 such that Re (a(u, u) + Ku 2 Y ) ≥ α u 2
V for all u ∈ V. We precede the proof by a lemma. The arguments are similar to [5, Theorem 4.2 (i)⇒ (ii)]. To be complete, we give them (in a more concise way). Lemma 4.5. If a is essentially real-coercive, then there exists a finite dimensional subspace

V 1 ⊂ V such that a is real-coercive on V ⊥ 1 × V ⊥ 1 .
Proof. Let P n be orthogonal projections of finite rank converging strongly to the identity. We claim that there exist α > 0 and n ∈ N such that (4.1) Re a(u, u)

+ P n u 2 V ≥ α u 2 V for all u ∈ V. In fact, if not, we find u n ∈ V such that u n V = 1 and (4.2) Re a(u n , u n ) + P n u n 2 V < 1 n .
We may assume that u n ⇀ u in V , taking a subsequence otherwise.

Then

P n u n ⇀ u as n → ∞. In fact, let v ∈ V . Then P n v → v. Hence ⟨P n u n , v⟩ V = ⟨u n , P n v⟩ V → ⟨u, v⟩ V .
Since P n u n V → 0 by (4.2), it follows that u = 0. Thus Re a(u n , u n ) → 0 by (4.2), contradicting essential real coerciveness. Thus (4.1) holds.

If we choose V 1 ∶= P n V , then V ⊥ 1 = ker P n and by (4.1) (4.3) Re a(u, u) ≥ α u 2 V for all u ∈ V ⊥ 1 .
Proof of Theorem 4.4. (i) ⇒ (ii) By Lemma 4.5 there exists a finite dimensional subspace V 1 of V such that (4.3) holds. By Proposition 4.1 b) two cases may occur. First case:

-Re a(u, u) ≥ α u 2 V for all u ∈ V ⊥ 1 . Since dim V ⊥ 1 = ∞, there exist u n ∈ V ⊥ 1 such that u n V = 1 and u n ⇀ 0. Since Re a(u n , u n ) ≤ -α ≤ 0,
this contradicts essential positive coerciveness. Thus this case is excluded, we are in the second case and (ii) is proved.

(ii) ⇒ (iii) Denote by P the orthogonal projection onto V 1 and by

Q ∶= Id -P the one onto V ⊥ 1 . Thus a(Qu, Qu) ≥ α Qu 2 V for all u ∈ V. Let b(u, v) = a(Qu, P v) + a(P u, Qv) + a(P u, P v) for all u, v ∈ V.
Then a(u, v) = a(Qu, Qv) + b(u, v) for all u, v ∈ V . Since P has finite rank, there exists a finite rank operator

K 1 ∶ V → V ′ such that b(u, v) = ⟨K 1 u, v⟩ for all u, v ∈ V, cf. [5, Theorem 4.2]. Then define K ∶ V → V ′ by ⟨Ku, v⟩ = -⟨K 1 u, v⟩ + α⟨P u, v⟩ V .
Then K has finite rank and Re (a(u, u) + ⟨Ku, u⟩) = Re (a(Qu, Qu)

+ α⟨P u, u⟩ V ) ≥ α( Qu 2 V + P u 2 V ) = α u 2 V . (iii) ⇒ (iv) is trivial. (iv) ⇒ (i) Let u n ⇀ 0 in V such that lim sup n→∞ Re a(u n , u n ) ≤ 0. Since K is compact, Ku n → 0 in V ′ and thus ⟨Ku n , u n ⟩ → 0. Thus lim sup n→∞ α u n 2 V ≤ lim sup n→∞ Re a(u n , u n ) ≤ 0. (v) ⇒ (i) The proof is the same since u n ⇀ 0 in V implies Ku n Y → 0 as n → ∞. (iii) ⇒ (v) Let j ∶ V → V ′ be the Riesz isomorphism. Then J = j -1 ○ K ∶ V → V is of finite rank and Re a(u, u) + Re ⟨Ju, u⟩ V ≥ α u 2 V . Let K 1 = 1 2 (J + J * ). Then Re ⟨Ju, u⟩ V = ⟨K 1 u, u⟩ V and K 1 is self- adjoint.
Since J is of finite rank, we can represent this operator by J = ∑ n k=1 ⟨., f k ⟩ V g k where the f k , g k are vectors in V . Hence J * is given by J * = ∑ n k=1 ⟨., g k ⟩ V f k . Consequently, K 1 is of finite rank. Thus there exist orthonormal vectors e 1 , ⋯, e n ∈ V and λ k ∈ R such that

K 1 u = n k=1 λ k ⟨u, e k ⟩ V e k . Choose λ = max{0, λ 1 , ⋯, λ n } and let K 2 = √ λP where P u ∶= n k=1
⟨u, e k ⟩ V e k defines an orthogonal projection of finite rank. Then

K 2 u 2 V = λ P u 2 V = λ⟨P u, u⟩ V = λ n k=1 ⟨u, e k ⟩ V 2 ≥ n k=1 λ k ⟨u, e k ⟩ V 2 = ⟨K 1 u, u⟩ V . Thus Re a(u, u) + K 2 u 2 V ≥ α u 2 V .
Next we want to characterize essential real coerciveness. It is obvious that -a is real-coercive if and only if a is real-coercive. Moreover, essential positive coerciveness is stronger than real coerciveness. Recall once more that we assume that dim V = ∞. Theorem 4.6. Assume that a is essentially real-coercive. Then either a is essentially positive-coercive or -a is essentially positive-coercive.

Proof. a) First we show that the two assertions are exclusive. Assume that a and -a are essentially positive-coercive. By Theorem 4.4 there exist α j > 0 and compact operators

K j ∶ V → V ′ (j = 1, 2) such that Re ((-1) j a(u, u) + ⟨K j u, u⟩) ≥ α j u 2
V . Adding these two inequalities, we deduce that

Re ⟨(K 1 + K 2 )u, u⟩ ≥ (α 1 + α 2 ) u 2 V for all u ∈ V. It follows that K 1 + K 2 ∶ V → V ′ is invertible and compact, which is impossible since dim V = ∞. b) By Lemma 4.5 there exist α > 0 and a finite dimensional subspace V 1 of V such that Re a(u, u) ≥ α u 2
V for all u ∈ V ⊥ 1 . By Proposition 4.1, two cases can occur :

Re a(u, u) ≥ α u 2 V for all u ∈ V ⊥ 1 or -Re a(u, u) ≥ α u 2 V for all u ∈ V ⊥ 1
In the first case a is essentially positive-coercive, by Theorem 4.4, and in the second case -a is essentially positive-coercive.

Asymptotic compactness and compact perturbation of forms

In this section we study when a semigroup approaches a finite dimensional semigroup as t → ∞. We call this property asymptotic compactness. Our main result is concerned with compact perturbation of forms for which we show that they preserve asymptotic compactness. This section is of independent interest. Now assume that -A generates a C 0 -semigroup S on a complex Banach space X. Assume that σ 1 is a compact and relatively open subset of σ(A). Then there exists a unique decomposition

(5.1) X = X 1 ⊕ X 2 ,
where X j are closed subspaces such that S(t)X j ⊂ X j , such that A 1 is bounded and σ(A 1 ) = σ 1 , σ(A 2 ) = σ(A) σ 1 , where -A j is the generator of S X j for j = 1, 2. We refer to [17, A-III Theorem 3.3]. The projection P σ 1 onto X 1 along (5.1) is called the spectral projection associated with σ 1 . If λ is an isolated point, we call P λ ∶= P {λ} the spectral projection associated with λ.

Let A be a closed operator on X. We say that A is a Fredholm operator if ker A and X range(A) have finite dimension. This implies that range(A) is closed in X. By There are several different definitions of the essential spectrum (see [START_REF] Edmunds | Spectral theory and differential operators. 2nd revised edition[END_REF] for 5 definitions). For example, in [START_REF] Reed | Methods of modern mathematical physics. I: Functional analysis[END_REF], the essential spectrum is the complement in σ(A) of the set of all isolated points in σ(A) with spectral projection of finite rank. For selfadjoint operators this coincides with our definition here, by Proposition 5.1, which also shows that the notion of spectral radius is independent of the definition.

ρ F (A) ∶= {λ ∈ C ∶ λ Id -A is a
For X a Banach space and T ∈ L(X), we let

T ess ∶= inf K∈K(X) T -K ,
where K(X) is the closed ideal of L(X) consisting of all compact operators. The Calkin algebra L(X) K(X) is a Banach algebra for the norm T ∶= T ess where T ↦ T ∶ L(X) → L(X) K(X) is the quotient mapping. As is well-known, one has

σ ess (T ) = σ( T ),
where σ( T ) denotes the spectrum of T in the Calkin algebra. We denote by r ess (T ) = sup{ λ ∶ λ ∈ σ ess (T )} the essential spectral radius of T . Now let S be a C 0 -semigroup on X and -A its generator. We denote by ω ess (A) ∶= inf{w ∈ R ∶ ∃M ≥ 0 such that S(t) ess ≤ M e wt for all t ≥ 0} the essential growth bound of S. Thus We call the semigroup S asymptotically compact if ω ess (S) < 0. Here we deviate from the terminology in [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF] and [START_REF] Nagel | One-parameter semigroups of positive operators[END_REF] where quasi-compact is used instead.

Recall that S is called uniformly exponentially stable if there exist ε > 0, M ≥ 1 such that S(t) ≤ M e -εt (t ≥ 0). Asymptotic compactness can be characterized as follows.

Proposition 5.2. Let S be C 0 -semigroup on X with generator A. The following assertions are equivalent: a) S is asymptotically compact; b) there exists a decomposition X = X 1 ⊕ X 2 where X j , j = 1, 2 are closed subspaces which are invariant under S such that dim X 1 < ∞ and S 2 is uniformly exponentially stable, where S 2 (t) = S(t) X 2 .

Proof. b) ⇒ a) Since S(t) ess ≤ S 2 (t) ≤ M e -εt , it follows that ω ess (A) < 0. a) ⇒ b) Since ω ess (A) < 0, one has r ess (S(1)) = e ωess(S) < 1. Let r ∈ (r ess (S(1)), 1) and note that σ 1 ∶= {λ ∈ σ(S(1)) ∶ λ > r} is finite and the spectral projection P for S(1) associated with σ 1 has finite rank. Then P X =∶ X 1 and X 2 ∶= ker P define a decomposition with the desired properties (cf. [17, A III Corollary 3.5]).

Thus, a semigroup S is asymptotically compact if and only if it converges to a finite dimensional semigroup as t → ∞, and this exponentially fast. This implies that the qualitative behaviour of S(t) when t → ∞ is determined by a finite dimensional system.

We would like to add the following property which is basically a corollary of Proposition 5. of Y invariant by S. Let S 2 = S Y 2 . Then T 2 X 2 ⊂ Y 2 . It follows from Lemma 5.6 that σ(S 2 ) ⊂ σ(T 2 ) ∪ {0}. Hence r(S 2 ) ≤ r(T 2 ) < r. Now let λ > r. We show that λ Id -S is a Fredholm operator. It is clear that ker(λ Id -S) ⊂ ker(λ Id -T ) has finite dimension. Since λ Id -S 2 is invertible,

(λ Id -T )Y ⊃ (λ Id -S 2 )Y 2 = Y 2 .
We show that Y 2 has finite codimension in Y (which in turn, implies that (λ Id -T )Y has finite codimension in Y ). Since X 2 has finite codimension there exist ϕ 1 , ⋯, ϕ n ∈ X ′ such that

X 2 = n ⋂ j=1 ker ϕ j .
This implies that Y 2 = ⋂ n j=1 ker ϕ j Y , which proves the claim. Hence ω ess (B) ≤ ω ess (A). b) Consider Z ∶= D(A) with the graph norm. Then T 1 (t) ∶= T (t) Z is a C 0 -semigroup which is similar to T . Its generator A 1 is the part of A in Z. Because of the similarity we have ω ess (A 1 ) = ω ess (A). It follows from the closed graph theorem that Z ↪ Y . Since T 1 (t) = T (t) Z = S(t) Z , it follows from a) that ω ess (A 1 ) ≤ ω ess (B).

Next we want to consider semigroups associated with a form. Let V, H be Hilbert spaces over C such that V ↪ d H. Let a ∶ V × V → C be a continuous sesquilinear form. As before we define an operator A on H by

D(A) ∶= {u ∈ V ∶ ∃f ∈ H, a(u, v) = ⟨f, v⟩ H for all v ∈ V }, Au ∶= f.
We call A the operator associated with a (on H) and write A ∼ a.

The form a is called H-elliptic if there exist w ≥ 0, α > 0 such that Re a(u, u) + w u 2 H ≥ α u 2 V for all u ∈ V.

Note that a continuous H-elliptic form is the same as a closed sectorial form in the terminology of Kato [START_REF] Kato | Perturbation theory for linear operators[END_REF]. If a is continuous and H-elliptic, then the associated operator A is msectorial, i.e., -A generates a holomorphic C 0 -semigroup S ∶ Σ θ → L(X) satisfying S(t) ≤ e w z for all z ∈ Σ θ for some w ∈ R and where θ ∈ (0, π 2], Σ θ = {re iα ∶ r > 0, α < θ}. Moreover, each m-sectorial operator can be obtained in this way (and the space V as well as the form a ∶ V × V → C such that a ∼ A are unique). We refer to [START_REF] Kato | Perturbation theory for linear operators[END_REF].

Moreover, there is a natural operator A on V ′ associated with a, namely by defining D(A) = V and ⟨Au, v⟩ = a(u, v).

Then also -A is the generator of a holomorphic C 0 -semigroup S on V ′ (which might no longer be quasi-contractive, see [START_REF] Arendt | Semigroups and evolution equations: functional calculus, regularity and kernel estimates[END_REF]). As usual we identify H with a subspace of V ′ by stipulating ⟨f, v⟩ ∶= ⟨f, v⟩ H for all f ∈ H, v ∈ V . Then S(t) H = S(t) (t ≥ 0). It follows from Theorem 5.5 that ω ess (A) = ω ess (A). This will be needed in the next perturbation result. This in turn is crucial for characterizing those operators which are associated with an essentially coercive form on H. Theorem 5.9. Let V, H be complex Hilbert spaces such that

V ↪ d H and let a ∶ V × V → C be a continuous H-elliptic form, A ∼ a. Let K ∶ V → V ′ be compact and define b ∶ V × V → C by b(u, v) = a(u, v) + ⟨Ku, v⟩.
(1) Then the form b is continuous and H-elliptic.

(2) Let B ∼ b. Then ω ess (B) = ω ess (A).

Proof. (1) We can assume that

Re a(u, u) ≥ α u 2 V (u ∈ V ),
where α > 0 (replacing a by a(., .) + w⟨., .⟩ H otherwise). Assume that b is not H-elliptic. Then there exists u n ∈ V such that u n V = 1 and

(5.2) Re a(u n , u n ) + Re ⟨Ku n , u n ⟩ + n u n 2 H < 1 n .
Passing to a subsequence, we may assume that

u n ⇀ u in V . Since K ∶ V → V ′ is compact, it follows that Ku n → Ku in V ′ . Hence ⟨Ku n , u n ⟩ → ⟨Ku, u⟩. Thus (5.2) implies that u n 2 H → 0 as n → ∞. Since V ↪ d H it follows that u = 0. Hence Re ⟨Ku n , u n ⟩ → 0 as n → ∞.
Thus (5.2) contradicts that Re a(u n , u n ) ≥ α > 0 for all n ∈ N.

(2) Let T be the C 0 -semigroup generated by A on V ′ , and S the semigroup generated by A + K on V ′ (where D(A + K) = D(A) = V ). It follows from Theorem 5.4 that ω ess (A) = ω ess (A + K). Moreover, by Theorem 5.5, ω ess (A) = ω ess (A) and ω ess (A + K) = ω ess (B). Thus ω ess (A) = ω ess (B).

Essentially positive coercive-forms and asymptotic compactness

In this section we show that the semigroup associated with a continuous elliptic form is asymptotically compact if (and basically only if) the form is essentially positive-coercive. We first prove that essential positive coerciveness implies already ellipticity. This could be derived from [9, Lemma 4.14] together with Theorem 4.4. However, the following proof is more direct.

Throughout this section we consider V, H complex, separable Hilbert spaces such that V ↪ d H and we let a ∶ V × V → C be sesquilinear and continuous. Proposition 6.1. If a is essentially positive-coercive, then a is Helliptic, i.e. there exist α > 0 and w ≥ 0 such that Re a(u, u) + w u 2 H ≥ α u 2 V for all u ∈ V. Proof. If a is not H-elliptic, then there exist u n ∈ V such that u n V = 1 and

(6.1) Re a(u n , u n ) + n u n 2 H < 1 n .
The continuity of a implies that Re a(u n , u n ) ≥ -M . Hence by (6.1),

u n 2 H ≤ 1 n M + 1 n .
It follows that lim n→∞ u n = 0 in H. Since V is reflexive this implies that u n ⇀ 0 in V . Now (6.1) yields a contradiction to essential positivecoerciveness.

Let a be an H-elliptic form with associated operator A on H. Denote by S the semigroup generated by -A on H. Thus S is holomorphic. We want to study the asymptotic behaviour of S(t) as t → ∞. Remark 6.2. If w = 0, i.e. if a is positive-coercive, then a is uniformly exponentially stable. In fact there exists c H > 0 such that

u H ≤ c H u V for all u ∈ V. Choose ε = α c 2 H . Then Re a(u, u) -ε u 2 H ≥ α u 2 V -εc 2 H u 2 V = 0 for all u ∈ V.
Thus Aε Id is accretive. It follows that the semigroup generated by -A + ε Id, i.e. (e εt S(t)) t≥0 is contractive. Thus S(t) ≤ e -εt for all t ≥ 0.

Thus positive-coercive forms lead to exponentially stable semigroups. We show now that essentially positive-coercive forms generate asymptotically compact semigroups. Theorem 6.3. Let a be an essentially positive-coercive form. Let A ∼ a and denote by S the semigroup generated by -A. Then S is asymptotically compact.

Proof. By Theorem 4.4 there exists a compact operator

K ∶ V → V ′ such that the form b is positive-coercive, where b(u, v) = a(u, v) + ⟨Ku, v⟩ for all u, v ∈ V.
Let B ∼ b on H and let T be the semigroup generated by -B. Then T is uniformly exponentially stable by Remark 6.2. Thus ω ess (T ) < 0. It follows from Theorem 5.9 that ω ess (S) = ω ess (T ). Thus S is asymptotically compact. Corollary 6.4. Let A ∼ a where a is an essentially positive-coercive form. Then there exists ε > 0 such that

σ ess (A) ⊂ {λ ∈ C ∶ Re (λ) ≥ ε}.
In particular, each λ ∈ σ(A) with Re (λ) < ε is an isolated point of the spectrum with finite dimensional spectral projection.

Next we answer the question which selfadjoint operators are associated with symmetric essentially coercive forms which remained open in Section 3. Corollary 6.5. Let A be a closed operator on H. The following assertions are equivalent.

(i) There exist V ↪ d H and a continuous, symmetric, essentially positive-coercive sesquilinear form a ∶ V ×V → C such that A ∼ a; (ii) A is selfadjoint and there exists ε > 0 such that

σ ess (A) ⊂ {λ ∈ C ∶ Re (λ) ≥ ε}.
In that case the form a in (i) is unique.

Proof. (i) ⇒ (ii) follows from Corollary 6.4 and the selfadjointness of A from Proposition 6.1 or Theorem 3.1. (ii) ⇒ (i): it follows from (ii) that there exists δ > 0 such that σ δ (A) ∶= {λ ∈ σ(A) ∶ Re (λ) ≤ δ} is finite and each λ ∈ σ δ (A) is an eigenvalue with finite dimensional eigenspace (see also [START_REF] Reed | Methods of modern mathematical physics. I: Functional analysis[END_REF]VII.3]). Denote by P the spectral projection associated with σ(A) ∩ (-∞, δ). Then P is selfadjoint of finite rank. The space P H has an orthonormal basis e 1 , ..., e n of eigenvectors of A, i.e., e k ∈ D(A) and Ae k = λ k e k for k = 1, ..., n. Let B be defined on

D(B) = D(A) by Bu = Au -∑ n k=1 λ k ⟨u, e k ⟩ H e k . Then B is selfadjoint and σ(B) ⊂ [δ, ∞). Thus there exist a Hilbert space V , V ↪ d H and a coercive, continuous, symmetric form b ∶ V × V → K such that B ∼ b. Define a ∶ V × V → K by a(u, v) = b(u, v) + n k=1 λ k ⟨u, e k ⟩ H ⟨e k , v⟩ H .
Then A ∼ a. Uniqueness follows from the fact that for each m-sectorial operator A there is a unique closed form a such that A ∼ a, see [14, VI. Theorem 2.7]. Corollary 6.6. Let A be a closed operator on H. The following assertions are equivalent.

(i) There exist V ↪ d H and a symmetric, essentially coercive form a ∶ V × V → C such that A ∼ a.

(ii) The operator A is selfadjoint and there exists δ > 0 such that

σ ess (A) ⊂ (δ, ∞) or σ ess (A) ⊂ (-∞, -δ).
Proof. If a is symmetric, then essentially coercive is the same as essentially real-coercive and this in turn is equivalent to a or -a being essentially positive-coercive by Theorem 4.4. Now Corollary 6.6 follows from Corollary 6.5.

A useful criterion for proving essential real-coerciveness is the following. Lemma 6.7. Let a ∶ V × V → C be of the form a = a 0b where a 0 is a real-coercive form and b is a continuous sesquilinear form satisfying

u n ⇀ 0 in V ⇒ lim sup n→∞ Re b(u n , u n ) ≤ 0.
Then a is essentially positive-coercive.

Proof. Let u n ⇀ 0 such that lim sup n→∞ Re a(u n , u n ) ≤ 0. Then, for some α > 0, lim sup

n→∞ α u n 2 V ≤ lim n→∞ Re a 0 (u n , u n ) = lim sup n→∞ (Re a(u n , u n ) + Re b(u n , u n )) ≤ 0. Remark 6.8. If the injection V ↪ H is compact, then a is H-elliptic if
and only if a is essentially positive-coercive (see Proposition 6.1 for one direction, the other is obvious). In that case, the associated semigroup S consists of compact operators and so ω ess (S) = -∞. We now give an example to show how Theorem 6.3 can be applied. In this example, the embedding of V in H is not compact. Example 6.9. Let Ω = R d ω be an exterior domain where ω is a bounded Lipschitz domain and

d ≥ 3. Let H = L 2 (Ω), V = H 1 (Ω),
and consider the form a given by

a(u, v) = Ω ∇u∇v + δ Ω uv + d j=1 Ω (b j u∂ j v + c j v∂ j u),
where the coefficients b j , c j are complex-valued in L d (Ω) ∩ L 2q (q-2) loc (Ω), where 2 < q < 2d (d-2) and where δ > 0. Then the form a is continuous and essentially positive-coercive. Let A ∼ a, S the semigroup generated by -A. Then S is asymptotically compact. Note that A is selfadjoint if b j = c j for j = 1, ⋯, d.

Proof. Since H 1 (Ω) ↪ L 2d (d-2) (Ω), the form a is continuous. By Lemma 6.7 it suffices to show the following. Let

u n ⇀ 0 in H 1 (Ω). Then ∫ Ω gu n ∂ j u n → 0 as n → ∞, where g ∈ L d (Ω) ∩ L 2q (q-2) loc (Ω). Let ε > 0 and let B be a ball large enough such that ∫ Ω B g d 1 d ≤ ε. Since (u n ) n is bounded in H 1 (Ω), there exists a constant c > 0 such that u n L 2d (d-2) ≤ c and ∂ j u n L 2 ≤ c for all n ∈ N. Thus Ω B gu n ∂ j u n ≤ c Ω B gu n 2 1 2 ≤ c Ω B g 2 d 2 2 d 1 2 Ω u n 2 d d-2 d-2 d 1 2
≤ c 2 ε for all n ∈ N.

Note that the embedding

H 1 (Ω ∩ B) ↪ L q (Ω ∩ B) is compact. Thus u n → 0 as n → ∞ in L q (Ω ∩ B). We estimate Ω∩B gu n ∂ j u n ≤ c Ω∩B gu n 2 1 2 ≤ c Ω∩B g 2 q q-2 q-2 2 1 2 Ω∩B u n 2 q 2 2 q 1 2 ≤ c g L 2q (q-2) (B∩Ω) u n L q (Ω∩B)
which converges to 0 as n → ∞. Thus

lim sup n→∞ Re Ω gu n ∂ j u n ≤ c 2 ε
where ε > 0 is arbitrary. This proves the claim. The remaining assertions follow from Theorem 6.3.

Remark 6.10. Another condition on the coefficients is b j , c j ∈ L ∞ (Ω) and lim

R→∞ ( b j L ∞ (Ω B R ) + c j L ∞ (Ω B R ) ) = 0,
where B R is the ball in R d of radius R. The proof is similar.

Our next goal is to prove the converse of Theorem 6.3. We need the following result which is a consequence of a characterization of operators with bounded H ∞ -calculus due to C. Le Merdy [START_REF] Merdy | The similarity problem for bounded analytic semigroups on Hilbert space[END_REF]. We refer also to the monography by M. Haase [START_REF] Haase | The functional calculus for sectorial operators[END_REF]Sec. 7.3.3]. Proposition 6.11. Let S be a quasicontractive holomorphic C 0 -semigroup on a Hilbert space H whose generator is -A. If Re λ > 0 for all λ ∈ σ(A), then there exist a Hilbert space V ↪ d H, a continuous realcoercive sesquilinear form a ∶ V × V → C, and there exists an equivalent scalar product [., .] on H such that A is associated with a on (H, [., .]).

Note that the assertion in the proposition says that

D(A) = {u ∈ V ∶ ∃f ∈ H such that a(u, v) = [f, v] for all v ∈ V }, Au = f.
To say that [., .] is an equivalent scalar product means that u ↦ [u, u] defines an equivalent norm on H.

Proof of Proposition 6.11. Since Re λ > 0 for all λ ∈ σ(A) and since the spectrum of A lies in a sector, there exists ε > 0 such that Re (λ) ≥ 2ε for all λ ∈ σ(A). This implies that -A + ε Id generates a holomorphic C 0 -semigroup T which is bounded on a sector. Since A + w Id is maccretive, it follows that A + w Id ∈ BIP (H) (see for example [START_REF] Merdy | The similarity problem for bounded analytic semigroups on Hilbert space[END_REF] for the definition of BIP (H)). By [START_REF] Arendt | Functional calculus, variational methods and Liapunov's theorem[END_REF]Corollary 2.4], it follows that Aε Id ∈ BIP (H). Now it follows from a result of Le Merdy [ Proof. Since S is asymptotically compact, H = X 1 ⊕ X 2 where X j are closed invariant subspaces, dim X 1 < ∞ and S 2 (t) ≤ M e -εt for all t ≥ 0 for some ε > 0, M ≥ 1, where S 2 (t) ∶= S(t) X 2 . Thus the spectrum of the negative generator of S 2 lies in the open right half-plane. Since S is quasi-contractive, also S 2 is quasi-contractive. It follows from Proposition 6.11 that there exist an equivalent scalar product

[., .] 2 on X 2 , V 2 ↪ d X 2 , a 2 ∶ V 2 × V 2 → C a continuous, real-coercive sesquilinear form such that a 2 ∼ A 2 on (X 2 , [., .]
), where -A j is the generator of S(.) X j , j = 1, 2.

Note that A 1 is a bounded operator. Defining

V ∶= X 1 ⊕ V 2 and [x 1 + x 2 , y 1 + y 2 ] ∶= ⟨x 1 , y 1 ⟩ H + [x 2 , y 2 ] 2 on H where x 1 + x 2 , y 1 + y 2 ∈ X 1 ⊕ X 2 and defining a(x 1 + x 2 , y 1 + y 2 ) = ⟨A 1 x 1 , y 1 ⟩ H + a 2 (x 2 , y 2 ) for x 1 + x 2 , y 1 + y 2 ∈ X 1 ⊕ V 2 ,
one obtains an essentially positive-coercive form on V (Theorem 4.4 (ii) is fulfilled). Then A ∼ a on (H, [., .]).

Essentially coercive forms and holomorphic semigroups

The purpose of this section is to give some information on the operator associated with an essentially coercive form using our results from Section 6. This will allow us to prove a converse version of Theorem 3.4. Throughout the section, V, H are complex Hilbert spaces, V ↪ d H and a ∶ V × V → C is a continuous and sesquilinear form. As previously, we denote by A the operator on H associated with a.

Theorem 7.1. If a is essentially coercive, then there exists θ ∈ R such that e iθ A is the negative generator of a holomorphic C 0 -semigroup, which is quasi-contractive on a sector and which is asymptotically compact. In particular, D(A) is dense in V .

Proof. There exists θ ∈ R such that e iθ a is essentially positive-coercive (Proposition 4.3). By Proposition 6.1, the form e iθ a is H-elliptic. Thus its associated operator e iθ A is m-sectorial. As is well-known, this implies that D(e iθ A) = D(A) is dense in V .

If a is accretive, we know from Subsection 3.2 that -A itself generates a contractive C 0 -semigroup S. In general S is not holomorphic. Here is an example.

Example 7.2. Let H = 2 , V = {x ∈ 2 ∶ ∑ ∞ n=1 λ n x n 2 < ∞} where 0 < λ 1 ≤ ⋯ ≤ λ n ≤ λ n+1 → ∞ equipped with the norm x 2 V = ∞ n=1 λ n x n 2 . Let a(u, v) = ∑ ∞ n=1 iλ n u n v n .
Then Re a(u, u) = 0, so a is accretive. One has S(t)u = (e -iλnt u n ) n∈N . Thus S is not holomorphic. In fact S extends to a unitary group.

However, the semigroup S associated with an essentially coercive, continuous and accretive form is always the boundary of a holomorphic C 0 -semigroup. To explain this result in detail, we recall the following.

For θ ∈ (0, π 2), as previously, let Σ θ = {re iα ∶ r > 0, α < θ} and let S ∶ Σ θ → L(X) be a holomorphic C 0 -semigroup with generator -A. If (7.1) sup

z∈Σ θ , z ≤1 S(z) < ∞,
then there exists a strongly continuous extension S ∶ Σ θ → L(X) and S ±θ ∶= (S(te ±iθ )) t≥0 are C 0 -semigroups with generator -e ±iθ A. We call S ±θ the boundary semigroups of S. Conversely, if A is a closed operator and θ ∈ (0, π 2) such that -e ±iθ A generate C 0 -semigroups S ± , then -A generates a holomorphic C 0 -semigroup S ∶ Σ θ → L(X) where 0 < θ < π 2 such that (7.1) holds and S ± are the boundary semigroups of S.

We refer to [START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF]Sec. 39] or [START_REF] Arendt | Boundary values of holomorphic semigroups[END_REF] for this and for further information. Now we can formulate a result which is a converse version of Theorem 3.4. Even though we cannot apply our results on the essential spectrum in the situation of Theorem 7.3 (since a might not be essentially positivecoercive), we note the following result. Proof. Since a is not coercive, there exist u n ∈ V such that u n V = 1 and a(u n , u n ) → 0 as n → ∞. We may assume that u n ⇀ u in V . Since a is essentially coercive, it follows that u ≠ 0

Theorem 7.3. Assume that V ≠ H, V ↪ d H, a ∶ V × V → C is con- tinuous,
. Let v ∈ V . Then Re a(u, v) = lim n→∞ Re a(u n , v) ≤ lim sup n→∞ (Re (a(u n , u n )) 1 2 (Re a(v, v)) 1 2 = 0.
(Here we used the accretivity of a). This implies that a(u, v) = 0 for all v ∈ V . Hence u ∈ D(A) and Au = 0. Remark 7.5. a) One cannot omit the hypothesis that a is accretive in Theorem 7.4, see the discussion at the end of Section 8. b) In the situation of Theorem 7.4, the semigroup S generated by -A may not be asymptotically compact. In fact, as is easy to see, a bounded C 0 -group is never asymptotically compact unless the underlying space is finite dimensional. Thus the semigroup in Example 7.2 gives what we want.

The semigroup S obtained in Theorem 7.1 is contractive on a sector Σ θ and asymptotically compact, i.e. lim t→∞ S(t) ess = 0. This implies automatically that lim t→∞ S(te iβ ) ess = 0 for all β ∈ (-θ, θ) as we will show now. However, as Example 7.2 shows, this is not true for β ∈ [-θ, θ]. Note that the boundary semigroup is obtained by the continuous extension of S to Σ θ for the strong operator topology and not the uniform topology.

Theorem 7.6. Let X be a complex Banach space θ ∈ (0, π 2] and let S ∶ Σ θ → L(X) be a holomorphic C 0 -semigroup such that S(z) ≤ M for all z ∈ Σ θ .

If lim t→∞ S(t) ess = 0, then for all 0 < β < θ, lim z →∞,z∈Σ β S(z) ess = 0.

Proof. a) We show that lim t→∞ S(tz) ess = 0 for all z ∈ Σ θ . For that we define

F ∶ Σ θ → C b ([0, ∞), L(X)) by F (z)(t) = S(tz). It follows from [3, Theorem A.7] that F is holomorphic. Consider the map q ∶ C b ([0, ∞), L(X)) → C b ([0, ∞), L(X) K(X))
given by q(f )(t) = f (t) where for T ∈ L(X), T is the image in L(X) K(X) by the quotient map. Then q is linear and bounded. Thus q ○F is holomorphic. Consider the Banach space C 0 ([0, ∞), L(X) K(X)) of all continuous functions g ∶ [0, ∞) → L(X) K(X) satisfying lim t→∞ g(t) L(X) K(X) = 0 and denote by Q ∶ C b → C b C 0 the quotient map. Then G ∶= Q ○ q ○ F is holomorphic. Since G(t) = 0 for all t > 0, it follows from the Uniqueness Theorem that G(z) = 0 for all z ∈ Σ θ ; i.e. lim t→∞ q(F (t)) = 0 in L(X) K(X)). Hence lim t→∞ S(tz) ess = lim t→∞ S(tz) L(X) K(X) = 0. b) Let 0 < β < θ and ε > 0. By a) there exists t 0 > 0 such that S(e ±iβ t) ess ≤ ε for all t ≥ t 0 . Let z = te iη , t ≥ t 0 , η ∈ [0, β]. Then S(te iη ) = S(e i(η-β) )S(te iβ ) ≤ M S(te iβ ) ≤ M ε.

If η ∈ [-β, 0] the argument is similar.

Selfadjoint operators revisited

In this section we reconsider Theorem 3.1 and give a complete characterization if j is injective.

Let V, H be Hilbert spaces such that V ↪ d H and let a ∶ V × V → C be a continuous sesquilinear form. We define the numerical range of a (with respect to H) by W (a) ∶= {a(u, u) ∶ u ∈ V, u H = 1}.

For λ ∈ C we define a λ ∶ V × V → C by a λ (u, v) = a(u, v) -λ⟨u, v⟩ H . This is consistent with the notations of Section 3.1 for j = Id. We first give the following characterization of coerciveness in terms of the numerical range (in H). , which implies that λ ∈ W (a). Moreover, the coercivity always implies the essential coercivity.

(ii) ⇒ (i) Assume that a λ is not coercive. Then there exist u n ∈ V such that u n V = 1 and a λ (u n , u n ) → 0. Passing to a subsequence we may assume that u n ⇀ u in V . It follows from (8.1) that u n H → 0. Consequently u = 0. Thus a λ is not essentially coercive.

If a is symmetric, then W (a) ⊂ R. So Proposition 8.1 applies to each λ ∈ C R and therefore, for such λ, it is equivalent to say that a λ is essentially coercive or coercive.

We denote by A the operator associated with a on H. From Theorem 3.1 we know that A is selfadjoint whenever a λ is essentially coercive for some λ ∈ C. Here is a characterization of the operators obtained in this way. Theorem 8.2. Let A be an operator on a Hilbert space H. The following assertions are equivalent:

(i) A is selfadjoint and semibounded; (ii) there exist a Hilbert space V ↪ d H and a symmetric, continuous form a ∶ V × V → C such that a λ is essentially coercive for some λ ∈ C and A ∼ a.

In that case the form in (ii) is unique.

Proof. (i) ⇒ (ii) By the Spectral Theorem, we may assume that H = L 2 (Ω, µ) with (Ω, Σ, µ) a measure space, and that A is a multiplication operator, i.e. there exists a measurable function m ∶ Ω → R such that Then a is a continuous, symmetric from and it is easy to see that A ∼ a. Now assume that A is bounded below. Then m ≥ -w µ-a.e. for some w ≥ 0. We show that a i is coercive. Let u n ∈ V such that H-elliptic, or equivalently, a is a sectorial closed form. By [14, Theorem 2.7], there is only one sectorial closed form to which A is associated.

a i (u n , u n ) = Ω m u n 2 dµ -i Ω u n 2 dµ → 0 as n → ∞.
We illustrate Theorem 8.2 by considering a lower bounded selfadjoint operator A on a Hilbert space H. Let

Proposition 2 . 1 .

 21 [START_REF] Arendt | Galerkin approximation in Banach and Hilbert spaces[END_REF] Corollary 4.5]] Let a ∶ V ×V → K be a continuous, essentially coercive form which satisfies uniqueness i.e.

Proposition 4 . 1 .

 41 a) The form a is coercive if and only if there exists θ ∈ R such that e iθ a is positive-coercive. b) The form a is real-coercive if and only if a or -a is positive-coercive. Now, as in the coercive case (see Section 2) we want to introduce topological properties which are weaker than real-coercive or positivecoercive. Definition 4.2. a) The form a is called essentially real-coercive if

  Fredholm operator} we denote the Fredholm resolvent set of A. It is an open subset of C and we denote by σ ess (A) ∶= C ρ F (A) the essential spectrum of A. The following property is remarkable (see [14, p. 243] or [16, 1.3.1]). Proposition 5.1. Let A be the negative generator of a C 0 -semigroup and let ω ⊂ ρ F (A) be an open connected set. If ω ∩ ρ(A) ≠ ∅, then ω ∩ σ(A) is discrete and P λ has finite rank for all λ ∈ ω ∩ σ(A).

ω

  ess (A) = lim sup t→∞ 1 t log S(t) ess = inf t>0 log S(t) ess , which implies that r ess (S(t)) = e tωess(A) for all t > 0, see [17, A III.].

  2 (cf. [17, B-IV Theorem 2.10]). Proposition 5.3. Let S be an asymptotically compact C 0 -semigroup with generator -A. Then the set σ -(A) ∶= {λ ∈ σ(A) ∶ Re λ ≤ 0} is finite and the spectral projection associated with σ -(A) has finite rank.

Remark 5 . 8 .

 58 The proof of Lemma 5.7 yields a stronger assertion, namely σess (S) ⊂ σess (T ) ∪ {0}, where σess (T ) = σ(T ) {λ ∈ σ(T ) ∶ λ is an isolated point with spectral projection of finite rank}. Proof of Theorem 5.5. a) Since T is holomorphic, T (1)X ⊂ D(A) ⊂ Y . One has S(t) = T (t) Y (since S(t)y = lim n→∞ Id + t n B -n y = T (t)y). It follows from Lemma 5.7 that e ωess(B) = r ess (S(1)) ≤ r ess (T (1)) = e ωess(A) .

1 ∶= 2 H

 12 15, Theorem 1.1 and Corollary 4.7] that there exists an equivalent scalar product [., .] on H such that -A + ε Id generates a holomorphic C 0semigroup T which is contractive for [., .] on a sector. By a result of Kato [14, VI § 2, Theorem 2.7] there are a Hilbert spaceV ↪ d H, a continuous H-elliptic form b ∶ V × V → C and θ ∈ [0, π 2) such that Aε Id is associated with b on (H, [., .]). Then A is associated with a ∶= b + ε[., .] on (H, [., .]). Since Re b(u, u) ≥ 0 for all u ∈ V , u 2 Re a(u, u) = Re b(u, u) + ε u defines a norm on V . Since b is H-elliptic, this norm is complete. Thus 1 is an equivalent norm on V . Hence there exists α > 0 such that Re a(u, u) = u 2 1 ≥ α u 2 V ;i.e. a is real-coercive.Theorem 6.12. Let S be an asymptotically compact quasi-contractive holomorphic C 0 -semigroup on H with generator -A. Then there exist an equivalent scalar product [., .] on H, a Hilbert space V ↪ d H and a continuous, essentially positive-coercive form a ∶ V × V → C such that A ∼ a on (H, [., .]).

  essentially coercive and accretive. Let A ∼ a and denote by S the C 0 -semigroup generated by A. Then S is holomorphic or S is the boundary semigroup of a holomorphic C 0 -semigroup.Proof. By Theorem 7.1 there exists θ ∈ [-π, π) such that -e iθ A generates a holomorphic C 0 -semigroup. Then θ ∈ (-π, π) because otherwise -A generates a C 0 -semigroup and hence A is bounded. This contradicts the assumption V ≠ H.Let B = -e iθ 2 A. Then -e ±iθ 2 B generates a C 0 -semigroup. Thus -B generates a holomorphic C 0 -semigroup T and the semigroup generated by -e -iθ 2 B = -A is a boundary semigroup of T .

Theorem 7 . 4 .

 74 Let V ↪ d H and let a ∶ V × V → C be continuous, accretive and essentially coercive. If a is not coercive, then 0 is an eigenvalue of A, where A ∼ a.

Proposition 8 . 1 . 2 H u 2 H 2 H

 81222 Let λ ∈ C. The following assertions are equivalent:(i) a λ is coercive; (ii) λ ∈ W (a) and a λ is essentially coercive. Proof. To say that λ ∈ W (a) is equivalent to the existence of β > 0 such that (8.1) a λ (u, u) ≥ β u 2H for all u ∈ V. If a λ is coercive, i.e. if there exists α > 0 such thata λ (u, u) ≥ α u 2 V for all u ∈ V, then a λ (u, u) ≥ α c for all u ∈ V, where u H ≤ c H u V for all u ∈ V . (i) ⇒ (ii)The coercivity of a λ implies (8.1) with β = α c

DΩ( 1 +

 1 (A) = {u ∈ L 2 (Ω, µ) ∶ mu ∈ L 2 (Ω, µ)}, Au = mu. Define V ∶= {u ∈ L 2 (Ω, µ) ∶ ∫ Ω m u 2 dµ < ∞}. Then V is a Hilbert space for the norm u 2 V ∶= m ) u 2 dµand V ↪ H. Since A is densely defined as a selfadjoint operator andD(A) ⊂ V , it follows that V is dense in H. Define a ∶ V × V → C by a(u, v) = Ω muvdµ.

Then u n 2 H

 2 = -Im a i (u n , u n ) → 0. Moreover, Ω m u n 2 dµ = Re a i (u n , u n ) → 0.

λ 1

 1 ∶= inf{⟨Au, u⟩ H ∶ u ∈ D(A), u H = 1} be the lower bound of A. Assume that there exists λ ess > λ 1 such that σ ess (A) = [λ ess , ∞). Denote by a the unique form of Theorem 8.2 such that A ∼ a. Then the following assertions hold. a) The form a λ is coercive if and onlyif λ ∈ C R or λ ∈ R, λ < λ 1 . b) For λ ∈ R, a λ is essentially coercive if and only if λ < λ ess . c) The lower bound λ 1 belongs to σ p (A). d) If λ ∈ (λ 1 , λ ess ) ∩ ρ(A), then a λ is essentially coercive but not coercive. Observe that Aλ Id ∼ a λ , but 0 ∈ σ p (Aλ Id) (cf. Theorem 7.4).Proof. a) follows from Proposition 8.1. b) follows from Corollary 6.4. c) follows from Theorem 7.1 and is well-known. d) is a consequence of b).

Acknowledgments: This research is partly supported by the Bézout Labex, funded by ANR, reference ANR-10-LABX-58.

Next we state a perturbation theorem due to Desch-Schappacher which will be needed later.

Theorem 5.4. [START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF]Theorem 3.7.25] Let A be the generator of a holomorphic C 0 -semigroup S and let K ∶ D(A) → X be compact (where D(A) carries the graph norm). Then A + K generates a holomorphic C 0 -semigroup S and ω ess (A + K) = ω ess (A).

We will also need the following interpolation result which is of independent interest. As before, if X, Y are Banach spces, we write Y ⊂ X if Y is a subspace of X; we write Y ↪ X if Y ⊂ X and if in addition the injection is contiunous.

Theorem 5.5. Let A be the generator of a holomorphic C 0 -semigroup T on the Banach space X. Let Y be a Banach space such that

Here B is defined by

In the theorem we merely ask that D(A) is a subspace of Y . But the closed graph theorem implies that the injection is automatically continuous if D(A) is considered with the graph norm.

For the proof of Theorem 5.5 we need some preparation.

Proof. Let r > r ess (T ). Then, by Proposition 5.1, the set M ∶= {λ ∈ σ(T ) ∶ λ > r} is finite and consists of isolated eigenvalues with finite dimensional spectral projection. Thus X = X 1 ⊕ X 2 where X j is a closed subspace, T X j ⊂ X j (j = 1, 2), dim X 1 < ∞ and r(T 2 ) < r where

We have shown that a i (u n , u n ) → 0 implies that u n V → 0 as n → ∞. This is equivalent to a i being coercive.

If A is bounded above, applying the argument to -A, we deduce that (-a) i = -a -i is coercive. Then also a -i is coercive and by Lemma 2.2 and Proposition 8.1 also a i is coercive.

(ii) ⇒ (i) We know from Theorem 3.1 that A is selfadjoint. It remains to show that A is semibounded. Since a λ is essentially coercive, it follows from Lemma 2.2 that a i is essentially coercive. By Proposition 4.3, there exists θ ∈ R such that e iθ a i is essentially positive-coercive and so H-elliptic by Proposition 6.1. Thus there exist w ≥ 0, α > 0 so that Re e iθ (a(u, u)

V for all u ∈ V. Consequently Re e iθ a(u, u) + (w + 1)) u 2 H ≥ α u 2 V for all u ∈ V. First case: Re e iθ = 0. Since V ↪ d H, it follows that the norms of V and H are equivalent. Thus V = H, i.e. A is a bounded operator. Second case: Re e iθ > 0. Then ⟨Au, u⟩ H ≥ -(w + 1) Re e iθ u 2 H for all u ∈ D(A). Thus A is bounded below. Third case: Re e iθ < 0. Then the second case implies that -A is bounded below, i.e. A is bounded above, which proves (i).

We now prove uniqueness. Let A be a selfadjoint operator which is bounded below, say ⟨Au, u⟩ H ≥ (-w + 1) u 2 H for all u ∈ D(A) and some w ≥ 0. Let A ∼ a where a ∶ V × V → C is a symmetric, continuous form, which is essentially coercive. Since, by Theorem 7.1, D(A) is dense in V , it follows that a(u, u) + w u 2 H ≥ u 2 H for all u ∈ V. Thus 0 ∈ W (a -w ). It follows from Lemma 2.2 that a -w is essentially coercive. Now Proposition 8.1 implies that a -w is coercive. Thus a is


S Hernandorena

Koeberle

MD Matthieu Piccoli
email: matthieu.piccoli@aphp.fr

MD Thomas Tannou
email: ttannou@chu-besancon.fr

MD Intza Hernandorena

MD Severine Koeberle

Ethical approach to the issue of confinement of the elderly in the context of the COVID-19 pandemic : prevention of frailty versus risk of vulnerability

Keywords: Âgisme, COVID-19, Fragilité, Gériatrie, Isolement social Ageism, COVID-19, Frailty, Geriatrics, Social isolation

Le contexte : COVID-19 et vulnérabilité sanitaire.

L'histoire de la médecine est marquée par de nombreuses crises sanitaires et épidémiques dont les principales ont suscité des changements sociétaux majeurs comme la peste, la variole, le choléra, la poliomyélite, la rougeole, Ebola, le VIH, ou encore la grippe espagnole. L'apparition des antibiotiques a permis d'endiguer une partie des épidémies bactériennes [START_REF] Debré | Vie et mort des épidémies[END_REF] (malgré l'émergence préoccupante de l'antibiorésistance [START_REF] Coignard | Antibiorésistance : la situation en France et dans le monde[END_REF]), mais ne permet pas d'éviter l'émergence de nouvelles épidémies virales [START_REF] Hervé | Les maladies émergentes et le franchissement des barrières d'espèces: implications anthropologiques et éthiques[END_REF].

Ce qui fait la spécificité des virus, c'est leur fonctionnement de parasite génétique qui implique à la fois la nécessité de survivre assez longtemps dans les cellules de l'hôte pour s'y répliquer et donc se transmettre et une capacité de mutation pour se protéger des phénomènes de mémoire immunologique. Cela conduit à une sélection naturelle des formes les plus contagieuses et suffisamment virulentes pour que l'épidémie se propage [START_REF] Domingo | Viral fitness: history and relevance for viral pathogenesis and antiviral interventions[END_REF][START_REF] Che | Quels éléments président à l'émergence d'une épidémie virale ? Peut-on la prévoir[END_REF].

La médecine, à travers les âges, a toujours essayé au mieux, selon ses moyens et ses connaissances, de protéger la santé. Mais, parfois, les logiques individuelles et sociétales s'entrechoquent et font émerger des conflits de valeurs [START_REF] Hervé | politique et santé[END_REF]. Ainsi, dans le champ de la protection des risques, notamment en Santé Publique, l'équilibre entre liberté d'un individu et protection d'une population, se fait parfois au prix de contraintes nécessaires [START_REF] Corvol | La prévention du risque en médecine: D'une approche populationnelle à une approche personnalisée[END_REF].

Dans le contexte actuel d'épidémie mondialisée à COVID-19 (maladie de l'agent viral pathogène Sars-CoV-2), nos systèmes de santé sont, pour la première fois depuis l'avènement d'une médecine technique et scientifique dite moderne, mis en tension par l'ampleur de la crise, popularisant les expressions « aplatir la courbe » [START_REF] Qualls | Community Mitigation Guidelines to Prevent Pandemic Influenza -United States, 2017[END_REF], distanciation sociale et confinement qui visent à étaler dans le temps le nombre de cas incidents simultanés, au prix d'une augmentation du nombre de cas et parmi ceux-là, des cas graves. Ce sont ces mesures qui, pour l'instant sont efficaces en matière de santé collectives, alors qu'il n'existe, à ce jour, aucun traitement curatif éprouvé à visée curative de la COVID-19. Les perspectives thérapeutiques actuelles reposent sur des traitements, parfois lourds et agressifs, des complications de cette maladie (maladies veineuses thrombo-emboliques, insuffisance respiratoire,...) et des séquelles dont nous ne mesurons pour l'instant pas l'ampleur.

Face à cette maladie d'évolution inconnue, d'allure imprévisible, l'accès aux soins de réanimation a été l'objet d'un conflit de valeurs entre la bienfaisance, la non-malfaisance, l'autonomie et la justice redistributive [START_REF] Beauchamp | Principles of biomedical ethics[END_REF]. Concernant la personne âgée, la discrimination sur le seul critère de l'âge (âgisme) ne permet pas toujours une réflexion concertée concernant le niveau de soin, incluant l'accès aux services de réanimation et peut engendrer une restriction d'accès systématisée ne permettant pas l'évaluation du rapport bénéfice/risque individuel. De la même manière, dans le champs des soins actifs, pour ces cas graves, lorsque les soins s'intensifient, que les moyens thérapeutiques se réduisent et que les perspectives raisonnables de guérison s'amenuisent, le respect de la dignité de la personne humaine, telle que signifiée dans la convention d'Oviedo [START_REF]Convention pour la protection des Droits de l'Homme et de la dignité de l'être humain à l'égard des applications de la biologie et de la médecine[END_REF] et l'équité dans l'accès aux soins imposent, dans le respect de l'autonomie de la personne, de stopper des actes n'ayant pour seule finalité que le maintien artificiel de la vie [START_REF]du 2 février 2016 créant de nouveaux droits en faveur des malades et des personnes en fin de vie[END_REF].

Aussi, pour l'épidémie de COVID-19, reconnue comme Urgence de Santé Publique de Portée Internationale le 30 janvier 2020 par l'OMS [START_REF]Déclaration sur la deuxième réunion du Comité d'urgence du Règlement sanitaire international (2005) concernant la flambée de nouveau coronavirus 2019[END_REF] puis comme pandémie le 12 mars 2020 [START_REF]L'OMS déclare que la flambée de COVID-19 constitue une pandémie[END_REF], le Haut Conseil en Santé Publique a rédigé des recommandations spécifiques pour les personnes « à risque de forme sévère » le 14 mars 2020 [START_REF]Avis provisoire Recommandations relatives à la prévention et à la prise en charge du COVID-19 chez les patients à risque de formes sévères[END_REF], tenant compte de l'ensemble de ces éléments et de l'impératif de protection des plus vulnérables : la distanciation sociale devenait le moyen de prévention privilégié.

En effet, il était déjà identifié que les comorbidités, qui sont corrélées à l'âge, sont un facteur de risque de mortalité surajouté aux syndromes respiratoires aigus liés à la COVID-19. On y retrouvait ainsi formulées des recommandations visant à éviter que le virus entre en contact avec cette -« des mesures d'exclusion concernant tous les établissements hébergeant des personnes à risque de forme grave de COVID-19 (hébergement de personnes âgées dépendantes, hébergement de personnes handicapées, hébergement de personnes précaires) de Il est important de rappeler que la vulnérabilité des personnes âgées fragiles, indépendamment de la situation liée au COVID, est une conséquence de :

-Un phénotype de fragilité [START_REF]Comment repérer la fragilité en soins ambulatoires[END_REF], qui peut être décrite par l'approche holistique de Gobbens et al, 2010 [START_REF] Gobbens | Determinants of Frailty[END_REF] comme la résultante d'une trajectoire de vie, associant à la fois une composante physique, psychologique et sociale. Le déséquilibre d'au moins une de ces composantes entraine une augmentation des événements indésirables, conduisant à une augmentation de la dépendance fonctionnelle, puis à une majoration des besoins en ressources de santé voir au décès précoce. On en retrouve d'autres descriptions, bien connues en Gériatrie [START_REF] Cesari | The frailty phenotype and the frailty index: different instruments for different purposes[END_REF].

-Plusieurs maladies, à des stades d'évolution divers, pouvant impacter l'indépendance fonctionnelle et conduire à une situation de dépendance d'autrui pour la réalisation d'actes de la vie quotidienne. Cela est illustré par le profil moyen des soins des personnes vivant en EHPAD, dont le niveau de « dépendance » est à mettre en parallèle avec les 7 ou 8 affections de longue durée avec lesquelles elles vivent, indépendamment de leur âge « civil ».

-Une altération neurocognitive dont l'impact sur les capacités de jugement et de décision peut conduire à une altération de leur autonomie décisionnelle, conduisant à majorer la dépendance sus-citée.

Cette situation de vulnérabilité, réduisant l'indépendance fonctionnelle et l'autonomie de décision conduit donc, de facto, à priver ces personnes d'une forme d'exercice de leur liberté, alors que la déontologie et l'éthique cherchent à la valoriser, a minima, de jure.

A cet état de fait se rajoute, dans le contexte pandémique actuel, une vulnérabilité sanitaire qui s'exprime en plusieurs temps, sans que l'on puisse estimer, a priori, la proportion de chacune de ces vagues ni la temporalité de leur survenue :

-Morbi-Mortalité directement liée à la COVID-19 (formes graves, séquelles respiratoires, complications des hospitalisations prolongées, autres séquelles pour l'instant inconnues, …) -Apparition de maladies aiguës nouvelles, négligées du fait du COVID (infarctus, AVC, cancer, …) -Déséquilibre des maladies chroniques, directement ou indirectement lié à la COVID-19, faute d'accès au programme de soins personnalisé prévu (réallocation des ressources au sein du système de santé, déprogrammation des soins non programmés, difficultés d'accès aux soins malgré des solutions de téléconsultations), mais aussi secondairement à la diminution des passages des aides à domicile habituellement en place, ne permettant plus l'assurance de la réalisation des aides aux activités de la vie quotidienne et limitant encore les contacts sociaux.

-Conséquences indirectes de la période critique à moyen/long terme : maladies liées ou aggravées par la précarité financière, conséquences psychiques de la crise (deuil pathologique, burnout, conséquences du confinement, …).

Ainsi, ce contexte de pandémie nous invite à prévenir les conséquences de chacune de ces quatre vagues, lorsque nous entendons protéger les personnes « vulnérables » et non à nous attarder uniquement à la première d'entre elles.

Conflits éthiques posés par le confinement des personnes âgées :

Il nous semble nécessaire de poser les tensions éthiques induites par le confinement des personnes âgées vulnérables à plusieurs niveaux pour étayer ce qui relève de la prévention et de la protection En effet, comme évoqué dans le modèle de Gobbens développé plus haut, l'impact du confinement est triple sur la fragilité des personnes âgées :

-Sur le plan physique : la réduction des activités physiques quotidiennes directes ou encadrées par des professionnels de santé sont fondamentales, tant les bénéfices de l'activité physique sont connues [START_REF] Etnier | A meta-regression to examine the relationship between aerobic fitness and cognitive performance[END_REF] : les conséquences précoces d'une telle réduction d'activité sur la sarcopénie, la chute et la majoration de la dépendance sont des phénomènes largement décrits dans la littérature [START_REF] English | Protecting muscle mass and function in older adults during bed rest[END_REF][START_REF] Yeung | Sarcopenia and its association with falls and fractures in older adults: A systematic review and metaanalysis[END_REF].

-Sur le plan psychologique : outre l'impact direct du confinement (anxiété, risque dépressif), la réduction des activités physiques et de la stimulation cognitive ont un impact majeur sur le risque psychologique (pour la personne [START_REF] Tomaka | The Relation of Social Isolation, Loneliness, and Social Support to Disease Outcomes Among the Elderly[END_REF] comme ses aidants [START_REF] Neri | Relationships between gender, age, family conditions, physical and mental health, and social isolation of elderly caregivers[END_REF]), pouvant s'exprimer sous forme de troubles psycho-comportementaux productifs (agitation, agressivité…) comme négatifs (retrait sur soi, comportements oppositionnels, …).

-Enfin, sur le plan social : l'isolement est le 3 ème facteur de mortalité en Europe après 75 ans, et renforce le sentiment d'inutilité sociale (impactant sur les composantes psychologiques) autant qu'il diminue les stimulations physiques, nutritives et sensoriels, impactant le 1 er point [START_REF] Ankri | Prévention des maladies après 75 ans: Intérêt d'agir sur les facteurs de risque[END_REF].

Ainsi, si l'intention bienfaisante de réduire le risque d'infection dans l'enjeu immédiat d'un confinement particulièrement restrictif des personnes âgées les plus fragiles ne fait pas de doute dans l'avis du HCSP mentionné ci-dessus, ses conséquences pourraient être paradoxalement contreproductives sur un plan médico-économique individuel et collectif. Restreindre l'autonomie dans son versant fonctionnel comme décisionnel majore encore le risque d'aggravation du déclin physique et psychique. Ceci aboutit à une nécessité de mobilisation plus importante du système de santé à tous les niveaux, impliquant une augmentation des durées moyennes de séjour, et donc, des risques de complications iatrogènes et nosocomiales [START_REF]Prévenir la dépendance iatrogène liée à l'hospitalisation chez les personnes âgées[END_REF] et donc de mortalité d'une population qu'il s'agissait initialement, de protéger.

Citons ainsi plusieurs programmes de prévention qui pourraient s'avérer utiles pendant cette période, pour identifier et suivre les personnes : le programme ICOPE de l'OMS [START_REF]Integrated care for older people: guidelines on community-level interventions to manage declines in intrinsic capacity[END_REF], le projet Vivifrail notamment porté par le Fonds européen de développement régional et le CHU de Toulouse [30] ou, dans ce contexte spécifique de confinement, l'outil ESOGER1 © , développé par le Centre d'Excellence sur la Longévité du RUISSS McGill [START_REF]Plateforme digitale d'évaluation sociogériatrique de premier niveau en période de distanciation sociale[END_REF] et transposé en France par l'intermédiaire de la Société Française de Gériatrie et Gérontologie [START_REF]COVID-19 : La SFGG adapte et déploie ESOGER1, un outil numérique canadien[END_REF].

Analyse des conflits éthiques à l'échelle des aidants et professionnels

A l'échelle des aidants professionnels, l'organisation territoriale est particulièrement hétérogène, se structurant entre des échelons d'organisation individuels et des organismes agréés de taille conséquente. Bien que les services décentralisés se soient réorganisés, les interventions restaient tributaires de moyens de transports qui pouvaient manquer à l'échelon collectif (par exemple, transports en commun réduits) et/ou individuel (par exemple, plusieurs membres d'un même foyer peuvent avoir besoin de se déplacer simultanément).

Outre les difficultés de transmission d'informations professionnelles et de formation, le Conseil

National Professionnel de Gériatrie a fait valoir que des équipements de protection individuelle venaient à manquer pour l'ensemble des métiers du grand âge, à domicile comme en institution [START_REF]Lettre ouverte au ministre de la santé demandant des masques pour les professionnels du grand âge[END_REF].

Cela a placé ainsi, comme l'exprimait le CCNE [START_REF]Réponse à la saisine du ministère des solidarités et de la santé sur le renforcement des mesures de protection dans les EHPAD et les USLD[END_REF], bon nombre de professionnels dans un « dilemme : se dévouer pour soigner, avec le risque pour soi-même et les autres d'être infecté par le soin que l'on prodigue ». Il existait ainsi une balance entre la bienfaisance liée à la réalisation de soins nécessaires, le risque de contamination de la personne aidée par les soins, en l'absence de protection et le risque professionnel pour le soignant, en réciprocité.

Cette question s'est sans doute posée également pour de nombreux intervenants au périmètre d'action mal reconnu, mais dont l'impact sur la vie sociale et l'aide quotidienne est majeur, par exemple, aide-ménagère, concierge, bénévoles et d'autres, dont l'intervention était plutôt pensée de gré à gré.

Pour les aidants, même si leurs déplacements pour "l'assistance aux personnes vulnérables" étaient autorisés à titre dérogatoire [START_REF]Attestation de déplacement dérogatoire[END_REF], leur situation est très hétérogène. Il s'agit dans presque 1 situation sur 2 d'un conjoint et dans près de 2/3 des cas, les aidants continuent à travailler [START_REF] Briard | Aider un proche : quels liens avec l'activité professionnelle ?[END_REF]. Compte-tenu des impératifs professionnels parfois rendus plus contraignants par les risques de perte de revenus ou des difficultés de déplacement évoquées plus haut, leur capacité à intervenir peut s'être également modifiée. Il existe également un risque pour la santé de l'aidant, souvent fragilisée (et souvent non abordée avec les professionnels de santé, la santé de la personne aidée étant jugée prioritaire) [START_REF]Les proches aidants : une question sociétale / accompagner pour préserver la santé[END_REF]. Aussi, la balance entre l'impératif de « prendre soin de l'autre » et « se protéger » (d'un point de vue sanitaire ou socio-économique) semble pouvoir se poser selon des termes comparables à ceux des professionnels.

Bien que des pistes immédiates aient été proposées [START_REF] Guedj | Lutter contre l'isolement des personnes agées et fragiles isolées en période de confinement[END_REF] et que les organisations territoriales se soient adaptées, notamment dans le renforcement du lien ville-hôpital avec la filière gérontologique s'appuyant sur l'ambulatoire [START_REF] Koeberle | COVID-19 outbreakorganisation of a geriatric assessment and coordination unit. A French example[END_REF], la reconnaissance véritable du statut d'aidant, une meilleure structuration de l'offre et une pérennisation d'un lien fort ville-hôpital sont des pistes qui apparaissent indispensables [START_REF] Libault | Concertation Grand âge et autonomie[END_REF].

Analyse des conflits éthiques à l'échelle de la société

A l'échelle de la société, il est du rôle d'un Etat de droit que de s'engager dans la protection des plus vulnérables de façon juste et équitable, selon la visée éthique de Paul Ricoeur « la vie bonne, avec et pour autrui, dans des institutions justes » [START_REF] Ricoeur | Soi-même comme un autre[END_REF]. A ce titre, eu égard aux données épidémiologiques disponibles relatives à la COVID-19, la mise en place très précoce de mesures de protections sanitaires ciblées sur ces populations est particulièrement attendue par la société. Respectant l'adage du « primum non nocere », la mise à l'écart du risque viral des populations les plus fragiles par des mesures "d'exclusion" semble être un impondérable. Pour autant, difficile de ne pas y voir un positionnement paternaliste, qui, s'il est appliqué de façon systématique et sans discernement, tranche avec les impératifs de justice et d'équité dans l'accès aux soins. En particulier, et de façon paradoxale quand il s'agit de personnes vivant en institution, on retrouve de manière saillante, le conflit possible entre la santé collective et la santé individuelle. Chacun étant possiblement un vecteur de la maladie, comment protéger une personne vulnérable, par exemple, ne pouvant se déplacer seule, de la contamination par une autre personne vulnérable, ou dont les capacités de mémorisation sont très atteintes et ne pouvant appliquer par elles-mêmes une mesure de distanciation sociale efficiente ?

A plus long terme, c'est la question du regard de la société sur ces personnes vulnérables qui se pose et, en filigrane, une forme d'âgisme latent, exprimant globalement une forme de paternalisme masquée sous une apparence de bienveillance. Cette tension éthique est particulièrement marquée par la question du déconfinement : face au risque pour ces personnes vulnérables, la question est-elle de leur proposer quelque chose de moins que les autres citoyens (c'est-à-dire les laisser confiner plus longtemps) ou plutôt de leur proposer des moyens supplémentaires (par exemple, des masques et autres équipements de protection individuels) ? Sous-jacent, se pose ainsi, plus largement, la question de la place de la personne âgée et/ou vulnérable dans la société qu'il semble nécessaire de considérer, non pas comme une charge, mais comme une valeur, une source d'échanges intergénérationnels et une manière de prendre en compte notre propre vulnérabilité, tel que cela a été développé avec l'éthique (et l'économie) du Care.

Cela revient, ni plus ni moins, à considérer que les personnes vulnérables (ou à risque de l'être), sont des citoyen.ne.s à part entière. Malgré un risque différent de développer une forme grave, elles sont concernées par la même problématique : la moitié de l'humanité vivante actuelle est en effet en train d'expérimenter une mesure de confinement. Ce rappel à une participation inclusive des personnes âgées dans la société était d'ailleurs l'objet d'une Déclaration issue du séminaire autour des droits fondamentaux des citoyens âgés [START_REF]Droits fondamentaux sur l'avancée en âge, l'accès aux droits et l'exercice des droits par le citoyen âgé[END_REF].

Focus spécifique autour des conflits éthiques pour la personne atteinte d'une pathologie neurocognitive.

La présence de troubles neurocognitifs peut conduire, notamment via les troubles mnésiques, exécutifs et émotionnels, à une altération des capacités de jugement et de la prise de décision et donc à une défaillance de l'adaptabilité à un environnement nouveau. Il en résulte une altération progressive de l'autonomie décisionnelle autant que de l'indépendance fonctionnelle. Cela fait naitre, par conséquent une contrainte éthique forte de l'ensemble du réseau de support et de soutien pour compenser cette altération de l'autonomie de la personne. A tous les niveaux (sociétal, institutionnel, personnel) les aidants, qu'ils soient familiaux, informels ou professionnels, ont le devoir de protéger au mieux les intérêts de la personne aidée, dans le respect de positions exprimées antérieurement, la facilitation de l'expression d'une autonomie relative dans un contexte de troubles neurocognitifs ou à défaut, dans la favorisation d'une autonomie relationnelle. Sur ce plan, la réflexion sur la liberté d'aller et venir en EHPAD (qui est un droit fondamental) a été longuement débattue, bien qu'il persiste toujours des points saillants : le sens de la déambulation de certaines personnes vivant avec des troubles neurocognitifs majeurs ne revêt pas une expression de pleine liberté en conscience, mais est souvent un symptôme d'anxiété et d'une déconnection partielle entre le corps physique et le corps pensant, qui conduit parfois à proposer une entrée dans une unité de vie protégée pour garantir la sécurité de la personne. Comme cela rentre en opposition avec son autonomie sociale, cela nécessite une procédure de discussion collégiale [START_REF]relatif à l'annexe au contrat de séjour dans les établissements d'hébergement sociaux et médico-sociaux pour personnes âgées[END_REF]. Mais cela occulte la problématique qui peut se poser aux personnes vivant avec les mêmes troubles à l'extérieur d'un EHPAD ou aux personnes vivant avec des troubles moindres ayant également un besoin de participation à la vie sociale de la cité, dans une optique de solidarité et d'échanges intergénérationnels.

Ainsi, la question du confinement en EHPAD et des modifications majeures de leur fonctionnement du fait de l'impact de l'épidémie sur les besoins sanitaires des résidents se doit d'être posée sous l'angle de la pleine réalisation des besoins physiologiques, sanitaires, sociaux et d'émancipation réelle de ces personnes particulièrement vulnérables, indépendamment de leur lieu de vie.

En effet, face à des symptômes qui modifient grandement le quotidien, l'institution vise à apporter la réalisation des besoins physiologiques et sanitaires primaires (alimentation, hygiène, logement, prise des médicaments … ), qui peuvent parfois être impossibles à réaliser au domicile, ainsi que des interactions sociales qui rompent l'isolement. La ritualisation peut aussi apporter une forme d'apaisement lorsque les repères dans le temps peuvent être difficiles à retrouver. Le besoin d'émancipation et de contact social, s'il peut être partiellement rempli par les usages numériques, est également tributaire de la fracture numérique. Il doit également s'envisager sous l'angle des besoins à remplir (l'outil au service de l'usage), plutôt que sous l'aspect d'une promotion de solutions uniques existantes et transposées, dans l'urgence, sans adaptation individuelle, véritable choix ni garde-fou : l'enjeu des usages numériques adaptés à la génération du grand âge étant de favoriser le contact social et l'inclusion des personnes vulnérables dans la société, pas de s'y substituer.

Apports de la réflexion éthique dans une sortie de crise

Trois paradigmes peuvent nous aider à apporter des solutions concrètes et pragmatiques face à cette crise : issus de réflexions antérieures, ils ont vocation à être toujours utilisés, avec comme objectifs principaux d'améliorer la santé et d'éviter la mortalité prématurée, ce d'autant lorsqu'elle est évitable :

-l'éthique de responsabilité, qui pense la responsabilité des conséquences de nos actes, choix et décisions, avant l'action -l'éthique de la sollicitude (du care), qui consacre le maximum de droits aux plus vulnérables, et que l'on peut développer à partir d'une citation d'Aristote « qui peut le plus peut le moins », visant à nous mettre en empathie avec la personne la plus vulnérable, de sorte d'obtenir, à défaut d'une solution universelle, une réponse la plus inclusive possible et de ne pas priver de libertés et de droits les plus vulnérables, mais plutôt de leur fournir des actions/moyens supplémentaires.

-l'éthique de la discussion, qui favorise l'émergence d'un consensus juste pour allouer les moyens aux objectifs. Ce troisième paradigme est celui qui est illustré dans l'état actuel du droit, quand il s'agit d'envisager une entrée en unité de vie protégée ou une limitation ou un arrêt de certains traitements (LAT), à savoir une procédure collégiale. Cette procédure, rappelons-le, tient compte de l'évaluation des bénéfices et risques, de l'avis de la personne, recueilli directement (lorsque cela est possible) ou indirectement à la lecture de ses directives anticipées, auprès de sa personne de confiance ou de ses proches.

On retrouve d'ailleurs une déclinaison de ces paradigmes dans les quatre principes qui inspirent la réglementation sur la protection des personnes vulnérables [START_REF]LOI no 2019-222 du 23 mars 2019 de programmation 2018-2022 et de réforme pour la justice[END_REF][START_REF]du 5 mars 2007 portant réforme de la protection juridique des majeurs[END_REF] Sur un plan sociétal, la mise en application de ces principes suppose qu'ils puissent être appliqués de manière équitable sur l'ensemble du territoire, c'est-à-dire qu'ils ne soient pas faussés par un manque de moyens qui rendrait d'emblée caduque une solution. Or c'est bien là l'enjeu éthique majeur : si tout ce qui était nécessaire, comme moyen matériel et humain pour le grand âge (et de surcroît dans un contexte de pandémie, en matière de protection individuelle et de moyens diagnostics), était disponible, il n'y aurait pas fondamentalement de questionnement éthique : tout le monde, y compris les personnes les plus vulnérables, pourrait jouir de ses droits et assumer ses devoirs en étant protégé et le choix ne serait pas celui d'une restriction de liberté individuelle ou de protection collective excessive des personnes vulnérables, choix qui nous sont imposés et contre lequel nous nous battons depuis plusieurs années. Cela passerait par des organisations bienveillantes, s'appuyant sur le maillage local du secteur sanitaire de proximité (avec médecin traitant, pharmacie, professions de santé, filière gérontologique dont la gériatrie de proximité et de recours, secteur médico-social et social à l'échelon intercommunal et départemental) pour mettre en oeuvre une éthique de la discussion en collégialité, s'appuyant en priorité sur l'avis de la personne ou, à défaut, de son entourage.

Si le socle théorique est désormais assez étoffé (loi d'adaptation de la société au vieillissement du 28 décembre 2015 [START_REF] Loi | 1776 du 28 décembre 2015 relative à l'adaptation de la société au vieillissement[END_REF], avis n°128 du Comité Consultation National d'Ethique sur les enjeux éthiques du vieillissement du 15 février 2018 [46], le rapport de Dominique Libault sur la concertation Grand âge et autonomie du 1 er mars 2019 [START_REF] Libault | Concertation Grand âge et autonomie[END_REF], le rapport de Myriam El Khomri en faveur de l'attractivité des métiers du grand âge du 29 octobre 2019 [START_REF] Khomri | Plan de mobilisation national en faveur de l'attractivité des métiers du grand âge 2020-2024[END_REF] ou le manifeste des gériatres en 15 mesures porté par le Conseil National Professionnel de Gériatrie du 15 novembre 2019 [START_REF]Le manifeste des gériatres en 15 mesures pour relever le défi de la transition démographique et de la crise des urgences[END_REF]), le temps est désormais à la mise en actes ! Profitons de cette crise pour affirmer dans les actes qu'on ne cesse pas d'être citoyen.ne quand on avance en âge, qu'on favorise la prise de parole des personnes concernées et que face à la possibilité de l'âgisme, les professionnels de santé et leur éthique (responsabilité, sollicitude et de la discussion) sont des avocats des droits des patients.

  Restreindre drastiquement voire interdire les visites dans les établissements d'hébergements collectifs » et « Interdire toute sortie aux personnes à risque résidant en hébergement collectif à l'extérieur de ces établissements » -« Contre-indiquer aux personnes à risque l'utilisation des moyens de transport collectifs » et « Demander à ces personnes de limiter les déplacements individuels aux seuls déplacements essentiels » Le 16 mars 2020, le Conseil Scientifique national COVID-19 reprenait à son compte ces recommandations [15] :

  de la personne dans le rôle de l'Etat face à une crise sanitaire et ce qui pourrait relever d'une tendance sociétale à une certaine forme d'âgisme et de paternalisme : -Au niveau individuel, dans la balance entre principe d'autonomie et principe de bienfaisance -Au niveau des aidants ou des professionnels, dans la balance entre bienfaisance, non malfaisance et risque professionnel -Au niveau sociétal, dans la balance entre principe de non-malfaisance et principe de justice Il convient néanmoins de ne pas méconnaître le niveau spécifique de la personne atteinte d'une pathologie neurocognitive ne lui permettant plus d'appréhender les enjeux d'une décision complexe à l'échelon individuel 1. Analyse des conflits éthiques à l'échelle de l'individu Si, comme le rappelaient tour à tour le Haut Conseil de Santé Publique, le Conseil Scientifique COVID-19 et le CCNE, les mesures de confinement, voire d'exclusion sont nécessaires, d'un point de vue sanitaire pour protéger les personnes âgées les plus vulnérables d'un risque de surmortalité immédiat lié à l'infection à COVID-19, il convient de s'interroger sur les conséquences possiblement paradoxales de ces mesures sur la santé de ces personnes que l'on cherche le plus à protéger.

  et qui pourraient être également utilisées dans toute situation où une vulnérabilité est suspectée : -Principe de nécessité : en découle le besoin d'attester objectivement qu'il existe une vulnérabilité particulière -Principe de subsidiarité : en découle le besoin d'attester qu'une alternative ne peut être envisagée (soit car inexistante, soit car déjà essayée et inefficace) -Principe de proportionnalité : en découle le besoin de recherche la moindre contrainte possible pour l'efficacité maximale -Principe d'individualisation : en découle le besoin de mise en adéquation des moyens liés à l'environnement de la personne et ses besoins spécifiques