Eric Amar

Bernard Chevreau

Isabelle Chalendar

SUBSPACES OF H p LINEARLY HOMEOMORPHIC TO l p

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction.

This note is inspired by the construction in [START_REF] Halanay | Subspaces of H ∞ and the study of contractions with spectral radius one[END_REF] of special weak*-closed subspaces in the algebra H ∞ (of bounded analytic functions in the open unit disc D). The last two named authors realized that it could be used to obtain weak* homeomorphic copies of ℓ ∞ in H ∞ in a very elementary fashion (at the expense of "forgetting" several additional technicalities needed by the author for later purposes). Then, the first author observed that another fast construction could be achieved via interpolating sequences using a deep result of P. Beurling. Thus below (Section 3) we present both types of construction. In fact a (relatively easy) characterization of weak*-continuous linear mappings from ℓ ∞ to H ∞ (Prop. 2.3) enables us to give a convenient description of "weak*-copies" of ℓ ∞ in H ∞ and, in particular, showing further on (Theorem 3.13) that any such copy is weak*complemented in H ∞ .

The starting point of our first construction is a countable separated set (i.e. consisting of isolated points) S := {a n ; n ∈ N} in the unit circle T. When the closure of this set has arc-length measure 0 the Rudin-Carleson interpolating theorem (see Remark 2 following Theorem 3.2) enables us to prove the existence of a weak*-closed subspace E of H ∞ whose predual has an ℓ 1basis (see below Section 2 for the terminology) consisting of evaluations at the points a n . Then, a straightforward application of a Paley-Wiener type of stability theorem for bases (in the predual of E) leads to the fact that any sequence (b n ) n in D with b n "close enough" to a n , is an interpolating sequence (i.e. the map h → (h(b n )) n from H ∞ is onto ℓ ∞ ) (see Theorem 3.8 and Theorem 3.9). In fact, this profusion of interpolating sequences in the "vicinity" of (a n ) n can be obtained directly from Carleson characterization of such sequences under the mere separation of the sequence (a n ) n ; moreover an idea from [START_REF] Amar | Extension de fonctions holomorphes[END_REF] leads to an explicit construction of a weak*-copy of ℓ ∞ for any of these interpolating sequences (cf. Theorem 3.11).

The above theme leads quite naturally to the question (already considered in the literature) of how to get copies of ℓ p in the Hardy space H p for 1 ≤ p < ∞ ; this question is related to the notion of Besselian and Hilbertian sequences and we briefly discuss it in Section 4. Along the way we give an elementary construction (coming from the first named author's thesis but never published elsewhere) of a Hilbertian basis which is not Besselian (the first such known example being due to Babenko [Bab]).

During the elaboration of this paper we discovered that certain of our results were already known, e.g., the construction of copies of ℓ ∞ via interpolating sequences is in Chap. VII of [Gar] or particular cases of more general ones (e.g., Theorem 3.13 which is a corollary of a deep result of [BesPelc]). However, in line with the above mentioned elementary character of the first construction (and starting point of this work), we have chosen to make this note as self-contained as possible, thus illustrating in (we hope) an easy fashion the interplay between some pieces of Banach space geometry and function theory. Some terminology and preliminary material are developed in Section 2.

Notations and Preliminaries.

As usual, A(D) (D open unit disk) denotes the disk algebra consisting of those functions continuous on the closed unit disk and analytic in D. For 1 ≤ p ≤ ∞ the Hardy space H p can be defined as the norm-closure (weak*-closure if p = ∞) of A(D) in L p (= L p (m)) (where m denotes the normalized Lebesgue measure on the unit circle T). Via Fatou's lemma the space H p can (and will) be identified as the space of analytic functions f in D satisfying ( f p :=)sup 0≤r<1 f r p < ∞ where f r is defined on T by f r (w) = f (rw) and the above weak*-closure refers to the duality

L ∞ (= L ∞ (m)) = (L 1 ) * .
In particular, H ∞ can be seen as the Banach algebra of bounded analytic functions in D (equipped with the "sup" norm:

h ∞ := sup z∈D |h(z)|) and, since H 1 0 is the preannihilator of H ∞ in the above duality, H ∞ is also the dual of the quotient space L 1 /H 1 0 . If [f ] denotes the equivalence class of f ∈ L 1 and h ∈ H ∞ we have thus [f ], h = f, h = f h (:= T f hdm = 1 2π 2π 0
f (e it )h(e it )dt).

Among the elements in the predual L 1 /H 1 0 of H ∞ we have the evaluations

E λ at points λ ∈ D ( E λ , h = h(λ)) ; in fact E λ = [P λ ] where P λ is the Poisson kernel at λ (P λ (z) = 1-|λ| 2 |1-λz| 2 ).
Of course these evaluations define (by restriction) weak*-continuous linear functionals on any weak*-closed subspace E of H ∞ and will thus be identified (keeping the same notation E λ ) with elements in the predual Q E of E (this predual Q E can be seen either as a quotient of L 1 or of L 1 /H 1 0 ). In some instances (see again Remark 2 after Theorem 3.2) evaluations at points of T make sense and are weak*-continuous.

We recall that a sequence (h n ) in H ∞ converges weak* to 0 if and only if it is bounded and converges pointwise to 0 on D.

We will also need some general duality facts. Given two Banach spaces X, Y, L(X, Y ) will denote the Banach space of bounded linear operators from X to Y. We recall the following fundamental and classical result (cf. [Ru], Theorem 4.14.). As an almost immediate corollary we have the following useful proposition. Proposition 2.2. Let X, Y be Banach spaces and M a (linear) subspace of X * . a) Suppose M is weak*-homeomorphic to Y * via a (linear) map T then M is weak*-closed (and T is norm-invertible). b) Conversely if a linear map T : Y * → X * is weak*-continuous and bounded below then it implements a weak*-homeorphism between Y * and the (weak* as well as norm-closed) subspace M = T (Y * ).

Proof

Everything is an immediate consequence of the classical duality result recalled above except the first part of a) for which (by virtue of this same result) we just need to show that T is bounded below. Suppose not: then there exists an unbounded sequence (y n ) n in Y * such that T y n → 0 ; consequently the sequence T y n converges weak* to 0 and so does the sequence (y n ) n in contradiction with its unboundedness.

Spaces of complex-valued sequences will intervene throughout this paper. We denote by ℓ 0 (resp. c 00 ) the linear space of all complex sequences (resp. finitely supported sequences) and by (ǫ n ) n∈N , (ǫ n = (δ n,j ) j∈N ) the canonical (algebraic) basis of c 00 . The Banach spaces c o , ℓ p , 1 ≤ p ≤ ∞ have their usual meaning.

We recall that a Schauder basis in a Banach space X (resp. a w*-Schauder basis in a dual space

X = (Q X ) * ) is a sequence (x n ) n in X such that any x ∈ X has a unique norm expansion x = Σ n α n x n (resp. weak*-convergent expansion x = Σ w * n α n x n , that is, for any L ∈ Q X we have L, x = n α n L, x n ) with (α n ) n ∈ ℓ 0 . Of course (ǫ n
) n is a Schauder basis for c 0 and ℓ p (for p < ∞) and a weak*-Schauder basis for ℓ ∞ . (The reader is referred to [Heil], [Si] for more details on bases.)

We recall also that two Schauder bases (resp. two w*-Schauder bases) (x n ) n in X and (y n ) n in Y are said to be equivalent (resp. w*-equivalent) if there exists A ∈ L(X, Y ) invertible (resp. w*-homeomorphism) such that Ax n = y n for all n. We will say that a Schauder basis in X is a c 0basis (resp. ℓ pbasis) if it is equivalent to the basis (ǫ n ) n (in the corresponding space and, of course, equivalence meaning w*-equivalence in the case p = ∞). Clearly X has an ℓ 1basis if and only if X * has an ℓ ∞basis. Thus our initial goal can be reformulated as finding weak*-closed subspaces with an ℓ ∞basis (or whose predual has an ℓ 1basis).

A first step in that direction is the following useful characterization of weak*-continuous linear maps from ℓ ∞ into H ∞ . Proposition 2.3. Let (g n ) n∈N be a sequence of functions in H ∞ . The following 3 conditions are equivalent:

(i) There exists a weak*-continuous linear map

T : ℓ ∞ → H ∞ such that T ǫ j = g j , j ∈ N. (ii) M := sup z∈D n∈N |g n (z)| < ∞.
(iii) There exists a Borel subset B of T of full measure (i.e. m(B) = 1) such that

M := sup z∈D n∈N |g n (z)| < ∞.
When these conditions are fullfilled we have M = M and T = S * with S (from

L 1 /H 1 0 into ℓ 1 ) defined by [f ] → ( T f g n ) n∈N and M = M = S (= T ).
Furthermore, in case the g n 's all belong to the disk algebra, they are equivalent to (ii)'

M 1 := sup z∈D n∈N |g n (z)| < ∞ (and M 1 = M ).
Proof The last assertion (equivalence of, say, (ii) and (ii)' under membership of the g n 's in the disk algebra) is obvious and stated here just for "the record".

(i) ⇒ (ii)

Let T be a weak*-continuous linear map from ℓ ∞ in H ∞ such that T ǫ j = g j , j ∈ N. It is straightforward to check that (as indicated at the (iii) of the proposition 2.3) T is the adjoint of the (bounded linear) map S (from L 1 /H 1 0 into ℓ 1 ) defined by [f ] → ( T f g n ) n∈N ; thus, for λ ∈ D, S(E λ ) = ( P λ g n ) n = (g n (λ)) n and we have n∈N |g n (λ)|(= S(E λ ) 1 ) ≤ S (since the -characters-E λ are of norm 1). Hence we have (ii) with M ≤ S .

(ii) ⇒ (iii) For each n there exists a subset ω n of T, of measure 0, such that at any w ∈ T\ω n , g n (w) is the limit of g n (z) as z approaches w nontangentially, g n (w) = NTlim z→w, z∈D g n (w). The set ω := n ω n is also of measure 0 and for z ∈ B := T\ω, n ∈ N we have

k≤n |g k (w)| = NT -lim z→w, z∈D k≤n |g k (z)| ≤ M. Therefore the sequence (g n ) n∈N satisfies (iii) with M ≤ M. (iii) ⇒ (i) Let c = (c n ) n ∈ ℓ ∞ ; the series n c n g n is absolutely convergent on B (indeed, for z ∈ B, n |c n g n (z)| ≤ M c ∞ ).
Hence, the sequence of partial sums (G N = n≤N c n g n ) is uniformly bounded and, via an application of the Lebesgue dominated convergence theorem to the equalities f,

G N = f G N , f ∈ L 1 , N ∈ N we obtain its weak*-convergence in L ∞ . Since H ∞ is weak*-closed in L ∞ the limit T c := n c n g n belongs to H ∞ . Moreover the (linear) map S o : [f ] → ( f, g n ) n = ( f g n ) n from L 1 /H 1 0 in (a priori) ℓ 0 is in fact ℓ 1 valued and bounded ; indeed n f g n ≤ n |f g n | ≤ |f | n |g n | ≤ M f 1 .

The duality relation

Sf, c = f, T c (f ∈ L 1 , c ∈ ℓ ∞
) yields the equality T = S * and the weak*-continuity of T. Note also that T = S ≤ M ; combined with the above inequalities M ≤ S and M ≤ M this proves the equality M = M and concludes the proof.

Thus, having a sequence (g n ) n satisfying Assertion (ii) in the above proposition we need "something more" to ensure the boundedness below of T. The idea is to have |g n | equal or close to 1 at some point a n or on some subset (these points or subsets being sufficiently "separated" for different values of n) and small elsewhere, 0 if possible at the a k 's for k = n (note that this idea clearly points the way towards interpolating sequences).

3. Constructions of weak*-copies of l ∞ in H ∞ .

An elementary construction.

It is based on the following simple lemma. Here D is the closed unit disk and for a subarc Γ of T with (distinct) endpoints w 1 , w 2 we denote by ∆ Γ the union of Γ with the open domain (contained in D) whose boundary consists of Γ together with the closed segment of endpoints w 1 , w 2 (in case Γ is a closed arc we exclude w 1 , w 2 from ∆ Γ ) .

Lemma 3.1. Let Γ be a proper subarc of T whose complement is not a singleton, a a point of Γ and δ > 0, then there exists a function g ∈ A(D) such that (i) g(a) = 1 and |g(z)| < 1 for z ∈ D\{a}, and (ii) |g| < δ on D\∆ Γ .

Proof

Take g(z) = ( āz+1 2 ) q (or ( 1 2-āz ) q ) with q sufficiently large.

We now give ourselves a sequence of open subarcs of T, (Γ n ) n∈N whose closures (in T) are pairwise disjoint, a sequence (a n ) n such that, for each n, a n ∈ Γ n and strictly positive scalars ǫ, δ n such that n∈N δ n < ǫ < 1. (We note that if we wish to start this procedure with a given sequence (a n ) n the only condition required on this sequence is that it be separated.)

We claim that the sequence given by application of the lemma (that is,

(g n ) n in the disk algebra A(D) such that, for each n, g n (a n ) = 1, |g n | < 1 on T\{a n }, and |g n | < δ n on T\Γ n ) satisfies Assertion (i) of the proposition 2.3 with M = 1 + ǫ. Indeed for z ∈ Γ n we have |g n (z)| ≤ 1 and for k = n |g k (z)| ≤ δ k ; hence k∈N |g k (z)| ≤ (1 + k =n δ k ) ≤ (1 + ǫ)
and this inequality is also satisfied for z ∈ T\ n∈N Γ n (since there we have

k∈N |g k (z)| ≤ ǫ). We now show that T is bounded below: Let c = (c n ) n ∈ ℓ ∞ and h := T c = w * n c n g n ; for a given n we have a) |c n | = max z∈∆n |c n g n (z)| and , for any z ∈ D, we have b) |c n g n (z)| ≤ |h(z)| + j =n |c j g j (z)|. Specializing to z ∈ ∆ Γn (in which case |g j (z)| < δ j for j = n) , we obtain |c n g n (z)| ≤ h + j =n δ j |c j | ≤ h + ǫ c
and, via a), |c n | ≤ h + ǫ c for all n, leading easily to the desired result:

(1 -ǫ) c ≤ T (c) . Let us summarize what we have shown. Theorem 3.2. The linear mapping T : c = (c n ) n≥N → T (c) = n≥N c n g n implements a weak* and norm-homeomorphism between ℓ ∞ and E = T (ℓ ∞ ) such that T ≤ 1 + ǫ, T -1 ≤ (1 -ǫ) -1 . Remarks 1. Observe that it is possible to choose ǫ so that the Banach-Mazur distance between ℓ ∞ and E = E ǫ , i.e., d BM (ℓ ∞ , E ǫ ) := inf { U U -1 ; U isomorphism}, be arbitrarily close to 1.
2. Note also that we can replace the functions g n by h n = u n g n where the u n satisfy

u n ∞ = 1 = |u n (a n )| without affecting the result (that is the map T : c = (c n ) → w * n c n h n still implements a weak*-homeomorphism between ℓ ∞ and T (ℓ ∞ )).
Let us illustrate this in the case where the closure F of the set {a n ; n ∈ N} has measure 0 (this will happen for instance when the sequence (a n ) n is convergent -which is in fact the situation in [START_REF] Halanay | Subspaces of H ∞ and the study of contractions with spectral radius one[END_REF]). This closed set F is thus a peak set for A(D) ; the (u n ) n can then be chosen in A(D) by virtue of the Rudin-Carleson interpolation theorem (cf. [Gam] Chap. 2, Theorem 6.12) so as to satisfy u n (a k ) = δ n,k . This will lead to h n (a k ) = δ n,k and T bounded below by one.

We observe that in this case the corresponding ℓ 1basis in Q E consists of evalutions E an at the points a n (i.e., E an , h = h(a n ), h ∈ H ∞ ) which, though not defined on all of H ∞ , are well-defined and weak*-continuous on T (ℓ ∞ ). [START_REF] Halanay | Subspaces of H ∞ and the study of contractions with spectral radius one[END_REF] (in fact already in [START_REF] Halanay | On the existence of invariant subspaces for some contraction with spectrum dominating an arc on the unit circle[END_REF]) was inspired by [Wo] which develops a general technique of producing subspaces of H p (1 ≤ p ≤ ∞) which are (isomorphic to) a direct sum of infinitely many copies of H p . This technique relies on constructing subspaces M which are ǫ-supported on a subarc J of T, that is,

The construction in

sup z∈T\J |f (z)| ≤ ǫ f ∞ for f ∈ M ; it is
clear that in our construction above, each (one-dimensional) subspace Cg n is δ nsupported on the corresponding Γ n and hence our space E above can be seen to be norm-isomorphic to ℓ ∞ (with a control on d BM (ℓ ∞ , E) similar to ours) as a consequence of [Wo], Lemma 1 (but weak*-topologies are not discussed there).

Since for each

z ∈ D the sequence (g n (z)) n is in ℓ 1 , the expansion for h = T (c) = n c n g n (z)
shows that E z , the evaluation at z, is well-defined on E and weak*-continuous, in fact E z = S -1 (g n (z)) n . The following continuity properties of the map z → E z , besides their intrinsic interest, will be useful later on. Here we denote by D the derived set of {a n ; n ∈ N} (that is, the set of accumulation points).

Proposition 3.3. The map z → E z is continuous on D\D and has nontangential limit at every point of T ; of course these limit properties transfer to any h in E individually.

Proof.

The last statement results from the inequality

|h(z) -h(w)| ≤ E z -E w h (h ∈ E, z, w ∈ D).
Since S is bicontinuous, proving the first statement amounts to prove the same continuity properties for the map

Φ : D ∋ z → S(E z ) = (g n (z)) n ∈ ℓ 1 .
Continuity of Φ on D\D: Let a ∈ D\D and τ > 0 ; by the definition of D and the fact that {a n ; n ∈ N} consists of isolated points there exists r > 0 such that the open disk D a,r intersects at most one ∆ Γn . Consequently we can choose an integer N such that n≥N δ n < τ /3 and, for n ≥ N, D a,r ∩ ∆ Γn = ∅ (hence

|g n (a) -g n (z)| < 2δ n for z ∈ D a,r ).
Thus starting with the inequality

Φ(a) -Φ(z) ≤ n<N |g n (a) -g n (z)| + n≥N |g n (a) -g n (z)| we obtain, for z ∈ D a,r Φ(a) -Φ(z) ≤ n<N |g n (a) -g n (z)| + 2τ /3.
Now the continuity (with respect to z) of the first term on the right hand side of this inequality enables us to pick η (0

< η < r) such that, for |z -a| < η, n<N |g n (a) -g n (z)| < τ /3, yielding for such z, Φ(a) -Φ(z) < τ.
Nontangential continuity of Φ: Let a ∈ T and, for 0 < r < 1, let St r the Stoltz domain defined by a and r (that is St r is the interior of the convex hull of rD ∪ {a} ) ; there is at most one n (say n 0 ) such that a ∈ ∆ rn ; elementary geometric considerations show that, as soon as the length of Γ n is smaller than 2 √ 1r 2 (and n = n 0 ) we have St r ∩ ∆ Γn = ∅. This done we proceed as before to conclude. 5. We note also that this construction provides (by restriction of the map T to c 0 ) a normisomorphism between c 0 and a subspace of the disk algebra, in other words a subspace of A(D) with a c 0basis. 6. Finally in the case when (a n ) n is convergent the set D is reduced to a singleton, say {a}, we observe that the subspace E does not seem to be "missing by much" its inclusion in the disk algebra A(D) : indeed only continuity at the point a might be lacking. Nevertheless this is enough to substantially "enlarge" the space E since (being isomorphic to ℓ ∞ ) it is not even separable.

A construction with interpolating sequences.

As is well-known, a sequence (b n ) n in D is said to be an interpolating sequence if the mapping

J : h ∈ H ∞ → (h(b n )) n ∈ ℓ ∞ is onto.
Here the construction goes even faster thanks to the following result of Pehr Beurling. Theorem 3.4. [Be] Given an interpolating sequence, (b n ) n in D, there exists a sequence of bounded analytic functions (β n ) n∈N having the following properties:

(i) β k (b j ) = δ k,j for all j, k ∈ N and (ii) sup z∈D n∈N |β n (z)| < M (where M is the interpolating constant).
Indeed, since Hypothesis (ii) is exactly Assertion (ii) of Proposition 2.3 we have already a weak*continuous linear map T from ℓ ∞ in H ∞ such that ∀n ∈ N, T ǫ n = β n . Moreover, for any c ∈ ℓ ∞ and any n ∈ N, we have

|c n | = |c n β n (a n )| = k c k β k (a n ) ≤ T (c) ∞
which shows that the map T is bounded below by 1. Thus, by Prop. 2.2, E := T (ℓ ∞ ) is weak*-closed and T, seen as a map from ℓ ∞ into E, is a weak*-homeomorphism whose inverse is obviously the restriction to the subspace E of the map J (defined at the beginning of this subsection). Moreover by (i) we can write for any a = (a n

) n ∈ ℓ 1 and c = (c n ) n ∈ ℓ ∞ a, c = n a n c n = j,k a j c k δ j,k = j a j E b j , T (c) .
Thus the "preadjoint" S (∈ L(Q E , l 1 )) of T is defined by S( j a j E b j ) = a : in other words, the predual Q E admits (E bn ) n as an ℓ 1basis and E admits (β n ) n as an ℓ ∞basis.

Summing up we have shown: Theorem 3.5. Let (b n ) n be an interpolating sequence in D and (β n ) n a sequence of H ∞ functions satisfying (i) and (ii) of Theorem 3.4 ; then the linear mapping

T : c = (c n ) n∈N → T (c) = n∈N c n β n implements a weak* and norm-homeomorphism between ℓ ∞ and E = T (ℓ ∞ ) (with inverse J |E ) such that T ≤ M, T -1 ≤ 1 ; the sequence (E bn ) n is an ℓ 1 -basis for Q E and (β n ) n an ℓ ∞ -basis for E.
It turns out that the reference [Be] is a preprint which is far from widely available. Thus we give below a proof of a weaker version (bound M 2 instead of M) based on a lemma of Drury [Dru] and a more classical formulation of results on interpolating sequences.

We start with a "finite" version of what is needed. This is in [START_REF] Amar | Interpolating sequences in the ball of C n[END_REF], Prop. 2.2, (with n = 1) where Drury's lemma is used along the lines of [Ber]. We include the proof for the sake of completeness. Lemma 3.6. Suppose the sequence (b j ) 1≤j≤n consisting of pairwise distinct elements of D is interpolating with constant M ; then we can find bounded analytic fonctions (β j ) 1≤j≤n such that (i)

β k (b j ) = δ k,j 1 ≤ k = j ≤ n and (ii) sup z∈D 1≤j≤n |β j (z)| < M 2 .
Proof.

Let λ be a primitive n-th roots of unity in C ; the hypothesis ((b j ) 1≤j≤n interpolating with constant M) ensures the existence of functions

(v j ) 1≤j≤n such that v k (b j ) = λ kj 1 ≤ j ≤ n and max 1≤j≤n v j ∞ ≤ M.
Define the functions ϕ j on D by

ϕ j (z) = 1 n n k=1 λ -kj v k (z), 1 ≤ j ≤ n,
the "Fourier transform" of the function v(•, z) := v k (z), where z ∈ D is a parameter, on the group G of the n-th root of unity with the measure

1 n n k=1 δ k .
Observe that

ϕ j (b l ) = 1 n n k=1 λ -kj v k (b l ) = 1 n n k=1 λ (l-j)k = δ j,l , 1 ≤ j = l ≤ n.
Clearly, the ϕ j 's belong to H ∞ (and ϕ j ∞ ≤ M) ; a straigthforward computation shows that, for any z in D,

|ϕ j (z)| 2 = 1 n 2 ( n k=1 |ϕ k (z)| 2 + 1≤k =l≤n λ j(l-k) A k,l ) with with A k,l = v k (z)v l (z).
Since, for given l = k (as already observed before), it follows that

n k=1 |ϕ k (z)| 2 = 1 n 2 n j=1 n k=1 |v k (z)| 2 ≤ M 2 .
The above equality can also be easily seen as a consequence of Plancherel theorem between the group on n-th roots of unity in C and its dual.

The proof of the lemma is thus completed by setting β j = (ϕ j ) 2 .

Thus, given an interpolating sequence (b n ) n∈N with constant M, for each n we have a finite sequence of analytic functions (β j,n ) 1≤j≤n satisfying (i) and (ii) of lemma 3.6. Now a standard procedure completes the proof of (the weak version of) P. Beurling's theorem.

Since this procedure requires successive extractions of subsequences it is convenient to introduce the following notation for (strictly increasing in our case) maps ϕ and ψ of N in itself:

ϕ ≺ ψ if there exists another strictly increasing sequence j : N → N such that ψ = ϕ • j (thus for any sequence whatsoever

(u n ) n the sequence (u ψ(n) ) n is a subsequence of the sequence (u ϕ(n) ) n (itself subsequence of (u n ) n ).
Note that all the β j,n are of norm bounded by M 2 . As a first step we extract from the bounded sequence (β 1,n ) n a subsequence (β 1,ϕ 1 (n) ) n which is weak*-convergent to say β 1 . Without loss of generality we may and do assume that ϕ 1 (1) ≥ 2 so that the subsequence (β 2,ϕ 1 (n) ) n of (β 2,n ) n is well-defined. As a second step we extract from (β 2,ϕ 1 (n) ) n a subsequence (β 2,ϕ 2 (n) ) n (that is, ϕ 1 ≺ ϕ 2 , with again, as precaution for the next step, ϕ 2 (1) ≥ 3 ) which is weak*-convergent to say β 2 . Continuing in this fashion we clearly obtain a sequence of strictly increasing maps

ϕ k from N to N such that ϕ 1 ≺ ϕ 2 ≺ • • • ≺ ϕ k ≺ • • • with, for all k, ϕ k (1) ≥ k + 1 and the sequence (β k,ϕ k (n) ) n is weak*-convergent to say β k .
We now show that the sequence (β j ) j satisfies the desired properties: let j, l in N ; by the initial definition of the (finite) sequence{β j,ϕ j (n) } 1≤j≤ϕ j (n) , we have β j,ϕ j (n) (b l ) = δ j,l , an equality which is of course preserved by taking the limit as n → ∞, yielding thus β j (b l ) = δ j,l .

Similarly we have for any given z ∈ D, 1≤j≤ϕ j (n) β j,ϕ j (n) < M 2 ; in particular for any integer N we will have, 1≤j≤N β j,ϕ j (n) < M 2 as soon as ϕ j (n) ≥ N, and taking the limit as n → ∞ we get 1≤j≤N |β j | < M 2 from which the proof is easily concluded.

Remark 3.7. Of course, the above construction is more "existential" than "explicit". However, as indicated in the introduction, in some situations it is possible to obtain a "concrete" construction (see Theorem 3.11).

Additional remarks.

When the elementary construction (developed in section 3.1) can be modified via the Rudin-Carleson interpolation theorem we obtain a weak*-copy E of ℓ ∞ whose predual Q E admits an ℓ 1basis consisting of point evaluations at the points a n (cf. Remark 2 following Theorem 3.2 above). It is interesting to observe that in such a case the predual Q E admits also ℓ 1bases of the form (E bn ) n with the b n 's in D. Thus E admits the system of functions biorthogonal to (E bn ) n , say (β n ) n , as an ℓ ∞basis and it follows easily that (b n ) n is an interpolating sequence. Before stating a precise result we recall a Paley-Wiener type stability theorem that will be the key to its proof. Theorem 3.8. ( [Si] Theorem 9.1, p. 84) Let (x n ) n be a Schauder basis in a Banach space X.

a) A sequence (y n ) n in X for which there exists a constant λ ∈]0, 1[ such that ∀α = (α j ) j ∈ c 00 , j α j (x jy j ) ≤ λ j α j x j is a basis (equivalent to the basis (x n ) n ). b) If in addition (x n ) n is an ℓ 1basis then there exists η > 0 such that (y n ) n is an ℓ 1basis whenever x ny n < η for all n.

Proof. a) The hypothesis implies that the linear map j α j x j → j α j (x jy j ) defined on the linear span of the x n 's extends into a bounded linear operator U ∈ L(X) satisfying U ≤ λ < 1. Consequently the operator A := I X -U is invertible and since Ax n = y n for all n we have the desired conclusion. b) Since (x n ) n is an ℓ 1basis there exists a constant ν such that for all α = (α j ) j ∈ c 00 we have α 1 ≤ ν j α j x j ; for α ∈ c 00 and (y n ) n such that x jy j ≤ η for all j, the inequalities

j α j (x j -y j ) ≤ η α 1 ≤ ην j α j x j
enable us to apply a) whenever ην < 1.

Theorem 3.9. Let E be a weak*-closed subspace of H ∞ consisting of functions continuously extendable to a subset Ω of D (Ω ⊃ D) such that the map Ω ∋ z → E z is valued in Q E and continuous. Suppose furthermore that Q E admits an ℓ 1basis of the form (E an ) n with the a n 's in T ∩ Ω. Then there exists (strictly) positive numbers τ n such that (i) the closed disks Dn := Dan,τn are pairwise disjoints, and

(ii) for any sequence (b n ) n chosen so that (for all n) b n ∈ D ∩ D n the sequence (E bn ) n is an ℓ 1basis for Q E (and consequently (b n ) n is an interpolating sequence).

(Of course in the context of this theorem the system (g n ) n in E, biorthogonal to (E an ) n is an ℓ ∞basis for E and the map implementing the weak*-homeomorphism between ℓ ∞ and E is given by T (γ) = j γ j g j .)

Proof.

(i) Since (E an ) n is an ℓ 1basis in Q E , we have inf n =m E an -E am > 0 ; this inequality combined with the continuity of the map z → E z at the points a k ensures that for all n, d(a n , {a m ; m = n} > 0.

(ii) : Immediate consequence of the continuity of the map z → E z combined with the above Theorem 3.8 (with

X = Q E , (x n ) n = (E an ) n , (y n ) n = (E bn ) n ).
Remark 3.10. This last argument is in fact a particular case of the Krein-Milman-Rutman stability theorem (cf. [Si] Theorem 10.3 p. 98) simplified here because we deal with the ℓ 1norm.

As announced in the introduction this rich supply of interpolating sequences in the vicinity of (a n ) n can be obtained under the mere hypothesis of separation of the a n 's on the unit circle. This is probably known to specialists in this area but for the sake of completeness we give a precise statement and prove it using the Carleson's characterization of interpolating sequences, namely a sequence

(b n ) n in D is interpolating if there exists m > 0 such that inf k∈N j =k b k -b j 1 -bk b j > m.
We will even exhibit a concrete linear extension operator associated to any such sequence (b n ) n close to (a n ) n (but of course the weak*-closed subspace of H ∞ -image of this operator-depends on the sequence (b n ) n while in Theorem 3.9 we obtain the same subspace). We will use the standard terminology B, B k , k = 1, 2, ... denote the usual Blaschke products, and,following [Am3] formula 7.6, p.48, we introduce the functions g k :

B(z) := n b n -z 1 -bn z , B k (z) := n =k b n -z 1 -bn z , k ∈ N
g k (z) = 1 -|b k | 2 1 -bk z B k (z) B k (b k ) , k ∈ N.
Theorem 3.11. Let (a n ) n be a separated sequence in T and let m ∈]0, 1[ ; then, there exists a sequence (η n ) n of strictly positive numbers such that any sequence

(b n ) n in D satisfying |b n -a n | <
η n for all n is an interpolating sequence with constant no smaller than m. Moreover the functions (g n ) n associated to the sequence (b n ) n by the above formula generate a weak*-copy of ℓ ∞ (with (g n ) n as an ℓ ∞basis).

Proof. We choose first positive numbers d n such that the closed disks Dan,dn are pairwise disjoint. Notational simplification for this proof: Da,d means D Da,d .

We get by induction on N a sequence of positive numbers τ n and a sequence of compact sets F n in Dn in the following fashion.

Set τ 1 = d 1 and F 1 = Da 1 ,τ 1 ; the map Θ : Da 2 ,d 2 ×F 1 ∋ (t, s) → ts 1 -ts is continuous and satisfies Θ(a 2 , s) (= 1) > m for all s. Thus the open set Θ -1 (]m, ∞[) contains the (compact) set {a 2 }×F 1 ; therefore, by a standard (and elementary) compacity argument, it contains a (compact) set F 2 of the form F 2 = Da 2 ,τ 2 with 0 < τ 2 (≤ d 2 ).

Suppose that τ 1 , τ 2 , .., τ N have been found such that for any s = (s 1 , .., s N ) ∈ F N := 1≤n≤N Dan,dn , we have

Θ N (s) := j =k, j,k≤N s j -s k 1 -sj s k > m.
The function

Θ N +1 : Da N+1 ,d N+1 ×F N ∋ (t, s) → ( 1≤j≤N t -s j 1 -ts j )×Θ N (s)),
is continuous and satisfies Θ N +1 (a N +1 , s) = Θ N (s) > m for all s in F N . The same continuitycompacity argument as above yields the existence of a (compact) set

F N +1 = Da N+1 ,τ N+1 ×F N such that, for any (t, s) ∈ F N +1 , Θ N +1 (t, s) (= 1) > m.
Thus the first statement of our theorem holds for any choice of the η n ' s satisfying η n ≤ τ n .

To prove the last assertion we just have to show that the functions (g k ) k satisfy Condition (ii) of Theorem 3.4 (indeed, as well as the functions (B k (z k (b k )) k , they satisfy Condition (i) which in turn ensures that the associated map

(c n ) n → n c n g n is bounded below). By the Carleson condition we have |B k (b k )| ≥ m (we assume of course that all sequences (b n ) n considered here satisfy |b n -a n | < τ n ) hence |B k (z k (b k ))| ≤ 1/m for all k in N and z in D.
Therefore, we just need to perform some work on the additional factor 1-|bn| 2 |1-bnz| to ensure (ii).

Firstly we note that for any z, t in D we have by geometrical observation the inequality 1 -bn z ≥

1 -|b n | which leads to 1-|bn| 2 |1-bnz| ≤ 2.
Next we let (ǫ n ) n another sequence of positive numbers such that Σ j ǫ j = ǫ < m. We apply again the standard continuity-compacity argument to the function ϕ : K n ×D an,τn ∋ (z, t) → 1-|t| 2 |1-tz| (obviously continuous and with value 0 on K n ×{a n }) to obtain η n > 0 (and of course η n < τ n ) such that ϕ is smaller than ǫ n on K n ×D an,ηn .

Now let z ∈ D : a) if z does not belong to the union of the disks D an,τn then, for each n,

|g n (z)| ≤ ǫ n /m (recall that |B n (b n )| > m)
and hence j |g j (z)| < ǫ ; b) otherwise z belongs to exactly one of these disks say D a k ,τ k ; in this case we get

j |g j (z)| = j =k |g j (z)| + |g k (z)| ≤ j =k ǫ j + 2/m.
Thus Condition (ii) is clearly satisfied ; this concludes the proof.

In the context of Theorem 3.9 one might think that the closure of the set {a n ; n ∈ N} is necessarily of measure 0. In fact it is easy to sketch examples showing that this idea is wrong.

Example 3.12. We start with a closed interval J = [α, β] of the real line and build in this interval a Cantor type set K = k≥1 J k of positive measure in the standard fashion, that is,

J k = J k-1 \ 2 k -1 n=2 k-1 I n (with J 0 = J) where, for 2 k-1 ≤ n < 2 k , I n =]b n , c n
[ is an open interval of length ρ k (with k>0 ρ k < βα to ensure that K has positive measure) removed in the usual "centered" way from each of the successive 2 k-1 intervals whose union is J k-1 . It follows easily from this classical construction that for any sequence (x n ) n satisfying b n ≤ x n ≤ c n (n ≥ 1) the endpoints b n , c n are in the closure of the set {x j ; j ≥ 1} ; thus this closure contains the set K (in fact it is equal to K {x j ; j ≥ 1}). Now choosing

x n in the open interval ]b n , c n [ we have d(x n , {x j ; j = n}) (= min{x n -b n , c n -x n } ) > 0.
Taking βα < 2π the sequence (a n ) n := (e ixn ) n will be one of the desired examples.

3.4. Weak*-copies of ℓ ∞ are w*-complemented.

As shown in the theorem 3.5, in the second type of construction the map T -1 (: E → ℓ ∞ ) is exactly the restriction to E of the interpolating map J :

H ∞ → ℓ ∞ J (h) := (h(b n )) n .
Observe also that the map P : H ∞ → H ∞ defined by P (h) := T (J (h)) is a weak*-continuous projection (norm-bounded by M 2 ) whose range is the space E and, consequently, E is weak*-complemented. This fact is no "accident" and is valid in a more general setting which we now describe. Let X denote an arbitrary Banach space X and let Z = X * . Note first that the map J above (where

X = L 1 /H 1 0 ) can be defined by J (h) = ( E b j , h ) j (where E b j is the element in L 1 /H 1 0 evaluation at b j ) .
We generalize this map by associating to any (bounded

) sequence K = (k j ) j of elements in X an "interpolation map" R (= R K ) from Z in ℓ ∞ defined by R(h) = ( k j , h ) j , h ∈ Z.
We also associate to the sequence K the (bounded linear) map

V (= V K ) from ℓ 1 into X defined by V (γ) = n γ n k n , γ ∈ ℓ 1 and note that V * = R.
We will say that K is an ℓ ∞interpolating sequence (ℓ ∞ -IS for short) if the map R K is onto (or equivalently if the map V K is bounded below).We will denote by X K the closure of the range of V K .

In this setting we have the following general result which is, in fact, with a slightly different formulation, the equivalence of Assertions 2 and 3 of Theorem 10 in [Di], page 48, (see below, Remark 3). Theorem 3.13. a) Let E be a (linear) subspace of Z weak*-homeomorphic to ℓ ∞ (via T ). Then (i) there exists a (bounded) ℓ ∞interpolating sequence K = (k j ) j such that T R K = I Z (and V K is bounded below) ;

(ii) the space E is weak*-complemented in Z and

X K is (norm-) complemented in X. b) Conversely if K = (k j ) j is an ℓ ∞ -interpolating sequence such that X K is complemented in X, say X = X K ⊕ Y then Y ⊥ is weak*-homeomorphic to ℓ ∞ .
Of course the case X = L 1 /H 1 0 gives the result announced by the title of this subsection. Proof. a) By Proposition 2.3 we can view T as a weak*-continuous, bounded below map from ℓ ∞ into Z with range E. Thus T = S * and S(∈ L(X, ℓ 1 )) is onto (by virtue of the classical duality result recalled in the introduction). Hence there exists a sequence

K = (k n ) n in X such that S(k n ) = ǫ n , n ∈ N.
Note that such a sequence is necesssarily biorthogonal to the sequence

(g n ) n = T (ǫ n ) n in Z ; indeed δ j,n = ǫ j , ǫ n = S(k j ), ǫ n = k j , T ǫ n = k j , g n . Moreover
, by the open mapping theorem, we may (and do) choose the sequence K = (k n ) n to be bounded. Then k n = V K (ǫ n ) for all n, and hence

SV K = I l 1 ; consequently, V K S (= V K SV K S) is a bounded projection in X whose range X K is thus norm-complemented in X. Similarly, R K T (= V * K S * = (SV K ) * = I ℓ ∞ )
and T V K is a weak*-continuous projection with range E which is therefore weak*-complemented in Z.

b) Here, since K is an ℓ ∞ -IS, V K implements an isomorphism between ℓ 1 and X K ; also we know, from the topological direct sum X = X K ⊕ Y, that X K is isomorphic to the quotient of X/Y. Therefore, by standard duality facts, the dual of X K is isomorphic to Y ⊥ .

Putting all this together : ℓ ∞ = (ℓ 1 ) * ∼ = X * K ≃ Y ⊥ we obtain the desired result.

Remarks 1. We come back briefly to the context of theorem 3.9 to point out that in this case the subspace E is also norm-complemented in the norm-closed subalgebra A Ω consisting of all the functions in H ∞ continuously extendable to D ∪ Ω. Indeed the map J (associated to (a n ) n ) makes sense on A Ω and so does the (norm-bounded) projection P.

2. The map T (from ℓ ∞ into H ∞ ) implementing the weak*-homeomorphism is usually called the (bounded) linear extension map (cf. [Gar] , Chap. VII, Section 2).

Thus in the more general setting above we will say that the sequence K = (k j ) j in X is a boundedly linearly extendable sequence (notation: BLES) in X for Z if R K admits a right inverse (i.e. there exists T ∈ L(ℓ ∞ , Z) such that RT = I ℓ ∞ ) (or equivalently if V K admits a left inverse).

It follows immediately from these definitions and characterizations that an ℓ ∞ -IS K such that X K = X is a BLES. Thus (via standard facts about duality of subspaces and quotient spaces)

a sequence K is an ℓ ∞ -IS in X for Z = X * if and only if it is a BLES in X K for Z = (X K ) * = Z/( ⊥ X K ).
One more important fact coming out right away from the above is that there exists an ℓ ∞ -IS in X for Z = X * if and only if X contains (isomorphically) a copy of ℓ 1 , that is, in turn by virtue of the celebrated Rosenthal's ℓ 1theorem (cf. [Ros] or [ Chap.1] in [NiPo]), if and only if there exists in X a bounded sequence (x n ) n which admits no weakly Cauchy subsequence. Recall that a sequence (y n ) n is weakly Cauchy if for any h in X * the numerical sequence ( y n , h ) n converges. Rosenthal's ℓ 1theorem says more precisely that a bounded sequence without weakly Cauchy subsequence admits a subsequence equivalent to the canonical basis of ℓ 1 . In particular we deduce from the above that there are such sequences in L 1 /H 1 0 . 3. Theorem 10 in Chapter V of [Di] (which, as mentioned there, is essentially Theorem 4 of [BesPelc] though the latter does not consider weak*-topology) states also (again, up to minor variations in the formulation) that Assertions a) and b) in Theorem 3.13 above are equivalent to the existence a copy of c 0 in X * . This is done via the (nice and rather deep) Bessaga-Pelcynski Selection principle and we refer to [Di], Chap. V, for the details. We limit ourselves to pointing out that any bounded linear map T : c 0 → X * admits a bounded linear extension T : ℓ ∞ → X * (just take T = S * with S = (T * ) |X ). (Of course, there is no reason why T bounded below would imply T bounded below.) 4. About ℓ pinterpolating sequences (1 ≤ p < ∞).

4.1. The case p = 1 ; Sidon sets.

From a theoretical point of view (and in terms of norm-isomorphism) the question of the existence of copies of ℓ 1 in a given Banach space Y is settled by the celebrated Rosenthal's ℓ 1 -Theorem recalled above.

Concretely, we need a bounded sequence G = (g j ) j in Y such that the associated map

V G : λ = (λ n ) n ∈ ℓ 1 → Σ j λ j g j is bounded below.
The Sidon sets provide a whole bunch of them "practically for free" in the disk algebra. Indeed, one definition of a Sidon set of integers is the following:

2. If K = (k j ) j is an ℓ p -BLES via T ∈ L(ℓ p , Z) then the sequence (g j ) j defined by g j = T ǫ j for j ∈ N, is biorthogonal to the sequence K and is a so-called p-Hilbertian sequence, that is, there exists a constant C such that j α j g j ≤ C α p , α = (α j ) j ∈ c 00 ‡ h .

Note also that, conversely, if (g j ) j is a p-Hilbertian sequence in Z then there exists T ∈ L(ℓ p , Z) such that T ǫ j = g j .

3. In case X is an Hilbert space then of course X K is (orthogonally) complemented, Z may be identified with X and Z with X K itself, and (since any bounded linear map from a subspace can be boundedly linearly extended to the whole space) the notions of IS and BLES coincide.

Like in the previous case a fundamental class of examples is provided by standard interpolation associated to sequences in the open unit disk for the Hardy spaces H p .

We briefly review this context:

The duality H p = (H p ′ ) * can be expressed by the mutual action

Φ g (f ) =:< f, g >= T f ḡ (:= 1 2π 2π 0 f (e it )ḡ(e it )dt), f ∈ H p , g ∈ H p ′ .
Note that in this approach g → Φ g is a conjugate linear isomorphism (isometric if p = 2) from H p ′ onto (H p ) * .

Starting with a sequence (z n ) n in D , we define the sequence K

= (k j ) j in H p ′ by k j (z) = (1 -|z j | 2 ) 1/p 1 1 -zj z .
Thus our map R K (:

H p → ℓ 0 ) is here given by R K (g) = ((1 -|z j | 2 ) 1/p g(z j
) that is exactly the map T p introduced in the development of interpolation theory in [ShSh].

The following proposition is essentially a reformulation of observation 2 above.

Proposition 4.5. A sequence K = (k j ) j in X is an ℓ p -BLES for Z iff it admits a biorthogonal p-Hilbertian sequence.

Once we have a bounded linear map from ℓ p into Z we just need this map to be bounded below to have an isomorphic copy of ℓ p in Z or in other words (we need) the existence of a constant c > 0 such that j α j g j ≥ c α p , α = (α j ) j ∈ c 00 ‡ b which can be also expressed by saying that the sequence (g n ) n is a p-Besselian sequence.

(We advert the reader that there is no universal agreement on the terminology regarding the notions of Hilbertian and Besselian sequences.) A sequence (g n ) n which is both p-Besselian and p-Hilbertian is called a p-Riesz sequence. In other words, a basis (in any Banach space) is a p-Riesz basis iff it is equivalent to the canonical basis of ℓ p .

Thus the above translates immediately in the following characterization of copies of ℓ p in a Banach space.

Theorem 4.6. A (closed) subspace M of Z is isomorphic to ℓ p if and only if it admits a p-Riesz basis.

The previous discussion might lead one to think that the existence of a p-Hilbertian sequence (g n ) n in Z which in addition is basic (that is, is a basis for the closed subspace M generated by this sequence) is enough to make M is isomorphic to ℓ p .

We now describe elementary examples showing that this is not the case. Proposition 4.7. ( 1 < p < ∞ ) In the space ℓ p there exists (i) a basis (u n ) n which is p-Hilbertian but non p-Besselian (hence non p-Riesz) ;

(ii) a basis (v n ) n which is p-Besselian but non p-Hilbertian; (iii) a basis which is neither p-Hilbertian nor p-Besselian.

The construction is based on the following finite dimensional computational lemma.

Lemma 4.8. (1 < p < ∞) In the space (C N +1 , p ) there exists a basis (e 0 , .., e N ) such that, denoting (f 0 , .., f N ) its dual basis, we have: a) it is, as well as its dual basis in (C N +1 , p ′ ) , norm bounded by 2 ; b) for λ ∈ C N +1 Proof (of lemma). Let (ǫ j ) 0≤j≤N denote the canonical basis in C N +1 . Setting e j = ǫ j + N +1 v, j = 1, .., N, e 0 = ρv, f 0 = ρ ′ v, f j = ǫ 0 + ǫ j with ρ = (N + 1) -1/p and ρ ′ = (N + 1) -1/p ′ ensures that 1 = e 0 , f 0 = e 0 p = f 0 p ′ = 1 and that (e j ) 0≤j≤N , (f j ) 0≤j≤N are dual bases in C N +1 (of course here, C N +1 is -algebraicallyidentified with its dual via the canonical bilinear form a, b = N j=0 a j b j ) . Now, the inequalities in a) are obvious for the f j 's and e 0 and follow from the following ones for the e j 's (j = 1, ..., N) : We now finish the proof of Proposition 4.7. Proof (of proposition).

Consider (with as usual (ǫ j ) j≥0 the canonical basis in ℓ p ) the standard decomposition of ℓ p into the (ℓ p ) direct sum of its (N + 1) dimensional ℓ p N +1 subspaces: ℓ p 2 = 1 k=0 ǫ k , .. , ℓ p N +1 = a N ≤k<a N+1 ǫ k , where a 1 = 0 and for N > 1 a N (= a N -1 + N) = (N +2)(N -1) 2 .

To each of these subspaces we apply the above lemma and denote (e k ) a N ≤k<a N+1 the basis thus obtained and (f k ) a N ≤k<a N+1 the dual one.

It is clear that (e k ) k≥0 is a basis in ℓ p which is p-Hilbertian (indeed for all λ ∈ c 00 , j λ j e j p ≤ 3 λ p ) with a dual bounded basis (namely (f k ) k≥0 ) and this basis (e k ) k≥0 cannot be p-Besselian : indeed, from c) in the previous lemma ; in fact rather from its proof) we get for each N an element λ (N ) (in ℓ p N +1 hence in c 00 ) such that λ (N ) p = N 1/p → ∞ as N → ∞, while j λ

(N ) j p is bounded.

Thus (e k ) k≥0 is an example for (i) and, by duality, (f k ) k≥0 is an example for (ii). As to (iii), we can just take w k = e k for a N ≤ k < a N +1 when N is even and w k = f k for a N ≤ k < a N +1 when N is odd.

Remark 4.9. The first example of a Hilbertian basis not Besselian was provided by Babenko [Bab] in a Hilbert space context ; the above one (again in the Hilbert space setting) comes from the first named author's thesis [Am2] but was never published elsewhere.

  Theorem 2.1. For S ∈ L(X, Y ) the following three assertions are equivalent : a) S has closed range; b) S * has (norm-) closed range; c) S * has weak*-closed range.

  exists λ ∈ C N +1 such that N j=0 λ j e j p < ρ N λ p with lim N →∞ ρ N = 0.

N

  +1 v p = (N + 1) -1/p ′ and (via Holder's inequality) N j=1 λ j ≤ N 1/p ′ λ p , it follows easily (using the triangular inequality) Finally with regard to c) , with λ = (0, 1, 1, .., 1) we get λ p = N 1/p , N → ∞, leading to the desired conclusion.

An infinite set Γ ⊂ Z is a Sidon set (here, for n ∈ Z, ǫ n is the function t → e int also identified as a function on the unit circle T) if there exists a constant M such that, for any α ∈ c 00 (Γ), j |α j | ≤ M j α j ǫ j ∞ (where c 00 (Γ)denotes the set of sequences with finite support in Γ). We have the following result.

Theorem 4.1. Let Γ ⊂ N a Sidon set ; then the (closed linear) subspace E Γ generated by the ǫ n , n ∈ Γ in the disc algebra A(D) is isomorphic to ℓ 1 ; moreover this isomorphism is a weak*homeomorphism (considering E Γ as a subspace of H ∞ ).

Proof.

The fact that Γ is a Sidon set means (with the above definition) that the (clearly valued in A(D) since here Γ ⊂ N) linear map V Γ : α → j∈Γ α j ǫ j is well-defined, bounded and bounded below. This proves the first assertion. To conclude one checks that V Γ , seen as a map from ℓ 1 in H ∞ , is the adjoint of the map S defined, for

for all n ∈ N and, in particular, for all n ∈ Γ ; in addition S is clearly linear, bounded and valued in c 0 (Γ). Thus V Γ is weak*-continuous and implements a weak*-homeomorphism from ℓ 1 (Γ) onto

Example 4.2. Among the Sidon sets contained in N we have the (set of values) of q-lacunary sequences, i.e., sets of the form {λ k ; k ∈ N} where the integers λ k satisfy λ k+1 λ k ≥ q for some (fixed) q > 1.

Remark 4.3. In view of the result in Subsection 3.4, it is natural to ask if one can obtain a weak*copy, say E, of ℓ 1 (= (c 0 ) * ) in H ∞ (= (L 1 /H 1 0 ) * ) with, in addition, E weak*-complemented in H ∞ . Following the same general and standard duality argument as in the proof of Theorem 3.13, if this were the case, one would obtain easily that the subspace ⊥ E has a complement (in L 1 /H 1 0 ) which is isomorphic to c 0 (everything so far is valid if we consider, instead of H ∞ (= (L 1 /H 1 0 ) * ), an arbitrary dual Banach space Z = X * ) ; but the space L 1 /H 1 0 , being of cotype 2 (cf. [Bou]) , cannot contain a copy of c 0 . Thus we cannot have a weak*, weak*-complemented copy of ℓ 1 in L 1 /H 1 0 .

4.2. The case 1 < p < ∞.. In this section X is a separable reflexive Banach space, Z = X * and Z * is identified with X. We denote by p ′ the conjugate of p ( 1 p + 1 p ′ = 1). The following definitions are natural adaptations of the p = ∞ case.

b) an ℓ pboundedly linearly extendable sequence for Z (ℓ p -BLES for short) if there exists T ∈ L(ℓ p , Z) such that ∀λ = (λ j ) j ∈ ℓ p , R • T (λ) = λ.

We list a few elementary observations and some examples before going on.

1. An ℓ p -BLES is an ℓ p -IS.


Stefan Mehedinteanu
email: s.mehedinteanu@yahoo.com

The Interaction of the Neutral Particles as the Merged Primordial Micro-black Holes With The Quarks of 7Be Can Explains the Cosmological Li Problem

A new mechanism for selectively reducing the lithium abundance, known as the cosmological lithium problem , while keeping other BBN predictions intact is proposed, respectively, the direct destroy of its precursor Be 7 4 7 3 Li , thus reducing its abundance.

by interaction with a new particle type. In the work, I will show that exist a such particle at keV T 50  during Big bang nucleosynthesis (BBN), and this could be a near fully decayed (by the release of its gravitons) primordially micro-black holes ( BH 

) that by merging with the quarks of the Be 7 nucleons, with its further releasing of gravitons, then directly destroy Be 7 as the precursor of

Introduction

There are three things which wait to be elucidated of very long time: namely, the black holes (BH) existence and structure, the gravitation, and how is created the Universe. Following the LIGO experiment, partially these things for the first time become more clearly, thus, firstly the existence of BHs, that theirs merging generates the gravitational waves (gravitons), that only these gravitons deform the space-time.

Mathematics is selectively real. We physicists love our equations. They are so powerful that we often become convinced they are real in themselves. It is that way that we become platonists making mathematics a kind of skeleton on which the flesh of the world is hung hang. But for Unger and Smolin, this reification of mathematics can lead physicists into dangerous territory where mathematical "beauty" and "elegance" get substituted for real information about the real world. As they put it: "Our mathematical inventions offer us no shortcut to timeless truth... They never replace the work of scientific discovery and of imagination. The effectiveness of mathematics in natural science is reasonable because it is limited and relative." Thus, like in the "motto", by the use of simple equations of physics before '75 and rationales we found the mainly waited answers to these hot questions.

In my recently works [START_REF] Mehedinteanu | The Timeline of Universe Based On the Gravitons Release Due of the Merging of the Primordial Micro-black Holes[END_REF], [START_REF] Mehedinteanu | The Quantization of the Gravitational Charges At the Origin of the Gravitational Waves From a Binary Black Hole Merger (LIGO)[END_REF], [START_REF] Mehedinteanu | What Is Gravity and How Is Embedded in Mater Particles giving them mass (not the Higgs field)?[END_REF] is proposed that the key of these is the graviton viewed as gravitational charge quantized as c M G Planck   and which participate as the inside constituent of any fundamental particle. For example, the electron (quark) which is in fact a sphere with a very small radius and inside this is distributed the momentum of photon energy in equilibrium with gravitational charge

Planck M G
. In effect, this particular model would states that the electron is in fact just a form of ''trapped light'', for that a collision  by instant evaporation (being of very small size) as has been considered in unanimity, in fact, these BHs  merge in pairs with release of gravitons as gravitational wave (GW) that deforms the space-time, after that, by capturing photons these remaining as matter particles [START_REF] Mehedinteanu | What Is Gravity and How Is Embedded in Mater Particles giving them mass (not the Higgs field)?[END_REF]. Thus, in the cited works above, this hypothesis is verified by being calculated the entirely timeline of Universe, and the LIGO experiment. Now, the BBN describes the production of the lightest nuclides-D , He However, the BBN success is not complete: the predicted value of the lithium abundance [5] Indeed, several astrophysical mechanisms of how the reduction of lithium may have come about have been proposed, none of which resolve the problem completely [START_REF] Goudelis | A light particle solution to the cosmic lithium problem[END_REF], [START_REF] Fields | The Primordial Lithium Problem[END_REF]. [START_REF] Goudelis | A light particle solution to the cosmic lithium problem[END_REF] the author suggests a new mechanism for selectively reducing the lithium abundance, while keeping other BBN predictions intact.

 2     e e .

Recently in this Letter

A variety of models involving light, weakly interacting particles have been extensively studied in recent years [20] cited in [START_REF] Goudelis | A light particle solution to the cosmic lithium problem[END_REF], including axions, axion-like particles (ALPs), and "dark" vectors. Here, they point out that the cosmological abundance of Li 7 can be reduced down to observed values if during its formation BBN is modified by the presence of light electrically neutral particles X that have substantial interactions with nucleons. Theirs abundance times the absorption cross section by either deuterium or Be without affecting other elements. This is because, unlike in many weak-scale solutions, the suggested mechanism does not inject any new neutrons into the primordial medium, and operates via direct destruction of Be 7 . A variety of particle physics realizations of this idea is possible, and in particular ALPs with small couplings to d-quarks represent a clear target of opportunity for upcoming searches at the intensity frontier.

In the following, will show that a such particle could be a BH  that merges with the others quarks of the nucleons, by the release of gravitons, thus, directly destroying Be 7 . These particles having near the same characteristics as ALPs, respectively keV T 50  , see below.

The primordially micro-black holes as Neutral particles

In Inflation models the scale leaving the horizon at a given epoch is directly related to the number ) ( N of e -folds of slow-roll inflation that occur after the epoch of horizon exit. Indeed, since

 dN k d  
as from [START_REF] Mehedinteanu | The Timeline of Universe Based On the Gravitons Release Due of the Merging of the Primordial Micro-black Holes[END_REF], and therefore ) ( ) ln(

 N k k end  , or, ] [m ke k N end 
where end k is the scale leaving the horizon at the end of slow-roll inflation, or usually ] [

1 1 m k k end    , the correct equation being ] [ 1   m e k k N end . When the wavelength ]) [ ( 1 m k  is large compared to the Hubble length ]) [ ( 1 m H 
, the distance that light can travel in a Hubble time becomes small compared to the wavelength, and hence all motion is very slow the pattern is essentially frozen in. In this new scenario of Universe evolution are reproduced of the known data. Also, along the entire Universe dynamics is verified the Einstein formula

2 mc E 
, or when all the created photons transform in mass of the particles. Since, the FLRW metric of the universe must be of the form To estimate the horizon entry we use some derivations done in [START_REF] Mehedinteanu | The Timeline of Universe Based On the Gravitons Release Due of the Merging of the Primordial Micro-black Holes[END_REF]. During Universe evolution , the horizon leave is when

1   leave leave leave H k a , ] [ 10 27 1 1 m H k leave leave      , s c H t leave leave 36 1 10 3 . 3     
at the Electroweak epoch Here, the Hubble constant is defined as resulting from the equations: a we obtain for outside the object (BH, planets, stars etc.)

] [ ] [ 2 2 3 4 3 4 3 2 3 2 2 2 2           m n a c G a c GE a c G a GM s H a a g g   ; (1)
, where:

g g E n   ;   g P g merging at n n     ;   3 4 3 a M  ; 2 Mc E  ; BH BH m n M    ; 1   H R .
Or, for inside of these objects:

] [ 3 4 3 4 ] [ 3 4 2 2 2 3 2 2 3 3 2 2        m c l n M G c G s a a G a a C merging at U       (2)
During Universe evolution at Electroweak epoch or Reheating due of the quantum fluctuations [START_REF] Mehedinteanu | The Timeline of Universe Based On the Gravitons Release Due of the Merging of the Primordial Micro-black Holes[END_REF]



in pairs and later by the trapping the thermal photons at Electro-Weak Interactions epoch , see [START_REF] Mehedinteanu | The Timeline of Universe Based On the Gravitons Release Due of the Merging of the Primordial Micro-black Holes[END_REF], [START_REF] Mehedinteanu | What Is Gravity and How Is Embedded in Mater Particles giving them mass (not the Higgs field)?[END_REF], here is not consume any graviton (not yet merging-the denominator being zero in eq. ( 1)), so, the added spacetime deformation is near zero. Just after BHs  generation, it appears theirs dividing into a lot of fractions which become as following: a fraction which will becomes matter particles (hadrons) via Quarks-Gluons-Plasma (EWi), see [START_REF] Mehedinteanu | The Timeline of Universe Based On the Gravitons Release Due of the Merging of the Primordial Micro-black Holes[END_REF], other for bosons ( . , , etc Z W 

) and neutrinos, other reserved for leptons (electrons), see [START_REF] Mehedinteanu | The Timeline of Universe Based On the Gravitons Release Due of the Merging of the Primordial Micro-black Holes[END_REF], other reserved for special electrically neutral particles (without photons trapping that that determines the electrical charges [START_REF] Mehedinteanu | What Is Gravity and How Is Embedded in Mater Particles giving them mass (not the Higgs field)?[END_REF]) (I call these "projectiles"), see below, the same for dark matter particles, see [START_REF] Mehedinteanu | The Timeline of Universe Based On the Gravitons Release Due of the Merging of the Primordial Micro-black Holes[END_REF]. pairs which not merge (so, into perfect equilibrium) go directly into Quantum Vacuum, from where these could be extracted (mainly by a Schwinger effect) as the real particles, that, by applying different external fields, like in case of experimental collisions (e-e, pp-p (LHC etc.), or instantly as  W in case of  -decay , see [1] as for the free neutrons decay ". The number of gravitons that has been released following BHs  merging is only In other words the contribution to the space-time deformation is due of . To mention that only this data set match the model. For the others epochs see the results done in [START_REF] Mehedinteanu | The Timeline of Universe Based On the Gravitons Release Due of the Merging of the Primordial Micro-black Holes[END_REF] During the calculation of the Universe timeline based on merging of the primordially micro-black holes ( BH  ) with release of theirs inside gravitons which deform the spacetime, is obtained the epoch of neutral particles production, where it was obtained the energy for a pure one graviton of BH  as from [START_REF] Mehedinteanu | What Is Gravity and How Is Embedded in Mater Particles giving them mass (not the Higgs field)?[END_REF],

   
J c C g 26 10       ; where m c m g C 3     with kg m g 43 10  
. Thus, the further Universe expansion could be when the a fraction of the primordial BH  particles of number calculated in [START_REF] Mehedinteanu | What Is Gravity and How Is Embedded in Mater Particles giving them mass (not the Higgs field)?[END_REF]

m H end   , K k T B X 9 10    ; s c H t end end 335 1    , ] [ 10 7 1 _ 1 1 m H k H N EWi laeve leae        we found 7 . 20  N
to match the iterations cycle:

N a H R T k m end end B g        1    .
The

  ∼ 1 sec to ∼ 3 min after the big bang.

 5 2

 2  , the value inferred from the atmospheres of Population II stars, this discrepancy, known as the cosmological lithium problem, from becoming a full-blown crisis for cosmology is the questionable interpretation of   obs H Li 7 as being the truly primordial value, unaltered by subsequent astrophysical evolution.

  , very weakly interacting particles with energy or mass of ~ 10 MeV and lifetimes of O(10 3 ) seconds can deplete Li Be 7 7 

ds

  is a three-dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature, or for small commoving time aHc dt 1, we can consider the distance as

  , the Friedmann equations are equivalent to this pair of equations:

  The next step it was when we consider the generation of the Virtual Micro-black-holes pairs (

		BH 	), which occur with a number density of approximately one per Quantum
	bubble, that means one per quantum bubble volume,	n	3     d	10 61	[ m	3 	]	; where
															P
															H
	Hubble constant	d			20 10 	m	, at Electroweak epoch (EW) when it is considered.
						H
	Theoretical arguments suggest that virtual black holes should have mass on the order of
	the Planck particles, each one having a graviton number	n		1   c   (	)		35 10	; where
															g	P	C
	the energy of the Planck particle is			9 10	J	; the Compton length is				m	c		1	;
															P	C	g
	and the mass of one graviton is	m		M	35 10		43  10	kg	. Therefore, per total in
														g	P
	Universe are created	n	U		61 10	35 10 		96 10	, so, the total energy of Universe is
									g
		26   10	96 10 		70 10	J	, that corresponds with others model of Universe evolution.
	u
	Now, my main supposition is that despite the elimination of	BH

  a huge number of the micro-black holes as Planck particles inverse process of a black-hole ) from micro-blacks holes created in Universe as the Planck particles during theirs merging (in pairs) that deforms the spacetime.

	Important notice
	"At singularity	10 (   27	10  20	) m	as following inflation theory when just are "born" the
		BHs
	  	a a 	  	2		3 G  8			3 c 	2
	P n are 10 2   J 70 10 9 26 J 10 end  a P , it results with  c  U E g   J 9 10 ; theirs energy being at horizon leave ; 1  leave P m H , kg 53 10 2 . 2   ; M U 62 61 3 1 10 1       P available BH n V H n  , or n P J a  leave BH  m  k 2 c BH  leave generated, when a m c C    ; with EW end BH EW 19 10 11        , GeV 19 10     BH ] [ 10 27 1 1 m H k leave leave      by iteration for 5 . 2  N ; in eq. (2); ] [ 10 2 . 1 1 26     m ; k end ] [ 10 20 1 m H    ; s t 27 10 3 . 3    ; ] [ 10 5 20 m R    ; ] [ 10 27 1 m H     ; l leave C 7 10 2 . 8   EW a ; where 96 10    P P g n n   as a "fix" number of gravitons escaped (an J ;
											g

the mysteriously particles which interact with nucleons are in fact the decayed microblack holes arrived at the time of BBN.

Conclusions

Again it is proved that the mechanism of merging of BHs  as applied to the calculation of timeline of Universe an for LIGO experiment is equally applicable to determine the production of these specially particles which following interaction with the quarks of