Clément Dubuisson

Keywords: 2010 Mathematics Subject Classification. primary 35P15, secondary 30C35, 47A75, 47B10 Fractional Schrödinger operator, complex perturbation, discrete spectrum, Lieb-Thirring type inequality

Notes on Lieb-Thirring type inequality for a complex perturbation of fractional Schrödinger operator.

NOTE ON LIEB-THIRRING TYPE INEQUALITIES FOR A COMPLEX PERTURBATION OF FRACTIONAL LAPLACIAN

CL ÉMENT DUBUISSON ABSTRACT. For s > 0, let H0 = (-∆) s be the fractional Laplacian. In this paper, we obtain Lieb-Thirring type inequalities for the fractional Schrödinger operator defined as H = H0 + V, where V ∈ L p (R d ), p ≥ 1, d ≥ 1, is a complex-valued potential. Our methods are based on results of articles by Borichev-Golinskii-Kupin [BGK09] and Hansmann [Han11].

INTRODUCTION.

The article by Abramov-Aslanyan-Davies [START_REF] Abramov | Bounds on complex eigenvalues and resonances[END_REF] gave rise to many papers devoted to the study of eigenvalues for complex perturbations of various selfadjoint operators. Recently, the fractional Laplacian (-∆) s , s > 0, received an increasing interest due to its numerous applications in applied mathematics and physics (see [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] for references).

For s > 0, the fractional Laplacian (-∆) s is defined with the help of the functional calculus applied to the nonnegative self-adjoint operator -∆. That is, (-∆) s is essentially self-adjoint on C ∞ c (R d ; C), and the domain of its closure is the fractional Sobolev space

W s,2 (R d , C) := {f ∈ L 2 (R d ), R d (1 + |ζ| 2s )| f (ζ)| 2 dζ < +∞},
where f is the Fourier transform of f (see [DPV12, Section 3.1]). By the spectral mapping Theorem, the spectrum of (-∆) s is R + = [0; +∞[. We consider the fractional Schrödinger operator

(1.1)

H = (-∆) s + V,
where V is the operator of multiplication by the complex-valued function V and we note H 0 := (-∆) s . In particular, the perturbed operator H is not supposed to be self-adjoint. We assume that V is a relatively compact perturbation of H 0 , i.e. dom(H 0 ) ⊂ dom(V ) and V (λ -H 0 ) -1 is compact for λ ∈ C\σ(H 0 ). The spectrum, the essential spectrum, and the discrete spectrum of H will be denoted by σ(H), σ ess (H), and σ d (H), respectively. Here, the discrete spectrum is the set of all eigenvalues which are discrete points of the spectrum whose corresponding eigenspaces (or rootspaces) are finite dimensional. Throughout this work, eigenvalues are counted according to their algebraic multiplicity. The essential spectrum of H is defined by σ ess (H) = {λ ∈ C, λ -H is not Fredholm}, where a closed operator is a Fredholm operator if it has a closed range and both its kernel and cokernel are finite dimensional. In our situation, σ ess (H) := σ(H)\σ d (H). For more details on these definitions, see [START_REF] Edmunds | Spectral theory and differential operators[END_REF]Chapter IX]. By Weyl's Theorem on essential spectrum (see [RS78, Theorem XIII.14]), we have

σ ess (H) = σ ess (H 0 ) = σ(H 0 ) = R + ,
and the only possible accumulation points of σ d (H) lie on σ ess (H).

Our interest in the present topic was motivated by the article of Frank-Lieb-Seiringer [FLS08] on Hardy-Lieb-Thirring inequalities for fractional Schrödinger operator with real-valued potential V . As an application, the authors give a proof of the stability of relativistic matter. In particular, for 0 < s < min{1; d 2 }, γ > 0, and

V -∈ L γ+d/2s (R d ), formula (5.11) in [FLS08] says that (1.2) λ∈σ d (H) |λ| γ ≤ C d,s,γ • V - γ+d/2s L γ+d/2s , where V -= max{0; -V } and C d,s,γ is defined at [FLS08, formula (5.11)].
In this paper, we obtain Lieb-Thirring type inequalities for the fractional Schrödinger operator H with complex-valued V . These inequalities give information on the rate of convergence of points from the discrete spectrum σ d (H) to the essential spectrum of H. The pertaining references on the subject are [FLLS06], [HS02], and [START_REF] Demuth | Eigenvalues of non-selfadjoint operators: a comparison of two approaches[END_REF].

We will assume a little more than V being a relatively compact perturbation of H 0 . Actually, we will suppose that V is relatively Schatten-von Neumann perturbation of the fractional Laplacian. Namely, let S p , p ≥ 1, be the Schatten-von Neumann class of compact operators, see Section 2.3 for further references on the subject. Saying that the potential V defined on R d is a relatively Schatten-von Neumann perturbation of H 0 means that dom(H 0 ) ⊂ dom(V ) and

(1.3) V (λ -H 0 ) -1 ∈ S p ,
for one (and hence for all) λ ∈ C\σ(H 0 ).

Hypothesis (1.3) is fulfilled provided V ∈ L p (R d )
< s ≤ d 2 and V ∈ L p (R d ) with p > d 2s .
Then, for τ > 0 small enough, we have

(1.4) λ∈σ d (H) d(λ, σ(H 0 )) p+τ |λ| α (1 + |λ|) β ≤ K • C p ω ω β-τ τ • V p L p ,
where the powers are

(1) α = min{ p+τ 2 ; d 2s }, (2) β = 2τ + 1 2 ( d s -p -τ ) + .
The constant K depends on d, p, s, and τ . Theorem 1.2. Let H be the fractional Schrödinger operator defined by (1.1) with s > d 2 and V ∈ L p (R d ) with p > 1. Then, for τ > 0 small enough, we have

(1.5) λ∈σ d (H) d(λ, σ(H 0 )) p+1-d 2s +τ |λ| α (1 + |λ|) β ≤ K • C p ω ω β-τ τ • V p L p ,
where the powers are

(1) α = 1 2 + 1 2 min{p -d 2s + τ ; 1}, (2) β = 2τ + 1 2 ( d 2s -p + 1 -τ ) + .
The constant K depends on d, p, s, and τ .

The above theorems essentially rely on complex analysis methods presented in [BGK09], while Theorem 1.3 is based on results of [Han11], obtained with the help of tools of functional analysis and operator theory.

Theorem 1.3. Let H be the fractional Schrödinger operator defined in (1.1) with s > 0 and V ∈ L p (R d ), with p > max{1; d 2s }. Then, for τ > 0, the following inequality holds

(1.6) λ∈σ d (H) d(λ, σ(H 0 )) p (1 + |λ|) d 2s +τ ≤ K • C p ω ω d 2s τ • V p L p ,
with K depending on d, p, s, and τ .

Let us assume that V is real-valued and V ∈ L p (R d ) for p > max{1; d 2s }. Then H = (-∆) s + V is a self-adjoint operator and σ d (H) lies on the negative real half-axis.

In [FLS08], the values of the parameter s are restricted to the range 0 < s < min{1, d 2 } due to the presence of the magnetic potential (see [FLS08, Section 2.1]). Setting γ = p -d 2s > 0, (1.2) becomes

(1.7)

λ∈σ d (H) |λ| p-d 2s ≤ C p,d,s V - p L p ,
where, as always, V -= min{V, 0}. Theorems 1.1 and 1.2 give Lieb-Thirring type inequalities for all positive values of s. In particular, for 0 < s ≤ d 2 , (1.4) becomes

(1.8)

λ∈σ d (H) |λ| max{ p+τ 2 ;p-d 2s +τ } ≤ C d,p,s,ω V p L p .
Rather expectedly, we see that (1.8) is slightly weaker than (1.7), but our results apply to a considerably larger class of potentials.

We continue with few words on the notation. The generic constants will be denoted by C, that is, they will be allowed to change from one relation to another. For two positive functions f, g defined on a domain Ω of the complex plane C, we write f (λ) ≈ g(λ) if there are constants

C 1 , C 2 > 0 such that C 1 f (λ) ≤ g(λ) ≤ C 2 f (λ), for all λ ∈ Ω. We write f (λ) g(λ) ( f (λ) g(λ)) if there is a positive constant C such that f (λ) ≤ C • g(λ) (f (λ) ≥ C • g(λ), respectively) for λ ∈ Ω.
The choice of the domain Ω will be clear from the context.

To compare Theorem 1.1 and Theorem 1.3, it is convenient to consider a sequence (λ n ) ⊂ σ d (H) which converges to λ ∈ σ ess (H). Recall that σ ess (H) = σ(H 0 ). We give details on the comparison between (1.4) and (1.6), the comparison between (1.5) and (1.6) being similar. Without the loss of generality, we assume d(λ n , σ(H 0 )) ≤ 1.

In the case λ ∈]0; +∞[, (1.6) is better than (1.4).

In the case λ = ∞, the term in (1.4) becomes

d(λ n , σ(H 0 )) p+τ |λ n | d 2s +2τ
, for n large enough, and (1.6) is again better than (1.4). When λ = 0, the situation is slightly more complicated. Choose τ > 0 small enough to guarantee α/τ ≥ 1. Then, (1.4) is better than

(1.6) provided Re(λ n ) ≤ 0 or |Im(λ n )| τ |Re(λ n )| α for Re(λ n ) > 0.
Inequality (1.6) is better than (1.4) in the opposite case, i.e., when Re(λ n ) > 0 and |Im(λ

n )| τ |Re(λ n )| α .
To sum up, we see that neither Theorem 1.1 nor Theorem 1.3 take the advantage over each other.

As concluding remark, we would like to mention that it is possible to consider complex matrix-valued potential V as in [Dub14a], devoted to the study of nonself-adjoint Dirac operators. Unlike the latter paper, the study of matrix-valued potentials is neither natural nor complicated in the present framework. The only difference with the scalar-valued case will be the presence of the constant n p/2 in (3.1) and (3.2), n being the size of the square matrix giving the matrix-valued potential.

At last, we say few words on the structure of the paper. We recall some known results and give references in Section 2. The key point of the proofs is the bound on the resolvent of H 0 , and it is proved in Section 3. In Sections 4 and 5 we prove Theorems 1.1 and 1.2, respectively. In Section 6 we deal with Theorem 1.3.

Acknowledgment: I thank Stanislas Kupin for his helpful comments on the subject.

PRELIMINARIES.

2.1. Theorem of Borichev-Golinskii-Kupin. The following theorem, proved in [BGK09, Theorem 0.2], gives a bound on the distribution of zeros of a holomorphic function on the unit disc D := {|z| < 1} in terms of its growth towards the boundary T := {|z| = 1}.

Theorem 2.1. Let h be a holomorphic function on D with h(0) = 1. Assume that h satisfies a bound of the form

(2.1) |h(z)| ≤ exp   K (1 -|z|) α N j=1 1 |z -ζ j | β j   ,
where |ζ j | = 1 and α, β j ≥ 0, j = 1, . . . , N .

Then, for any 0 < τ < 1, the zeros of h satisfy the inequality

h(z)=0 (1 -|z|) α+1+τ N j=1 |z -ζ j | (β j -1+τ ) + ≤ C • K,
where C depends on α, β j , ζ j and τ .

Above, x + = max{x, 0}.

Conformal maps.

Let ϕ a be a conformal map sending D to the resolvent set of the operator H 0 , ρ(H 0 ) = C\R + . For a > 0, it is given by the relation

(2.2) ϕ a : z → λ := -a z + 1 z -1 2 ,
and the inverse map going from C\R + to D is

ϕ -1 a : λ → z := √ λ -i √ a √ λ + i √ a .
Later in the paper, we will have to compare the distance from λ = ϕ a (z) to the boundary of ρ(H 0 ), ∂ρ(H 0 ) = R + , and the distance from z to ∂D = T. The results of this kind are called distortion theorems.

Proposition 2.2 (Distortion between C\R + and D). Let, as above, λ = ϕ a (z). We have

(2.3) a • d(z, T) • |z + 1| |z -1| 3 ≤ d(λ, R + ) ≤ 8a • d(z, T) • |z + 1| |z -1| 3 ,

and

(2.4)

√ a 4 • d(λ, R + ) |λ|(a + |λ|) ≤ d(z, T) ≤ 4 √ a • d(λ, R + ) |λ|(a + |λ|) .
Proof. The first inequality is a direct application of Koebe distortion Theorem [Pom92, Corollary 1.4] to the map ϕ a , so the proof is omitted.

For the second one, we have

(2.5) |z + 1| = 2 |λ| | √ λ + i √ a| , |z -1| = 2 √ a | √ λ + i √ a| .
On the other hand,

| √ λ+i √ a| 2 = |λ|+a+2 √ a Im( √ λ), and, since Im( √ λ) ≥ 0, we obtain |λ| + a ≤ | √ λ + i √ a| 2 ≤ | √ λ| + | √ a| 2 ≤ 2(a + |λ|).
Going back to inequalities (2.3), we get (2.4).

2.3.

F (λ) := (λ + a)(a + H) -1 V (λ -H 0 ) -1 ,
where a is large enough to guarantee that (a + H) is invertible. The coming Proposition 2.3 implies that

V (λ -H 0 ) -1 ∈ S p for λ ∈ ρ(H 0 ), provided V ∈ L p (R d ) and p > max{1; d 2s }. For λ ∈ ρ(H 0 ), we have F (λ) ∈ S p and F holomorphic in ρ(H 0 ), therefore the holomorphic function of interest is, for all λ ∈ ρ(H 0 ), (2.7) f (λ) := det p (Id -F (λ)),
where p = min{n ∈ N, n ≥ p}.

In particular, the zeros of f are the eigenvalues of H (counted with algebraic multiplicities), and, for A ∈ S p , we have

(2.8) |det p (Id -A)| ≤ exp Γ p A p Sp .
We also use the well known inequality from [Sim05, Theorem 4.1], which we call Birman-Solomyak inequality; some authors prefer to call it Kato-Seiler-Simon inequality. Observe that this inequality holds true for 1 < p ≤ 2 by duality of the case p ≥ 2.

Proposition 2.3. Let V ∈ L p (R d ) complex-valued with p > max{1; d 2s }, and s > 0. Assume that λ ∈ ρ(H 0 ).
Then V (λ -H 0 ) -1 ∈ S p , and

V (λ -H 0 ) -1 p Sp ≤ (2π) -d V p L p • (λ -| • | 2s ) -1 p L p .

BOUND ON THE RESOLVENT.

In this section, we bound the expression (λ -| • | 2s ) -1 L p appearing in Proposition 2.3. The difficulty is to obtain the "right" bound when s > d 2 . Indeed, when 0 < s ≤ d 2 , we simply adapt the proof from [START_REF] Demuth | On the discrete spectrum of non-selfadjoint operators[END_REF] to dimensions d ≥ 1. The proof of the bound when s > d 2 requires more work. We will repeatedly use the following elementary inequalities.

Lemma 3.1.

(1) Let a, b ≥ 0, and α > 0, then

min{1; 2 α-1 }(a α + b α ) ≤ (a + b) α ≤ max{1; 2 α-1 }(a α + b α ).
(2) In particular, with α = 2, for a, b ≥ 0, we have

a 2 + b 2 ≤ a + b ≤ √ 2 a 2 + b 2 . We recall that v d-1 = 2π d-1 2 Γ( d-1 2 ) for d ≥ 2 and it is convenient to put v 0 = 2 for d = 1. Proposition 3.2. Let λ = λ 0 + iλ 1 ∈ C\R + . Set δ = d 2s -1. For 0 < s ≤ d
2 and p > d 2s , we have

(3.1) (λ -| • | 2s ) -1 p L p ≤ v d-1 2s • M 1 • |λ| d 2s -1 d(λ, σ(H 0 )) p-1 ,
where

M 1 = max K 2 ; +∞ 0 t δ dt (t 2 + 1) p 2
, and K 2 is defined in (3.7) and depends on d, p, and s. For s > d 2 and p > 1, we have

(3.2) (λ -| • | 2s ) -1 p L p ≤ v d-1 2s • N 1 d(λ, σ(H 0 )) p-d 2s ,
where

N 1 = max +∞ 0 t δ (t 2 + 1) p 2 dt; 1 0 t δ dt + 2 +∞ 0 dt (t 2 + 1) p 2 .
Proof. We start with the polar change of variables

(λ -|x| 2s ) -1 p L p = v d-1 +∞ 0 r d-1 |r 2s -λ| p dr,
and we put

(3.3) I = +∞ 0 r d-1 |r 2s -λ| p dr = +∞ 0 r d-1 |(r 2s -λ 0 ) 2 + λ 2 1 | p 2
dr.

First, we assume that λ 0 < 0, that is, d(λ, σ(H 0 )) = |λ|. In (3.3), we use (r 2s -λ 0 ) 2 ≥ r 4s + λ 2 0 , and we make the change of variables t = r 2s |λ| , so

I ≤ +∞ 0 r d-1 (r 4s + |λ| 2 ) p 2 dr = 1 2s • |λ| d 2s |λ| p +∞ 0 t d 2s -1 (t 2 + 1) p 2 dt. (3.4)
The integrals in (3.4) converge since p > d 2s > 0. Hence, for λ 0 < 0,

(3.5) I ≤ 1 2s • +∞ 0 t d 2s -1 (t 2 + 1) p 2 dt • |λ| d 2s -1 d(λ, σ(H 0 )) p-1 .
Second, we assume λ 0 ≥ 0 and λ 1 > 0 (since (λ -|x| 2s ) -1 = ( λ -|x| 2s ) -1 ). In (3.3), we obtain with the help of the change of variables t = r 2s -λ 0 λ 1 ,

I = 1 2sλ p-1 1 +∞ - λ 0 λ 1 (λ 1 t + λ 0 ) d-1 2s (λ 1 t + λ 0 ) 1 2s -1 (t 2 + 1) p 2 dt = 1 2sλ p-1 1 +∞ - λ 0 λ 1 (λ 1 t + λ 0 ) d 2s -1 (t 2 + 1) p 2 dt. (3.6) We distinguish two cases: 0 < s ≤ d 2 and s > d 2 . If s = d 2 , for Re(λ) ≥ 0, bound (3.1) is obvious from (3.6). Now assume that 0 < s < d 2 and put δ = d 2s -1. Since λ 1 > 0, -λ 0 λ 1 < 0, we have +∞ - λ 0 λ 1 (λ 1 t + λ 0 ) δ (t 2 + 1) p 2 dt = 0 - λ 0 λ 1 (λ 1 t + λ 0 ) δ (t 2 + 1) p 2 dt + +∞ 0 (λ 1 t + λ 0 ) δ (t 2 + 1) p 2 dt.
In the first integral on the right-hand side of this equality, we use that λ 1 t + λ 0 ≤ λ 0 . As for the second term, we observe that, by Lemma 3.1, (λ

1 t + λ 0 ) δ ≤ C d,s (λ 1 t) δ + λ δ 0 .
Here, C d,s = max{1; 2 δ-1 }. Hence, we have

I ≤ 1 2sλ p-1 1 λ δ 0 0 - λ 0 λ 1 1 (t 2 + 1) p 2 dt + +C d,s λ δ 1 +∞ 0 t δ (t 2 + 1) p 2 dt + C d,s λ δ 0 +∞ 0 1 (t 2 + 1) p 2 dt ≤ K 1 2sλ p-1 1 • λ δ 0 + λ δ 1 ,
where

K 1 = max (1 + C d,s ) R + 1 (t 2 + 1) p 2 dt; C d,s R + t δ (t 2 + 1) p 2
dt . Then, putting C d,s = max{1; 2 1-δ }, we have by Lemma 3.1,

I ≤ K 1 2sλ p-1 1 • C d,s (λ 0 + λ 1 ) δ ≤ K 1 2sλ p-1 1 • C d,s ( √ 2) δ |λ| δ .
Consequently, for λ 0 ≥ 0,

(3.7) I ≤ K 2 2s • |λ| d 2s -1 d(λ, σ(H 0 )) p-1 ,
where

K 2 = K 1 • C d,s • 2 δ/2
. Recalling (3.5), we obtain inequality (3.1) in the case s ≤ d 2 . Let us turn now to the case s > d 2 . Suppose again that λ 1 > 0. We recall (3.6)

I = 1 2sλ p-1 1 +∞ - λ 0 λ 1 (λ 1 t + λ 0 ) δ (t 2 + 1) p 2 dt.
Since -1 < δ = d 2s -1 < 0, we cannot use the previous bound. Making the change of variables u = t + λ 0 λ 1 , we obtain

+∞ - λ 0 λ 1 (λ 1 t + λ 0 ) δ (t 2 + 1) p 2 dt = λ δ 1 +∞ - λ 0 λ 1 (t + λ 0 λ 1 ) δ (t 2 + 1) p 2 dt = λ δ 1 +∞ 0 u δ (u -λ 0 λ 1 ) 2 + 1 p 2 du.
The last integral is bounded independently of λ, i.e.,

+∞ 0 u δ (u -λ 0 λ 1 ) 2 + 1 p 2 du ≤ 1 0 u δ du + +∞ 1 1 (u -λ 0 λ 1 ) 2 + 1 p 2 du ≤ 1 0 u δ du + R 1 (u 2 + 1) p 2 du.
Indeed, in the first inequality, we use (u -λ 0 λ 1 ) 2 + 1 ≥ 1 when 0 ≤ u ≤ 1, and u δ ≤ 1 when u ≥ 1 (since δ < 0 and p > 1). Hence, for λ 0 ≥ 0,

(3.8) I ≤ K 3 2s • λ δ 1 λ p-1 1 = K 3 2s • 1 d(λ, σ(H 0 )) p-d 2s , with K 3 = 1 0 u δ du + 2 +∞ 0 du (u 2 + 1) p 2
. Recalling (3.5), the proof of (3.2) is finished.

4. PROOF OF THEOREM 1.1.

Reminding (2.7), we have f (λ) = det p (Id -F (λ)) for λ ∈ ρ(H 0 ) = C\R + , where

F (λ) := (λ + a)(a + H) -1 V (λ -H 0 ) -1 ∈ S p , p ≥ 1.
Inequality (2.8) implies that

log(|f (λ)|) ≤ Γ p (λ + a)(a + H) -1 V (λ -H 0 ) -1 p Sp .
From [Han10, Lemma 3.3.4], we know the following bound on (-a -H) -1 for some a > 0. Then, there exists ω ≥ 1, depending on d, p, s, and V , such that for any a ≥ ω

(4.1) (-a -H) -1 ≤ C ω |ω -a| ,
where

C ω = (1 -V (-ω -H 0 ) -1 ) -1
. By Proposition 2.3, we get the next inequality for p > 1 and λ ∈ C\R

+ log |f (λ)| ≤ Γ p (2π) d • C p ω |ω -a| p • |λ + a| p V p L p (λ -| • | 2s ) -1 p L p . (4.2) Since 0 < s ≤ d 2 , from (3.1), we have (4.3) log |f (λ)| ≤ K 1 C p ω |ω -a| p • V p L p • |λ + a| p |λ| d 2s -1 d(λ, σ(H 0 )) p-1 ,
where (4.4)

K 1 = Γ p (2π) d • v d-1 2s • M 1
and M 1 is defined in (3.1). We now transfer the above inequality on D in order to apply Theorem 2.1. That is, we consider the function g(z) = f • ϕ a (z), where ϕ a is defined by (2.2). It is clearly holomorphic on D. By definition (2.2), we see |λ + a| = 4a|z| |z -1| 2 . So, Proposition 2.2 applied to inequality (4.3) gives

log |g(z)| ≤ K 1 C p ω |ω -a| p • V p L p • (4a) p |z| p a d 2s -1 |z + 1| d s -2 |z -1| 3(p-1) |z -1| 2p |z -1| d s -2 a p-1 d(z, T) p-1 |z + 1| p-1 ≤ K 2 a d 2s |ω -a| p • V p L p • |z| p d(z, T) p-1 |z + 1| p-d s +1 |z -1| d s -p+1 , (4.5) with K 2 = 4 p K 1 C p
ω . Now, by Theorem 2.1, we have for all τ > 0, (4.6)

g(z)=0 (1 -|z|) p+τ |z -1| ( d s -p+τ ) + |z + 1| (p-d s +τ ) + ≤ C • K 2 a d 2s |ω -a| p • V p L p ,
where K 2 is defined above and C depends on d, p, s, and τ . There are three cases to consider: Case 1: d 2s < p < d s , Case 2: p = d s , and Case 3: p > d s .

For each case we will transfer the relation (4.5) back to ρ(H 0 ) = C\R + . Recalling Proposition 2.2, we see

1 -|z| = d(z, T) ≥ √ a 4 • d(λ, σ(H 0 )) |λ| 1/2 (a + |λ|) , |z + 1| 2 ≥ 2|λ| a + |λ|
, and |z -1| 2 ≥ 2a a + |λ| .

Then we will integrate the resulting inequality with respect to a ∈ [ω; +∞[ to get to a sharper bound. This integration follows the idea of Demuth, Hansmann, and Katriel (see [START_REF] Demuth | On the discrete spectrum of non-selfadjoint operators[END_REF] or [START_REF] Demuth | Eigenvalues of non-selfadjoint operators: a comparison of two approaches[END_REF]). Since the computations are similar for all above cases, we give the details of the proof in the first case and present only the main steps in the remaining cases.

g(z)=0 (1 -|z|) p+τ |z -1| d s -p+τ ≤ C • K 2 a d 2s |ω -a| p • V p L p ,
We transfer this inequality back to ρ(H 0 ), i.e.,

(1 -|z|) p+τ |z -1| d s -p+τ ≥ a d 2s +τ 2 5p 2 + 3τ 2 -d 2s • d(λ, σ(H 0 )) p+τ |λ| p+τ 2 (a + |λ|) d 2s + p 2 + 3τ 2 , so (4.8) λ∈σ d (H) d(λ, σ(H 0 )) p+τ |λ| p+τ 2 (a + |λ|) d 2s + p 2 + 3τ 2 a -τ • K 2 |ω -a| p • 2 5p 2 + 3τ 2 -d 2s • V p L p .
We can now perform the integration with respect to a ∈ [ω; +∞[. All terms in this relation are positive, so we can permute the sum and the integral by the Fubini Theorem. Consequently, we obtain

λ∈σ d (H) d(λ, σ(H 0 )) p+τ |λ| p+τ 2 +∞ ω |ω -a| p a -1 (a + |λ|) d 2s + p 2 + 3τ 2 da +∞ ω da a 1+τ • V p L p .
Obviously, (a + |λ|)

d 2s + p 2 + 3τ 2 da ≥ +∞ ω |ω -a| p (a + |λ|) d 2s + p 2 +1+ 3τ 2 da ≥ (|λ| + ω) p+1 (|λ| + ω) d 2s + p 2 +1+ 3τ 2 +∞ 0 t p dt (t + 1) d 2s + p 2 +1+ 3τ 2 .
So, for d 2s < p < d s , (4.9)

λ∈σ d (H) d(λ, σ(H 0 )) p+τ |λ| p+τ 2 (ω + |λ|) d 2s -p 2 + 3τ 2 ≤ C • K 1 C p ω I 1 τ ω τ • 2 δ 1 • V p L p ,
where K 1 is defined in (4.4), C is defined in (4.6),

I 1 = +∞ 0 t p dt (t + 1) d 2s + p 2 +1+ 3τ 2 ,
and

δ 1 = 9p 2 + 3τ 2 -d 2s .
4.2. Case 2: p = d s . Reminding (4.6), we obtain (4.10)

g(z)=0 (1 -|z|) p+τ |z -1| τ |z + 1| τ ≤ C • K 2 a d 2s |ω -a| p • V p L p .
Furthermore, we have

(1 -|z|) p+τ |z -1| τ |z + 1| τ ≥ a p 2 +τ 2 2p+τ • d(λ, σ d (H 0 )) p+τ |λ| p 2 (a + |λ|) p+2τ
, so, for all 0 < τ < 1,

(4.11) λ∈σ(H) d(λ, σ(H 0 )) p+τ |λ| p 2 (a + |λ|) p+2τ a -τ • K 2 |ω -a| p • 2 2p+τ • V p L p .
After integration, we find (4.12)

λ∈σ d (H) d(λ, σ(H 0 )) p+τ |λ| p 2 (ω + |λ|) 2τ ≤ C • K 1 C p ω I 2 τ ω τ • 2 δ 2 • V p L p ,
where K 1 is defined in (4.4), C is defined in (4.6), I 2 = +∞ 0 t p (t + 1) p+1+2τ dt, and δ 2 = 4p + 3 2 τ . In (4.12), we used 1 ≤ 2 τ /2 , to obtain a clear statement of δ j in Remark 4.1.

g(z)=0 (1 -|z|) p+τ |z + 1| p-d s +τ ≤ C • K 2 a d 2s |ω -a| p • V p L p .
Then, we get

(1 -|z|) p+τ |z + 1| p-d s +τ ≥ a p+τ 2 2 3 2 (p+τ )+ d 2s • d(λ, σ(H 0 )) p+τ |λ| d 2s (a + |λ|) 3p 2 -d 2s + 3τ 2 ,
and, for 0 < τ < p -d s , (4.14)

λ∈σ d (H) d(λ, σ(H 0 )) p+τ |λ| d 2s (a + |λ|) 3 2 (p+τ )-d 2s a d 2s -p+τ 2 • K 2 |ω -a| p • 2 3 2 (p+τ )+ d 2s • V p L p .
We integrate the above inequality

λ∈σ d (H) d(λ, σ(H 0 )) p+τ |λ| d 2s +∞ ω |ω -a| p a p+τ 2 -d 2s -1-τ (a + |λ|) 3 2 (p+τ )-d 2s da V p L p τ ω τ .
As before, we do the change of variables t = a -ω |λ| + ω , and we distinguish two

cases: if 1 2 (p-d s -2-τ ) < 0, we use the bound [(|λ|+ω)t+ω] ≤ (|λ|+ω)(t+1), and if 1 2 (p -d s -2 -τ ) ≥ 0, we apply the bound from below [(|λ| + ω)t + ω] ≥ (|λ| + ω)t.
We present the case 1 2 (p -d s -2 -τ ) ≥ 0 in details, the other cases are analogous and are omitted. We see

+∞ ω |ω -a| p a p 2 -d 2s -1-τ 2 (a + |λ|) 3 2 (p+τ )-d 2s da = = (|λ| + ω) p+1 (|λ| + ω) 3 2 (p+τ )-d 2s +∞ 0 t p [(|λ| + ω)t + ω] p 2 -d 2s -1-τ 2 (t + 1) 3 2 (p+τ )-d 2s dt ≥ (|λ| + ω) p 2 -d 2s -1-τ 2 (|λ| + ω) p 2 -d 2s -1+ 3τ 2 +∞ 0 t 3p 2 -d 2s -1-τ 2 (t + 1) 3 2 (p+τ )-d 2s dt. So, if 1 2 (p -d s -2 -τ ) ≥ 0, we obtain (4.15) λ∈σ d (H) d(λ, σ(H 0 )) p+τ |λ| d 2s (ω + |λ|) 2τ ≤ C • K 1 C p ω I 3 τ ω τ • 2 δ 3 • V p L p ,
where K 1 is defined in (4.4), C in (4.6),

I 3 = +∞ 0 t 3p 2 -d 2s -1-τ 2 (t + 1) 3 2 (p+τ )-d 2s
dt, and

δ 3 = 7 2 p + d 2s + 3 2 τ . When 1 2 (p -d s -2 -τ ) < 0, we obtain (4.16) λ∈σ d (H) d(λ, σ(H 0 )) p+τ |λ| d 2s (ω + |λ|) 2τ ≤ C • K 1 C p ω I 4 τ ω τ • 2 δ 4 • V p L p ,
where K 1 is defined in (4.4), C is defined in (4.6), I 4 = +∞ 0 t p (t + 1) p+1+2τ dt, and δ 4 = δ 3 = 7 2 p + d 2s + 3 2 τ . In relations (4.9), (4.12), (4.15), and (4.16), we use the bound ω + |λ| ≤ ω(1 + |λ|), because ω ≥ 1, and we come to inequality (1.4). Thus, the proof in the case 0 < s ≤ d 2 is finished. Remark 4.1. In the above inequalities, one has

I j = +∞ 0 t p+ 1 2 (p-d s -2-τ ) + (t + 1) p+1+2τ + 1 2 max{ d s -p-2τ ;0;p-d s -2-τ } dt, δ j = 7p 2 + 3τ 2 + min{p; d s } - d 2s ,
where j = 1, . . . , 4.

PROOF

K 4 = Γ p (2π) d • v d-1 2s • N 1 ,
where N 1 depends on d, p, and s only. As before, we have

log |g(z)| ≤ K 4 C p ω |ω -a| p • 4 p a d 2s |z| p d(z, T) p-d 2s |z -1| 3d 2s -p |z + 1| p-d 2s .
We set K 5 = 4 p K 4 C p ω . Applying Theorem 2.1, we have

g(z)=0 (1 -|z|) p-d 2s +1+τ |z -1| ( 3d 2s -p-1+τ ) + |z + 1| (p-d 2s -1+τ ) + ≤ (5.2) ≤ C K 5 a d 2s |ω -a| p • V p L p .
The separation in different cases with respect to p and d 2s is clear from the following picture (Figure 1). The x-axis represents p and the y-axis represents d 2s . There are four straight lines given by y = 1, x -y -1 = 0, -x + 3y -1 = 0, and x -3y -1 = 0. 5.1. Case 1: p -d 2s -1 ≥ 0 and 3d 2s -p -1 < 0. We find (5.3)

λ∈σ d (H) d(λ, σ(H 0 )) p-d 2s +1+τ |λ|(a + |λ|) 3 p+τ 2 -3d 4s + 1 2 a -1 2 (p+1-3d 2s +τ ) |ω -a| p • V p L p .
It remains to integrate with respect to a on [ω; +∞[. We do it in the same way as in the case 0 < s ≤ d 2 . From (5.3), we obtain

λ∈σ d (H) d(λ, σ(H 0 )) p+1-d 2s +τ |λ| +∞ ω |ω -a| p a 1 2 (p-3d 2s -1-τ ) (a + |λ|) 3 p+τ 2 -3d 4s + 1 2 da V p L p τ ω τ .
Hence, if p -3d 2s -1 > 0, and τ > 0 is small enough,

(5.4)

λ∈σ d (H) d(λ, σ(H 0 )) p+1-d 2s +τ |λ|(ω + |λ|) 2τ ≤ C • K 4 C p ω I 5 τ ω τ • 2 δ 5 • V p L p ,
where K 4 is defined in (5.1), C in (5.2),

I 5 = +∞ 0 t 1 2 (3p-3d 2s -1-τ ) (t + 1) 3 p+τ 2 -3d 4s + 1 2
dt, and

δ 5 = 1 2 (7p + 5 -3d 2s + 3τ ). Otherwise, if p -3d
2s -1 ≤ 0, we have (5.5)

λ∈σ d (H) d(λ, σ(H 0 )) p+1-d 2s +τ |λ|(ω + |λ|) 2τ ≤ C • K 4 C p ω I 6 τ ω τ • 2 δ 6 • V p L p ,
where K 4 is defined in (5.1), C is defined in (5.2), I 6 = +∞ 0 t p (t + 1) p+1+2τ dt, and δ 6 = δ 5 = 1 2 (7p + 5 -3d 2s + 3τ ).

5.2. Case 2: p -d 2s -1 < 0 and 3d 2s -p -1 < 0. We have (5.6)

λ∈σ d (H) d(λ, σ(H 0 )) p-d 2s +1+τ |λ| 1 2 (p+1-d 2s +τ ) (a + |λ|) p+1-d 2s +τ a -1 2 (p+1-3d 2s +τ ) |ω -a| p • V p L p .
Integrating this inequality gives (5.7)

λ∈σ d (H) d(λ, σ(H 0 )) p+1-d 2s +τ |λ| 1 2 (p+1-d 2s +τ ) (ω + |λ|) 1 2 ( d 2s -p+1+3τ ) ≤ C • K 4 C p ω I 7 τ ω τ •2 δ 7 • V p L p ,
where K 4 is defined in (5.1), C in (5.2),

I 7 = +∞ 0 t p dt (t + 1) 1 2 (p+ d 2s +3+3τ )
, and

δ 7 = 2(2p + 1 -d 2s + τ ). We recall that 0 < p -d 2s < 1, hence d 2s -p + 1 > 0.
5.3. Case 3: p -d 2s -1 < 0 and 3d 2s -p -1 ≥ 0. This time, we have (5.8)

λ∈σ d (H) d(λ, σ(H 0 )) p-d 2s +1+τ |λ| p+1+τ 2 -d 4s (a + |λ|) p+1 2 + d 4s + 3τ 2 a -τ |ω -a| p • V p L p .
After integration, the previous inequality becomes (5.9)

λ∈σ d (H) d(λ, σ(H 0 )) p+1-d 2s +τ |λ| 1 2 (p+1-d 2s +τ ) (ω + |λ|) 1 2 ( d 2s -p+1+3τ ) ≤ C • K 4 C p ω I 8 τ ω τ •2 δ 8 • V p L p ,
where K 4 is defined in (5.1), C in (5.2),

I 8 = +∞ 0 t p dt (t + 1) 1 2 (p+3+ d 2s +3τ )
, and δ 8 = 9 2 p + 5 2 -7d 4s + 3 2 τ . As before, 0 < p -d 2s < 1, and so d 2s -p + 1 > 0. To make the statement of the theorem more transparent in Case 2, we use

1 (1 + |λ|) τ ≥ 1 (1 + |λ|) 3τ 2
. This gives the power β in relation (1.5). Finally, since ω ≥ 1, we bound ω + |λ| ≤ w(1 + |λ|). The proof of Theorem 1.2 is finished.

Remark 5.1. In the above inequalities, one has

I j = +∞ 0 t p+ 1 2 (p-3d 2s -1-τ ) + (t + 1) p+1+2τ + 1 2 max{p-3d 2s -1-τ ;0; d 2s +1-p-τ } dt, δ j = 2(2p + 1 - d 2s + τ ) - 1 2 max{p - d 2s -1 + τ ; 0; 3d 2s -p -1 + τ },
where j = 5, . . . , 8. Theorem 6.1. Let A be a normal bounded operator and B be an operator such that B -A ∈ S p for some p ≥ 1. Suppose also that σ(A) is convex. Then the following inequality holds

LIEB-THIRRING

λ∈σ d (B) d (λ, σ(A)) p ≤ B -A p Sp .
Hansmann has another result in this direction (see [Han13, Cor. 1]). Since in our situation the spectrum σ(A) is convex, it gives no improvement.

We will apply Theorem 6.1 to (-a -H) -1 and (-a -H 0 ) -1 , and so we introduce the parameter a > 0. As in previous section, we need a distortion result. Introduce a conformal map g : C\R + → C\[-1 a , 0] defined by

g(λ) = -1 a + λ .
Below, we denote by λ and µ the variables in C\R + and C\[-1 a , 0], respectively. Proposition 6.2. For λ ∈ C\R + , we have the following bound

d g(λ), - 1 a , 0 ≥ 1 2 √ 5 • d (λ, R + ) (a + |λ|) 2 .
Proof. We obtain a bound for the function g : C\[a; +∞[→ C\[0; 1 a ] defined by g(λ) = 1 λ and then compose it by the translation T : λ → λ+a, that is g = -g •T . After some technical computations (see [START_REF] Dubuisson | Study of the discrete spectrum of complex perturbations of operators from mathematical physics[END_REF]), we find The claimed inequality follows.

As in proofs of Theorems 1.1 and 1.2, we use an integration with respect to the parameter a, to improve the rate of convergence in the left-hand side of inequality (1.6). This trick is borrowed from Theorem 5.3.3 in [START_REF] Demuth | Eigenvalues of non-selfadjoint operators: a comparison of two approaches[END_REF]. We recall that ω is defined in (4.1). 6.2. Proof of Theorem 1.3. We put A = (-a -H 0 ) -1 which is normal and B = (-a -H) -1 which is bounded, for a > ω, so that A and B exist. We know that B -A = BV A ∈ S p , hence we can apply Theorem 6.1. For p ≥ 1, it gives where K 1 is defined in (6.2). The next step of the proof is the integration with respect to parameter a. Since the computations are similar to the integration performed in Section 4, the technical details are omitted. We obtain from (6.3) Using ω + |λ| ≤ ω(1 + |λ|), the proof of Theorem 1.3 is complete.

  and p > max{1; d 2s } (see Proposition 2.3). We denote by d(z, Ω) := inf w∈Ω |z -w| the distance between z ∈ C and Ω ⊂ C. As usual, x + = max{0; x}. The main results of the present article are the following theorems. The constants ω and C ω are defined in (4.1). Theorem 1.1. Let H be the fractional Schrödinger operator defined by (1.1) with 0

4. 1 .

 1 Case 1: d 2s < p < d s . For 0 < τ < d s -p, relation (4.6) becomes (4.7)

  τ . In the left-hand side of this relation, we use the bound a ≤ a + |λ| and we make the change of variables t = a -ω |λ| + ω . Hence, we come to +∞ ω |ω -a| p a -1

4. 3 .

 3 Case 3: p > d s . For 0 < τ < p -d s , relation (4.6) becomes (4.13)

FIGURE 1 .

 1 FIGURE 1. The different cases So, we have three different cases to consider: Case 1: p -d 2s -1 ≥ 0 and 3d 2s -p -1 < 0, Case 2: p -d 2s -1 < 0 and 3d 2s -p -1 < 0, and Case 3: p -d 2s -1 < 0 and 3d 2s -p -1 ≥ 0. Below, the computations are similar to the case s ≤ d 2 , so they are omitted.

  BOUND USING A THEOREM FROM [Han11]. 6.1. Hansmann's Theorem and conformal mapping. The following theorem is the key ingredient for the proof of (1.6). It is proved in [Han11].

  , [a; +∞[) |λ| • (a + |λ|).

(6. 1 )d

 1 µ∈σ d (B) d (µ, σ(A)) p ≤ B -A p Sp .For p > max{1; d 2s }, we bound the right-hand side of inequality (6.1) with the help of Proposition 2.3, and the inequalities (4.1) and (3.5)B -A p Sp ≤ (2π) -d (-a -H) -1 p • V p L p • (-a -|x| 2s ) + 1) p/2 dt. Then µ = (-a -λ) -1 = g(λ) ∈ σ d (B) if and only if λ ∈ σ d (H), hence µ∈σ d (B) d (µ, σ(A)) p = {g(λ),λ∈σ d (H)} d (g(λ), σ(A)) p (λ, σ(H 0 )) p (a + |λ|) 2p .The last inequality results from Proposition 6.2. Thus, we obtain (6.3)λ∈σ d (H) d(λ, σ(H 0 )) p (a + |λ|) 2p ≤ (2 √ 5) p K 1 • C p ω • a d 2s -p |ω -a| p • V p L p ,

.

  λ∈σ d (H) d(λ, σ(H 0 )) p +∞ ω a p-d 2s -1-τ |ω -a| p (a + |λ|) 2p da V p L p τ ω τ .Hence, assuming first that p -d/2s > 1, we come to+∞ ω a p-d 2s -1-τ |a -ω| p da (a + |λ|) 1) 2p dt. When p -d/2s < 1, we have +∞ ω a p-d 2s -1-τ |a -ω| p da (a + |λ|) Hence λ∈σ d (H) d(λ, σ(H 0 )) p (ω + |λ|) d 2s +τ ≤ (2 √ 5) p • K 1 C p ω I τ ω τ • V p L p ,where K 1 is defined in (6.2), andI = +∞ 0 t p+(p-d 2s -1-τ ) + (t +1) p+ d 2s +1+τ +(p-d 2s -1-τ ) + dt.

  Schatten classes and determinants. One can find the definitions and properties of Schatten classes and regularized determinants related to these classes in [DHK09] or [Dub14a]. For detailed discussion and proofs, see the monographs by Gohberg-Krein [GK69] and Simon [Sim77].

	Let us consider the following operator
	(2.6)

INSTITUT DE MATH ÉMATIQUES DE BORDEAUX UNIVERSIT É DE BORDEAUX 351, COURS DE LA LIB ÉRATION F-33405 TALENCE CEDEX E-mail address: clement.dubuisson@math.u-bordeaux.fr


Alexey Ozerov
email: alexey.ozerov@technicolor.com

C ¸agdas ¸bilen
email: cagdas.bilen@technicolor.com

Patrick Pérez
email: patrick.perez@technicolor.com

MULTICHANNEL AUDIO DECLIPPING

Keywords: Multichannel audio, audio declipping, fullrank spatial model, nonnegative tensor factorization, generalized expectation-maximization

Audio declipping consists in recovering so-called clipped audio samples that are set to a maximum / minimum threshold. Many different approaches were proposed to solve this problem in case of singlechannel (mono) recordings. However, while most of audio recordings are multichannel nowadays, there is no method designed specifically for multichannel audio declipping, where the inter-channel correlations may be efficiently exploited for a better declipping result. In this work we propose for the first time such a multichannel audio declipping method. Our method is based on representing a multichannel audio recording as a convolutive mixture of several audio sources, and on modeling the source power spectrograms and mixing filters by nonnegative tensor factorization model and fullrank covariance matrices, respectively. A generalized expectationmaximization algorithm is proposed to estimate model parameters. It is shown experimentally that the proposed multichannel audio declipping algorithm outperforms in average and in most cases a stateof-the-art single-channel declipping algorithm applied to each channel independently.

INTRODUCTION

Audio declipping consists in recovering so-called clipped audio samples that are set to a maximum / minimum threshold. This signal clipping may happen due to limits of the acquisition system or during audio post-processing. Audio declipping belongs as well to a larger family of audio inpainting problems, as recently formulated by Adler et al. [START_REF] Adler | Audio inpainting[END_REF], where the goal is to recover any missing audio samples, assuming their locations within the observed signal are known. Audio declipping is a well known problem and several approaches were proposed in the past [START_REF] Janssen | Adaptive interpolation of discrete-time signals that can be modeled as autoregressive processes[END_REF][START_REF] Abel | Restoring a clipped signal[END_REF][START_REF] Godsill | A Bayesian approach to the restoration of degraded audio signals[END_REF]. Moreover, since the publication of [START_REF] Adler | Audio inpainting[END_REF], where audio inpainting is posed as a general inverse problem, audio declipping has regained interest and several new improved methods were proposed [START_REF] Kitić | Consistent iterative hard thresholding for signal declipping[END_REF][START_REF] Siedenburg | Audio declipping with social sparsity[END_REF][START_REF] Harvilla | Efficient audio declipping using regularized least squares[END_REF][START_REF] Kitić | Sparsity and cosparsity for audio declipping: a flexible non-convex approach[END_REF][START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF]. Some of these methods are based on local sparse/cosparse models [START_REF] Adler | Audio inpainting[END_REF][START_REF] Kitić | Consistent iterative hard thresholding for signal declipping[END_REF][START_REF] Kitić | Sparsity and cosparsity for audio declipping: a flexible non-convex approach[END_REF] and others on more structural models such as social sparsity [START_REF] Siedenburg | Audio declipping with social sparsity[END_REF] or nonnegative matrix factorization (NMF) [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF].

All the audio declipping approaches mentioned above [START_REF] Adler | Audio inpainting[END_REF][START_REF] Janssen | Adaptive interpolation of discrete-time signals that can be modeled as autoregressive processes[END_REF][START_REF] Abel | Restoring a clipped signal[END_REF][START_REF] Godsill | A Bayesian approach to the restoration of degraded audio signals[END_REF][START_REF] Kitić | Consistent iterative hard thresholding for signal declipping[END_REF][START_REF] Siedenburg | Audio declipping with social sparsity[END_REF][START_REF] Harvilla | Efficient audio declipping using regularized least squares[END_REF][START_REF] Kitić | Sparsity and cosparsity for audio declipping: a flexible non-convex approach[END_REF][START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF] are designed for single-channel (mono) audio signals. However, the majority of audio recordings nowadays are multichannel, typically stereo. This concerns not only professionally produced audio recordings, but also the user-generated audio captures, since more and This work was partially supported by ANR JCJC program MAD (ANR-14-CE27-0002). more devices (e.g., cameras, smart-phones and tablets) are provided with several microphones. A straightforward and naive approach to declip multichannel audio would consist in applying a singlechannel declipping algorithm to each channel independently. However, such a simple strategy would suffer from the following limitations. First, even though the same algorithm is applied to each channel, this is done independently and the reconstruction errors might be uncorrelated over channels, thus leading to spatial inconsistency in the resulting declipped multichannel audio. Second, this approach does not exploit possible correlations between channels, which are usually strong, and thus leads to a suboptimal performance. Both issues could be potentially addressed by an approach modeling and exploiting these correlations. Let us give a simple example. An audio source within a stereo recording can be clipped only in the left channel and not in the right one. As such, it seems very beneficial to use the right channel in order to reconstruct the clipped parts in the left one. This is somehow in line with approaches proposed for image desaturation (an equivalent problem in image processing) [START_REF] Zhang | Estimation of saturated pixel values in digital color imaging[END_REF][START_REF] Mansour | Color image desaturation using sparse reconstruction[END_REF], where a non-clipped color channel is used to reconstruct a clipped one.

To the best of the authors knowledge none of the existing audio declipping approaches is suitably designed for multichannel audio, i.e., none of them exploits the inter-channel correlation for an improved and spatially consistent declipping result. In this paper we propose such an approach for the first time. We model the multichannel audio to be declipped as a convolutive mixture of several audio sources, which is a usual assumption in audio source separation [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF][START_REF] Sawada | Underdetermined convolutive blind source separation via frequency bin-wise clustering and permutation alignment[END_REF]. We model then (in the short-time Fourier transform [STFT] domain) the source power spectra with an Itakura-Saito (IS) nonnegative tensor factorization (NTF) model [START_REF] Févotte | Notes on nonnegative tensor factorization of the spectrogram for audio source separation : statistical insights and towards self-clustering of the spatial cues[END_REF][START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF] and their convolutive mixing with full-rank covariance matrices [START_REF] Duong | Underdetermined reverberant audio source separation using a fullrank spatial covariance model[END_REF]. This is precisely the modeling used for informed source separation in [START_REF] Liutkus | Low bitrate informed source separation of realistic mixtures[END_REF], though here we target a completely different application. The overall modeling leads to a multivariate Gaussian distribution of the mixture STFT, which is a linear transform. Both the Gaussian modeling and the transform's linearity make possible handling time domain losses (due to the clipping) in an optimal way. The proposed approach is essentially an extension of our previous work on single-channel audio declipping using single-source NMF model [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF] and on joint single-channel audio declipping and source separation using multisource NTF model [START_REF] Bilen | Joint audio inpainting and source separation[END_REF]. Note that in the single-channel case, in order to perform audio declipping only, one does not need caring about a possible multi-source nature of an audio recording [START_REF] Adler | Audio inpainting[END_REF][START_REF] Janssen | Adaptive interpolation of discrete-time signals that can be modeled as autoregressive processes[END_REF][START_REF] Abel | Restoring a clipped signal[END_REF][START_REF] Godsill | A Bayesian approach to the restoration of degraded audio signals[END_REF][START_REF] Kitić | Consistent iterative hard thresholding for signal declipping[END_REF][START_REF] Siedenburg | Audio declipping with social sparsity[END_REF][START_REF] Harvilla | Efficient audio declipping using regularized least squares[END_REF][START_REF] Kitić | Sparsity and cosparsity for audio declipping: a flexible non-convex approach[END_REF][START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF], and a multi-source assumption may be needed when the source separation is targeted as well [START_REF] Bilen | Joint audio inpainting and source separation[END_REF]. However, in the multichannel case we consider here, a multi-source assumption becomes essential even for declipping only, since different sources may be clipped differently in different channels due to their different mixing characteristics. As a byproduct of this multi-source assumption, the proposed approach may be used without any modification for joint multichannel audio declipping and source separation. However, we do not investigate this feature here and leave it for a further study.

The rest of the paper is organized as follows. The proposed approach, including modeling, model estimation via a generalized expectation-maximization (GEM) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF], and signal reconstruction via Wiener filtering, is presented in Section 2. Section 3 is devoted to experiments, where the proposed multichannel declipping algorithm is compared with an NMF-based single-channel declipping algorithm [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF] applied independently to each channel. In Section 4 conclusions are drawn and some further research directions are given.

PROPOSED APPROACH

Initial assumptions

Let us consider a multichannel audio signal x it , where i = 1, . . . , I and t = 1, . . . , T are channel and time indices, respectively. 1 It is assumed that the signal is clipped everywhere except on so-called observation support (OS) Ξ ⊂ {1, . . . , I} × {1, . . . , T }. In other words, the values of x it are observed for (i, t) ∈ Ξ and are clipped and unknown for (i, t) ∈ Ξ {1, . . . , I} × {1, . . . , T }\Ξ . The goal of audio declipping is to recover the clipped samples x it , (i, t) ∈ Ξ , given the observed ones.

Model

It is assumed that the multichannel audio signal is a mixture of J audio sources as

x it = J j=1 y ijt , (1)
where y ijt are the samples of the source images, i.e., of the contributions of sources into the mixture, and j = 1, . . . , J is the source index. It is important to highlight that, while in audio source separation [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF][START_REF] Sawada | Underdetermined convolutive blind source separation via frequency bin-wise clustering and permutation alignment[END_REF] the mixing equation ( 1) is always introduced within the problem formulation, here it is a part of the modeling assumptions.

In the windowed time domain, the time domain signals are taken in overlapping (usually non-rectangular) frames of length M . Mixture and source images become {x imn } and {y ijmn } respectively, with n = 1, . . . , N being the frame index and m = 1, . . . , M the sample location within the frame. The OS within the framed representation corresponding to Ξ in time domain is a set Ξ ⊂ {1, . . . , I} × {1, . . . , M } × {1, . . . , N } whose restrictions to n-th frame and to a couple of i-th channel and n-th frame read

Ξ n = {(i, m)|(i, m, n) ∈ Ξ } and Ξ in = {m|(i, m, n) ∈ Ξ }, respec- tively.
The Short-Time Fourier Transfrom (STFT) coefficients of the time domain signals x it and y ijt can be obtained by applying the complex-valued unitary Discrete Fourier Transform (DFT) matrix, U ∈ C F ×M , to the windowed time domain counterparts, yielding xin = Ux in and yijn = Uy ijn where each vector represents the samples within a frame, for example

x in = [x imn ]m=1•••M and xin = [x if n ] f =1•••F ,
with index f representing the frequency index of the Fourier transform coefficients. 1 Throughout this paper the time-domain signals will be represented by letters with two primes, e.g., x , framed and windowed time-domain signals will be denoted by letters with one prime, e.g., x , and complex-valued shorttime Fourier transform (STFT) coefficients will be denoted by letters with no primes, e.g., x.

With the above assumptions the time domain mixing equation ( 1) can be rewritten in the STFT domain as

x f n = J j=1 y jf n , (2)
where x f n and y jf n are channel-wise vectors defined as

x f n = [x if n ]i=1•••I and y jf n = [y ijf n ]i=1•••I .
It is then assumed that the latent source images are modeled as in [START_REF] Liutkus | Low bitrate informed source separation of realistic mixtures[END_REF] with an IS-NTF spectral model [START_REF] Févotte | Notes on nonnegative tensor factorization of the spectrogram for audio source separation : statistical insights and towards self-clustering of the spatial cues[END_REF][START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF] and a full-rank covariance spatial model [START_REF] Duong | Underdetermined reverberant audio source separation using a fullrank spatial covariance model[END_REF]. More precisely, each complex-valued vector y jf n is assumed following a zero-mean circular complex Gaussian distribution as

y jf n ∼ Nc(0, R jf v jf n ), (3)
where R jf is a complex-valued positive definite Hermitian matrix (the full-rank spatial model) and v jf n > 0 is a variance having the following low-rank NTF structure

v jf n = K k=1 q jk w f k h nk , (4)
with

Q = [q jk ] j,k , W = [w f k ] f,k and H = [h nk ] n,k being, respec- tively, J × K, F × K and N × K nonnegative matrices. The full set of model parameters to be estimated is θ = {Q, W, H, [R jf ] j,f }.

Model estimation and signal reconstruction

Let us first introduce few notations. Let x in [x imn ] m∈Ξ in denote the observed (non-clipped) windowed samples in the i-th channel and the n-th frame. Let x n [x T 1n , x T 2n , . . . , x T In ] T denote the concatenation over all channels of the observed samples in the n-th frame.

Estimation of the signal

One can write the posterior distribution of each source image timefrequency vector y jf n given the corresponding observed frame x n and model θ as y jf n |x n ; θ ∼ Nc(ŷ jf n , Σy jf n y jf n ) with ŷjfn and Σy jf n y jf n being, respectively, posterior mean and posterior covariance matrix, each of which can be computed by Wiener filtering [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF] as2 ŷjfn = Σ H

x n y jf n Σ -1

x n x n x n , (5)
Σy jf n y jf n = Σy jf n y jf n -Σ H x n y jf n Σ -1 x n x n Σ x n y jf n , (6)
given the definitions

Σy jf n y jf n R jf v jf n , (7)
Σ x n y jf n U(Ξ n ) H Σy jn y jf n , (8)
Σ x n x n U(Ξ n ) H j Σy jn y jn U(Ξ n ), (9)
where

• Σy jn y jn diag [R jf (k, l)v jf n ] f k,l ,
• Σy jn y jf n is an IF × I matrix formed by columns of Σy jn y jn with index in {f, f +F, f +2F, . . . , f 3 , and

+(I -1)F }, • U(Ξ n ) diag [U(Ξ in )] i is an IF × I|Ξ n | matrix
• U(Ξ in ) is the F × |Ξ in | matrix formed by columns from U with index in Ξ in . An estimate of declipped multichannel signal in the STFT domain can be reconstructed by summing up source image estimates computed in (5), i.e., xfn = j ŷjfn . Note that if the source separation is needed as well, it can be immediately achieved by simply keeping the source images estimates.

Model estimation

Model parameters are estimated via a GEM algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] that is based on multiplicative update (MU) rules [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF]. This is an iterative procedure, where the expectation step consists in computing the conditional expectations of empirical latent source image covariances Cy jf n y jf n E y jf n y H jf n |x n ; θ , given the observations and the model, as

Cy jf n y jf n = ŷjfn ŷH jf n + Σy jf n y jf n , (10)
where ŷjfn and Σy jf n y jf n are computed as in ( 5) and ( 6). Model parameters are then updated during maximization step as

R jf = 1 N n 1 v jf n Cy jf n y jf n , (11) pjfn
= 1 I tr R -1 jf Cy jf n y jf n , (12)
and

q jk ← q jk f,n w f k h nk pjfn v -2 jf n f,n w f k h nk v -1 jf n , (13)
w f k ← w f k j,n h nk q jk pjfn v -2 jf n j,n h nk q jk v -1 jf n , (14)
h nk ← h nk j,f w f k q jk pjfn v -2 jf n j,f w f k q jk v -1 jf n , (15)
where equations ( 13) -( 15) are MU rules minimizing the IS divergence [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF] between tensors P = [p jf n ] j,f,n and V = [v jf n ] j,f,n as in [START_REF] Févotte | Notes on nonnegative tensor factorization of the spectrogram for audio source separation : statistical insights and towards self-clustering of the spatial cues[END_REF][START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF].

Handling clipping constraint

In clipping, the original unknown signal is known to have its magnitude above clipping threshold outside the OS, and so should have the reconstructed signal frames. However, this constraint is not straightforward to handle within the above-described modeling, since the posterior distribution of source images (i.e., y jf n given x n and θ) is no longer Gaussian under this constraint. We have found in [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF] that the following approach (referred to as covariance projection in [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF]) is quite efficient. The following is done for each frame n. If after Wiener filtering (5) at least one windowed time domain sample in {x imn }i,m does not satisfy the constraint, all the estimated samples not obeying the constraint are projected on the corresponding clipped signal and start to be considered as observed, i.e., their indices are added to the OS. Then, the Wiener filtering is recomputed over and over by applying the same strategy. This iterative process stops once all reconstructed samples obey the constraint. Once it is done, the GEM algorithm goes its usual way. Note that the above constraint handling process is re-started from the beginning at each GEM iteration, i.e., the OS is reset to its initial state at the beginning of each iteration. Here we apply exactly the same strategy, since its adaptation to the multichannel case does not present any difficulty.

Generalizing single-channel approaches and blind source separation

Note that the proposed multichannel framework including both the modeling and the GEM algorithm generalizes single-channel NMF/NTF-based frameworks [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF][START_REF] Bilen | Joint audio inpainting and source separation[END_REF] we previously proposed. More precisely, for I = 1 and J = 1 the multichannel framework coincides exactly with the NMF-based single-channel declipping [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF] up to some trivial component-wise scaling (1 × K matrix Q) and some trivial frequency-wise scaling (1 × 1 matrices R 1f ). Moreover, for I = 1 it coincides exactly with the NTF-based single-channel joint declipping and source separation approach [START_REF] Bilen | Joint audio inpainting and source separation[END_REF] up to a trivial source and frequency-wise scaling (1 × 1 matrices R jf ). Finally, it is interesting to remark that when there is no clipping, i.e., Ξ = ∅, the proposed multichannel framework reduces to a blind source separation (BSS) approach that is quite similar to, though not exactly coincides with, the BSS methods described in [START_REF] Arberet | Nonnegative matrix factorization and spatial covariance model for underdetermined reverberant audio source separation[END_REF][START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF].

EXPERIMENTS

Dataset

Since so far there is no available dataset for mutichannel declipping, we have created a small one ourselves using four convolutive stereo (I = 2) mixtures of three sources (J * = 3)4 from the dev1 development dataset of the "Determined and over-determined speech and music mixtures" task of the 2010 Signal Separation Evaluation Campaign (SiSEC2010) [START_REF] Araki | The 2010 signal separation evaluation campaign (SiSEC2010): -Audio source separation[END_REF]. To cover various cases we have chosen both speech and music signals, live and synthetic convolutive mixtures, different reverberation times (130 ms and 250 ms), and different microphone spacings (1 m and 5 cm). More specifically, we have chosen the following sequences (mixtures):

• Sequence 1: dev1_female3_liverec_130ms_1m

• Sequence 2: dev1_male3_synthconv_130ms_5cm

• Sequence 3: dev1_nodrums_synthconv_250ms_1m

• Sequence 4: dev1_wdrums_liverec_250ms_5cm

For each 10 sec length sequence sampled at 16 kHz we kept only the first 5 sec length half. Then, following [START_REF] Siedenburg | Audio declipping with social sparsity[END_REF], each multichannel signal was scaled to have maximum amplitude of 1 in time domain. Finally, each signal was clipped at clipping thresholds 0.7 and 0.9, which corresponds to a rather moderate clipping. 5

Evaluation metric

To evaluate the performance we use the SNRm metric [START_REF] Siedenburg | Audio declipping with social sparsity[END_REF], i.e., the signal to noise ratio (SNR) computed only on the clipped part. This metric was used to evaluate single-channel declipping [START_REF] Siedenburg | Audio declipping with social sparsity[END_REF][START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF], but it extends very easily to the multichannel case as follows:

SNRm = 10 log 10 I i=1 x i,orig ( Ξ i ) 2 I i=1 x i,orig ( Ξ i ) -x i,est ( Ξ ) 2 , (16)

Metric & Method

Clipping level = 0.9 Clipping level = 0. 1. SNRm improvement (dB) for the proposed multichannel algorithm (for J = 1, 2, 3, 4 and 6) over the single-channel NMF-based algorithm [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF] applied to each channel independently (top rows). SNRm improvement (dB) for the single-channel algorithm [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF] over the clipped signal (bottom row).

where x i,orig is the i-th channel of the original time domain signal, x i,est is the i-th channel of the estimated signal, Ξ i = {1, . . . , T } \ Ξ i (with Ξ i = {t|(i, t) ∈ Ξ }) is the set of time indices where the signal is lost due to clipping in the i-th channel, and . denotes the 2 norm of a vector.

Parameters

For the proposed approach, the STFT is computed using a halfoverlapping sine window of 1024 samples (64 ms) and the proposed GEM algorithm is run for 50 iterations. Following [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF] the total number of components, K, is set to 20 for music signals and 28 for speech signals. For comparison, declipping on each channel is performed independently with the NMF based declipping algorithm in [START_REF] Bilen | Audio declipping via nonnegative matrix factorization[END_REF] with the number of components, STFT parameters and number of iterations being the same as the proposed approach. The proposed multichannel declipping algorithm is used with number of sources set to J = 1, 2, 3, 4, 6 in order to observe the change of performance with respect to the number of sources within the model. Before running the GEM algorithms all the model parameters are initialized with random values while assuring that the entries of Q, W and H are all nonnegative and the covariance matrices R jf are all positive definite and Hermitian.

Simulation results

The declipping performance of the proposed algorithm for different numbers of sources is presented in Table 1 along with the baseline result of declipping each channel independently. The results of the proposed algorithm is shown in terms of how much quality (measured in SNRm) is increased with respect to the baseline result, hence positive values represent an improvement while negative values represent a decrease in quality.

The first thing to observe from the results in Table 1 is that at small J (1 or 2), the performance of the proposed approach is consistently better in average than the baseline. This is expected since the multichannel declipping algorithm exploits the correlation among the two channels and therefore provides a better approximation. As the number of sources increases however, the performance can be seen to be unstable, worse than the baseline more frequently. This is due to the fact that when performing multichannel declipping one must approximate the spatial covariance matrices, {R jf } jf , and as the number of sources increases, the accurate estimation of correlation matrices gets increasingly harder and more sensitive to initial-ization. This is in line with the application of similar multichannel models and algorithms to convolutive BSS [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF], where it is observed that the algorithms are very sensitive to initialization of parameters. More surprisingly this even affects the case J = J * , so when the number of sources is correctly set in the algorithm, the performance is not always necessarily better than the performance with smaller J. However, the best average performance for both clipping levels (0.9 and 0.7) is achieved with J = 2, which proves the importance of considering a multi-source modeling for multichannel declipping. Finally, whatever the value of J, there is always at least one test sequence in our simulations for which the performance of the proposed multichannel algorithm is worth than that of its single-channel counterpart. In our opinion this may be related to the above mentioned sensitivity of the proposed multichannel algorithm to parameters initialisation. As such, we believe that initializing the model parameters with a more hand-crafted (not random) initialization may lead to a better declipping performance, as it is observed for convolutive BSS [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF].

CONCLUSION

In this paper we have presented an audio declipping algorithm that can perform declipping on multichannel audio by exploiting the correlations between the channels in addition to the low rank NTF structure of the power spectrum of the audio signals. The proposed algorithm not only takes into account the correlations among the channels, but also the multi-source structure of the audio signals as well. It is shown that the proposed algorithm performs better than simply performing declipping on each channel when the number of sources in the estimation model is kept small. It is seen that when a larger number of sources is considered, the performance of the proposed method may drop due to large number of parameters to be estimated. Finally, while outperforming the baseline in average, the proposed approach does not always show a consistent improvement for all test sequences. We believe that this is due to its sensitivity to the parameters initialization. Future work will include investigation of hand-crafted model parameters initialization strategies, investigation of the proposed approach in the context of joint declipping and convolutive BSS, as well as a more efficient implementation to enable faster estimation.

7

  Seq. 1 Seq. 2 Seq. 3 Seq. 4 Average Seq. 1 Seq. 2 Seq. 3 Seq. 4 Average

	SNRm improvement (dB) J = 1	1.30	3.84	2.89	0.58	2.15	3.34	-3.60	0.23	4.77	1.18
	for the proposed	J = 2	3.12	9.20	2.70	-0.86	3.54	4.17	0.85	0.01	2.87	1.98
	multichannel algorithm	J = 3	3.80	2.43	2.81	-0.18	2.21	7.13	-4.68	1.83	1.86	1.53
	over the single-channel	J = 4	-0.89	-8.50	1.57	-0.28	-2.02	5.04	-3.77	-0.54	2.00	0.68
	algorithm [9]	J = 6	-6.16	-19.84	8.52	1.84	-3.90	3.34	0.20	-0.78	1.45	1.05
	SNRm improvement (dB) for
	the single-channel algorithm [9]	17.83	34.42	12.66	7.67	18.15	16.09	17.71	17.06	9.99	15.21
	over the clipped signal
	Table

a H represents the conjugate transpose of the vector (or matrix) a.

diag ([B i ] i ) with B i (i = 1, . . . , I) being matrices represents the corresponding block diagonal matrix.

We denote the true number of sources in the mixture as J * , since in this work we make a distinction between the true number of sources J * and the number of sources J in the model.

The choice of keeping only 5 second length signals and considering only moderate clipping is related to the fact that the computational load of the proposed approach is quite heavy so far. In case of moderate clipping a quite small portion of frames is really clipped, and for non-clipped frames our approach may be very efficiently optimized.