aa r X i v : . [ m a t h . F A ] F e b WEAKLY SEPARATED BESSEL SYSTEMS OF MODEL SPACES
ALBERTO DAYAN
Abstract.
We show that any weakly separated Bessel system of model spaces in theHardy space on the unit disc is a Riesz system and we highlight some applications tointerpolating sequences of matrices. This will be done without using the recent solutionof the Feichtinger conjecture, whose natural generalization to multi-dimensional modelsub-spaces of H turns out to be false. Introduction
Let H be the Hardy space on the unit disc D , that is, the reproducing kernel Hilbertspace of those power series centered at the origin with square-summable Taylor coefficients.Its kernel s is the well-studied Szegö kernel s w ( z ) := 11 − wz w, z ∈ D and its multiplier algebra can be identified isometrically with H ∞ , the algebra of boundedholomorphic functions on D . A key role for the study of the function theory and thehyperbolic geometry of the unit disc is played by inner functions , which are those boundedanalytic functions on the unit disc with an unimodular radial limit almost everywhere onthe unit circle. Given an inner function Θ , one can define the associated model space H Θ := H ⊖ ΘH as the orthogonal complement in the Hardy space of all multiples of Θ in H . A greattreatment of the main properties of model spaces, together with their interactions withoperator theory on spaces of analytic functions, can be found in [8].Any function in H that vanishes with multiplicity m at a point λ in D is divisible in H by a Blaschke factor , i.e., an inner function of the form b mλ , where b λ ( z ) := λ − z − λz z ∈ D . Therefore the model space associated to b mλ is m -dimensional and it is spanned by thekernels at λ that represent up to m − derivatives of any function of H at λ , that is,(1.1) H b mλ = span (cid:26) s λ , ∂ s λ ∂w , . . . , ∂ m − s λ ∂w m − (cid:27) . Since a model space is a subspace of H generated by an inner function, it comes naturalto ask whether function theoretical properties of a sequence of inner functions (Θ n ) n ∈ N translate to Euclidean properties for the sequence ( H Θ n ) n ∈ N . Out of the many results thatconstitute such a valuable dictionary between operator theory and function theory, one ofthe most significant for the purpose of this note can be found in [10, Th. 3.2.14]: Date : February 9, 2021.The author was partially supported by National Science Foundation Grant DMS 1565243.
Theorem 1.1.
Let (Θ n ) n ∈ N be a sequence of inner functions such that Q ∞ n =1 Θ n convergesuniformly on any compact subset of D . The followings are equivalent: (i): For any bounded sequence ( φ n ) n ∈ N in H ∞ there exists a function φ in H ∞ suchthat (1.2) φ − φ n ∈ Θ n H , n ∈ N ; (ii): There exists a C ≥ such that, for any sequence ( h n ) n ∈ N of unit vectors in H such that h n belongs to H Θ n for any n in N and for any ( a n ) n ∈ N in l , (1.3) C X n ∈ N | a n | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ∈ N a n h n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C X n ∈ N | a n | ; (iii): There exists a positive δ such that, for any z in D , (1.4) sup n ∈ N Y j = n | Θ j ( z ) | ≥ δ. A sequence of closed subspaces ( H n ) n ∈ N of a Hilbert space H that satisfies (ii) is calleda Riesz system , and the least C for which (1.3) holds is the Riesz bound of the sequence.If the least C such that the right hand side of (1.3) holds is finite then ( H n ) n ∈ N is a Besselsystems with
Bessel bound C . On the other hand, condition (iii) is a function theoreticalproperty on the inner functions (Θ n ) n ∈ N whose importance can not be overstated, beingequivalent, thanks to Carleson’s corona Theorem, [4], to the existence of two boundedsequences ( f n ) n ∈ N and ( g n ) n ∈ N in H ∞ such that f n Θ n + g n · Y j = n Θ j = 1 n ∈ N . Moreover, condition (iii) is related in [11, Lec. IX] to separation conditions on the subspacesin ( H Θ n ) n ∈ N , being equivalent to asserting that the sine of the angle between any modelspace and the closure of the linear span of all the others is uniformly bounded below:(1.5) inf n ∈ N sin H Θ n , span j = n { H Θ j } ! > . If (1.5) holds, we say that ( H Θ n ) n ∈ N is strongly separated , whereas weak separation will correspond to an uniform bound from below for the angle between any pair of distinctmodel spaces:(1.6) inf n = j sin( H Θ n , H Θ j ) > . Suppose now that Θ n = b m n λ n , for some sequence ( λ n ) n ∈ N in the unit disc and some sequence ( m n ) n ∈ N of positive integers. Condition (i) of Theorem 1.1 becomes then an interpolationproperty : for any bounded sequence ( φ n ) n ∈ N in H ∞ there exists a bounded analytic func-tion φ that, for any n in N , agrees with φ n at λ n up to its m n − st derivative. Theorem 1.1is therefore a great example of how the deep interconnection between operator theory andfunction theory greatly helps the studying and the understanding of interpolating sequences .Let H k be a reproducing kernel Hilbert space of analytic functions on a domain D of C d ,and let M k be its multiplier algebra. A sequence Z = ( z n ) n ∈ N in D is interpolating for M k if for any bounded sequence ( w n ) n ∈ N in C there exists a function φ in M k such that φ ( z n ) = w n for any n in N . Intuitively, an interpolating sequence is a separated sequence, EAKLY SEPARATED BESSEL SYSTEMS OF MODEL SPACES 3 as we need to be able to specify the values of φ arbitrarily at the nodes ( z n ) n ∈ N . It is alsonot surprising that the separation conditions we will look at depend on the kernel k : Z isa weakly separated sequence if there exists a positive M such that, for any n = j in N ,there exists a function φ n,j whose norm in M k doesn’t exceed M and that separates z n and z j , that is, φ n,j ( z n ) = 1 φ n,j ( z j ) = 0 . A celebrated work of Carleson,[4], characterized interpolating sequences for H ∞ : Theorem 1.2.
A sequence
Λ = ( λ n ) n ∈ N is interpolating for H ∞ if and only if it is weaklyseparated and the measure µ Λ := X n ∈ N (1 − | λ n | ) δ λ n satisfies the embedding condition (1.7) || f || L ( D ,µ Λ ) ≤ C Λ || f || H f ∈ H . A measure µ on a domain D that embeds continuously L ( D, µ ) into a reproducingkernel Hilbert space H k on D is called a Carleson measure for H k . One can find in[4] a characterization of Carleson measures for H that involves a one-box condition, andhence the hyperbolic geometry of the unit disc. It turns out that (1.7) holds if and onlyif the sequence of all lines through the kernels ( s λ n ) n ∈ N is a Bessel system,[1, Prop. 9.5],highlighting once again a correspondence between the hyperbolic geometry of the unitdisc and the Euclidean geometry of the Hardy space. Moreover, such a characterizationof Carleson measures in terms of Bessel systems allows to extend Theorem 1.2 to somemultiplier algebra other than H ∞ . For example, a class of multiplier algebras for which ananalogous of Theorem 1.2 holds is the one associated with complete Pick kernels . One ofthe most important properties that connects interpolating sequence to the study of relatedHilbert spaces is the fact that, for any multiplier φ of H k , any kernel function k z in H k isan eigenfunction of the adjoint of the multiplication operator M φ :(1.8) M ∗ φ ( k z ) = φ ( z ) k z z ∈ D, as a straightforward computation using the property of adjoints shows. In particular, if M φ is a contraction and φ ( z n ) = w n for any n in N , then the linear map T from S Z := span n ∈ N { k z n } to itself given by(1.9) T ( k z n ) := w n k z n n ∈ N is a contraction. A reproducing kernel Hilbert space is said to have the Pick property if the existence of such a contraction T is also a sufficient condition for the existence of afunction φ in the unit ball of M k such that φ ( z n ) = w n . In particular, this says that M ∗ φ is an isometric extension of T to H k , which implies that any two disjoint sets of points Z and Z can be separated by a function in M k of norm at most M if and only if the anglebetween S Z and S Z in H k is bounded below by /M :(1.10) sup (cid:8) || φ || M k (cid:12)(cid:12) φ | Z = 1 , φ | Z = 0 (cid:9) = 1sin( S Z , S Z ) . Moreover,[1, Th. 9.19], this implies that Z is interpolating if and only if the sequence oflines through the kernels ( k z n ) n ∈ N is a Riesz system.Since (1.9) being a contraction is equivalent to the infinite matrix (1 − w n w j ) k z j ( z n ) n, j ∈ N ALBERTO DAYAN being positive semi-definite, one can extend the Pick property to the case of matrix-valued functions in H k , by defining H k to have the s × t Pick property if whenever z , . . . , z N arepoints in D and W , . . . , W N are s × t matrices such that ( Id − W ∗ n W j ) k z j ( z n ) ≥ then there exists a multiplier φ in the closed unit ball of M ( H k ⊗ C t , H k ⊗ C s ) := ( φ = ( φ l,r ) | l = 1 , . . . , s, r = 1 , . . . t, sup f =0 || φ f || H k ⊗ C s || f || H k ⊗ C t < ∞ ) such that φ ( z i ) = W i , for i = 1 , . . . , N . We say that H k has the complete Pick property if it has the s × t Pick property for any positive integers s and t . The Hardy space H has the complete Pick property, as well as some of its natural generalizations, such as thereproducing kernel Hilbert spaces H s on D , − ≤ s ≤ , defined by the kernels k sw ( z ) := ∞ X n =0 ( n + 1) s ( wz ) n z, w ∈ D and the Drury-Arveson space H d on the d -dimensional unit ball B d defined by the kernel b w ( z ) := 11 − h z, w i , z, w ∈ B d . For instance, see [1, Ch. 7]. In a recent work, [2], Aleman, Hartz, M c Carthy and Richterextended Theorem 1.2 by showing that any weakly separated sequence Z on a domain D such that the sequence of lines through the kernels ( k z n ) n ∈ N is a Bessel system is aninterpolating sequence for M k , provided that H k has the complete Pick property. This isdone by using the recent positive answer to the Feichtinger conjecture , which states thatany Bessel system of one-dimensional subspaces is the disjoint union of finitely many Rieszsystems. It has been shown, [5] [13], that the Feichtinger conjecture is equivalent to manyother conjectures in operator theory, including the Paving conjecture, who had been provedby the well-known work of Marcus, Spielmann and Srivastava [7].In [6], the author asked whether the positive answer to the Feichtinger conjecture can beextended to multi-dimensional model spaces of H , that is, if any Bessel system of modelspaces is the disjoint union of finitely many Riesz systems. We show in Section 3.2 thatthis is not the case, though any Bessel systems of model spaces satisfying (1.6) is in fact aRiesz system: Theorem 1.3.
Any weakly separated Bessel system of model spaces in H is a Riesz system. The motivation for Theorem 1.3 is the study of interpolating sequences of matrices introduced by the author in [6]. As Section 3 explains in details, we say that a sequenceof square matrices A = ( A n ) n ∈ N with eigenvalues in D is interpolating if, for any boundedsequence ( w n ) n ∈ N in C , there exists a bounded holomophic function φ such that φ ( A n ) = w n Id. If P n is, for any n in N , the minimal polynomial of A n , then(1.11) B n ( z ) := P n ( z ) P n (cid:0) z (cid:1) z ∈ D EAKLY SEPARATED BESSEL SYSTEMS OF MODEL SPACES 5 is a Blaschke product with zeros at the eigenvalues of A , and any H function that vanishesat A n is a multiple of B n . Let H = ( H n ) n ∈ N be the sequence of model spaces associatedto ( B n ) n ∈ N . The author extended in [6, Th. 6.6] Theorem 1.2 to sequences of matricesof uniformly bounded dimensions, by showing that A is interpolating if and only if H is a weakly separated Bessel system. Theorem 1.3 can be rephrased to drop the extraassumption on the sizes of the matrices in A : Theorem 1.4. A is interpolating if and only if the sequence H is a weakly separated Besselsystem. Section 2 deals with the proof of Theorem 1.3. Section 3 provides a brief summary of thecontent of [6] and gives an argument for Theorem 1.4, together with an explicit example ofan interpolating sequence of matrices. We also give in Section 3.2 an example of a sequenceof matrices whose associated sequence of model spaces ( H n ) n ∈ N is a Bessel system whichcan not be written as the disjoint union of finitely many Riesz systems, giving a negativeanswer to a question posed by the author in [6].The author would like to thank John M c Carthy for the valuable suggestions given duringall the conversations that led to this work.2.
The Proof of the Main Result
The first main tool for the proof of Theorem 1.3 can be found in [11, Lec. IX], andrelates the sine of the angle between two model spaces H Θ and H Θ with the constant inCarleson corona Theorem: Theorem 2.1.
There exists a constant c ≥ such that, for any Θ and Θ inner functionson D such that inf z ∈ D max {| Θ ( z ) | , | Θ ( z ) |} = δ ≥ then δ c ≤ sin( H Θ , H Θ ) ≤ cδ. We are also going to use the following re-statement of the Bessel system condition:
Proposition 2.2.
A sequence ( H n ) n ∈ N of closed sub-spaces of a Hilbert space H is a Besselsystem with Bessel bound M if and only if, for any sequence ( h n ) n ∈ N of unit vectors suchthat h n belongs to H n for any n in N , (2.1) sup || x || =1 X n ∈ N | h x, h n i | = M . Proof.
The idea of the proof comes from [1, Prop. 9.5]. Choose for any n in N a unit vector h n in H n , and suppose first that (2.1) holds. Then, for any ( a n ) n ∈ N in l , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ∈ N a n h n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup || x || =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h x, X n ∈ N a n h n i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup || x || =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ∈ N h x, h n i a n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M X n ∈ N | a n | , ALBERTO DAYAN thanks to (2.1) and Cauchy-Schwartz’s inequality.Conversely, let M be the Bessel bound for the sequence ( H n ) n ∈ N , and fix a unit vector x in H . Then set a n = h x, h n i , and observe that X n ∈ N | h x, h n i | = h x, X n ∈ N a n h n i≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ∈ N a n h n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M X n ∈ N | a n | ! = M X n ∈ N | h x, h n i | ! . This concludes the proof. (cid:3)
Remark 2.3.
Fixed x in H , we can choose ( h n ) n ∈ N so that the sum in (2.1) attains itsmaximum. We can actually maximizes each term of the sum, by setting h n to be theorthogonal projection onto H n of x , divided by its norm: f n ( x ) := P H n ( x ) || P H n ( x ) || . Proposition 2.2 then says that ( H n ) n ∈ N has a finite Bessel bound M if and only if(2.2) sup || x || =1 X n ∈ N (1 − dist ( x, H n )) = sup || x || =1 X n ∈ N | h x, f n ( x ) i | = M Lastly, we are going to use the one dimensional case of [6, Th. 5.1]. For any x in aHilbert space H let ˆ x denote its normalization x/ || x || . Theorem 2.4.
For any inner function Θ on D , dist H ( ˆ s z , H Θ ) = | Θ( z ) | z ∈ D . We are now ready to prove Theorem 1.3:
Proof of Theorem 1.3.
Let ( H Θ n ) n ∈ N be a weakly separated Bessel system. We will showthat (1.4) holds, and Theorem 1.1 will conclude the proof. Thanks to (2.2), for any fixed z in D there exists a positive integer n z that minimizes the distance between ˆ s z and H n : dist H ( ˆ s z , H n z ) = min n ∈ N dist H ( ˆ s z , H n ) , which thanks to Theorem 2.4 becomes | Θ n z ( z ) | = min n ∈ N | Θ n ( z ) | . Therefore, by Theorem 2.1 and weak separation we have that inf n = n z | Θ n ( z ) | > , which implies that Y n = n z | Θ n ( z ) | = sup n ∈ N Y j = n | Θ j ( z ) | EAKLY SEPARATED BESSEL SYSTEMS OF MODEL SPACES 7 is bounded below uniformly on z if and only if X n = n z (1 − | Θ n ( z ) | ) = X n = n z (1 − dist ( ˆ s z , H Θ n )) is uniformly bounded on z , which is true thanks to Remark 2.3. (cid:3) Observe that the proof of Theorem 1.3 used a weaker version of the Bessel systemcondition, in which the sup in (2.1) is taken only on normalized kernel functions, ratherthan on all unit vectors in H . It remains open for us whether such a weaker conditionis enough to characterize Bessel systems of model spaces, and a positive answer for thespecial case in which each Θ n is a Blaschke product would be of great interest for us, asthis is the case that we consider when we apply Theorem 1.3 to interpolating sequences ofmatrices, as we will see in Section 3: Question 2.5.
Is any sequence of model spaces ( H Θ n ) n ∈ N in H satisfying (2.3) sup z ∈ D X n ∈ N | h ˆ s z , f n ( ˆ s z ) i | < ∞ a Bessel system? Is it true if Θ n is, for any positive integer n , a Blaschke product? Remark 2.6.
Question 2.5 has a positive answer whenever Θ n = b λ n is, for any positiveinteger n , a degree-one Blaschke factor at a point λ n , and therefore whenever H Θ n is theline spanned by the Szegö kernel at λ n , [9, Ch. VI, Lemma 3.3]. Moreover, a positiveanswer to Question 2.5 would give a function theoretical characterization for the Besselsystem condition for ( H Θ n ) n ∈ N , as (2.3) is equivalent to sup z ∈ D X n ∈ N − | Θ n ( z ) | < ∞ Interpolating Matrices
The motivation for Theorem 1.3 is the study of interpolating sequences of matrices . Theauthor asked in [6] whether some well known characterizations for interpolating sequencesfor H ∞ such as Theorem 1.2 extend to sequences of square matrices A = ( A n ) n ∈ N , withoutassuming any restriction on the sequence of their dimensions. The fact that a squarematrix might have a non trivial algebraic structure invariant under holomorphic functions(its eigenspaces, for example), makes an interpolation problem using matrices a bit trickierthan the classic one: given two points λ and w in D there is no function φ in H ∞ thatmaps A = (cid:20) λ λ (cid:21) to W = (cid:20) w w (cid:21) , although both A and W are bounded in the operator norm and the constant function w isa contraction in H ∞ that sends the spectrum of A to the spectrum of W . Since choosingbounded targets in the operator norm makes even a one-point interpolation problem im-possible to solve via bounded analytic functions, in order to define interpolating matricesone has to identify a target with a bounded sequence in H ∞ , [6, Def. 1.1]: ALBERTO DAYAN
Definition 3.1 (Interpolating Matrices) . A sequence A = ( A n ) n ∈ N of square matriceswith spectra in the unit disc is interpolating for H ∞ if for any bounded sequence ( φ n ) n ∈ N in H ∞ there exists a φ in H ∞ such that φ ( A n ) = φ n ( A n ) , n ∈ N . Equivalently, [6, Th. 4.1], one can choose diagonal targets, and define A to be interpo-lating if for any bounded sequence ( w n ) n ∈ N in C there exists a bounded analytic function φ such that φ ( A n ) = w n Id, n ∈ N . Here an analytic function on the unit disc is applied to a square matrix via the
Riesz-Dunford functional calculus , hence the assumption on the spectra of the matrices in A .In order to characterize interpolating sequences of matrices, a rather trivial yet importantobservation is that, for any pair of similar matrices M and N with spectra in D and forany holomorphic function f on the unit disc then f ( M ) and f ( N ) are similar as well, andthe matrix that performs both similarities is the same, M = P − N P = ⇒ f ( M ) = P − f ( N ) P, as an elementary computation using the power series of f shows. As a consequence, wecan assume without loss of generality that each matrix of the sequence A is in its Jordancanonical form : if, for any positive integer n , λ n, , . . . , λ n,k n are the eigenvalues of A n ,then A n = diag( J n, , . . . , J n,k n ) , where J n,j is a Jordan block of size m n,j J n,j = λ n,j . . . λ n,j . . . ... ... ... ... ... . . . λ n,j
10 0 0 0 λ n,j . Since, for any function f holomorphic in D , f ( A n ) = diag( f ( J n, ) , . . . , f ( J n,k n )) , where f ( J n,j ) = f ( λ n,j ) f ′ ( λ n,j ) f ′′ ( λ n,j )2 . . . f ( mn,j − ( λ n,j )( m n,j − f ( λ n,j ) f ′ ( λ n,j ) . . . f ( mn,j − ( λ n,j )( m n,j − ... ... ... ... ... . . . f ( λ n,j ) f ′ ( λ n,j )0 0 0 0 f ( λ n,j ) , one realizes that a holomorphic function vanishes at A n if and only if it vanishes at itseigenvalues with the right multiplicity. Specifically, the multiplicity of λ n,j as a zero of f must be the maximal size of a Jordan block of A n associated to λ n,j . In particular, anyfunction in H that vanishes at A n is a multiple of the Blaschke product and the functionin (1.11) can be re-written as B n = k n Y j =1 b m n,j λ n,j . EAKLY SEPARATED BESSEL SYSTEMS OF MODEL SPACES 9
Hence, for any n in N , the subspace H n :=H ⊖ { f ∈ H | f ( A n ) = 0 } =span ( s λ n,j , ∂ s λ n,j ∂w , . . . , ∂ m n,j − s λ n,j ∂w m n,j − (cid:12)(cid:12)(cid:12)(cid:12) j = 1 , . . . , k n ) (3.1)containing all the interpolation information of the matrix A n is in fact a model space: H n = H B n n ∈ N . Each H n can be seen also as a kernel at the matrix A n . More precisely, let M be a m × m square matrix with eigenvalues in the unit disc, and let H M be the associated model spacein H . Let us define, for any u and v in C m , the H function K M ( u, v )( z ) := X n ∈ N h v, M n u i C m z n z ∈ D . Then, thanks to the definition of the inner product in H ,(3.2) h f, K M ( u, v ) i H = h f ( M ) u, v i C m f ∈ H . Equation 3.2 works as a reproducing property for the collection X M := { K M ( u, v ) | u, v ∈ C m } . In particular, since f ( A ) = 0 if and only if the right hand side of (3.2) vanishes for any u and v , we have that X M coincides with the model space H M . Moreover, (3.2) implies thatthe collection of function in X M is linear in v and conjugate-linear in u , and that for any φ in H ∞ (3.3) M ∗ φ ( K M ( u, v )) = K M ( u, φ ( M ) ∗ v ) u, v ∈ C m , extending (1.8) to this matrix setting. Another analogy with the scalar case comes fromseparation: thanks to (3.3) and since H has the Pick property, [1, Th.5.20], (1.10) extendsby saying that A is weakly separated if the sequence ( H n ) n ∈ N is weakly separated or,equivalently, if there exists a positive M such that, for any pair of distinct positive integers n and j , there exists a bounded analytic function φ n,j whose H ∞ norm doesn’t exceed M and that separates A n and A j , that is, φ n,j ( A n ) = Id φ n,j ( A j ) = 0 . Following the same idea, if ( H n ) n ∈ N is strongly separated we say that A is stronglyseparated , and the Pick property, together with (3.3), makes it equivalent to assertingthe existence of a bounded sequence ( φ n ) n ∈ N in H ∞ that separates each A n with the restof the sequence, i.e., φ n ( A j ) = δ n,j Id.
The scalar case has an even more geometric viewpoint on separation via bounded analyticfunctions: given two points z and w in the unit disc there exists a function φ whose H ∞ norm doesn’t exceed M that separates z and w (and hence the sine of the angle between s z and s w is bounded below by /M ) if and only if their pseudo-hyperbolic distance ρ ( z, w ) := | b w ( z ) | is bounded below by /M . This extends to the matrix case by looking at the action of theadjoint of the multiplication by a Blaschke product on different model spaces: Lemma 3.2.
Let A and A be two square matrices corresponding to the Blaschke products B and B , and let H and H be the associated model spaces. Then the sine of the anglebetween H and H is equal to δ > if and only if the restriction of M ∗ B to H is boundedbelow by δ , that is, inf x ∈ H || M ∗ B (ˆ x ) || = sin( H , H ) . Proof.
The sine of the angle between H and H is equal to δ if and only if the least H ∞ norm of a function φ such that ( M ∗ φ ) | H = 0 and ( M ∗ φ ) | H = Id H is /δ . Any such a φ vanishes on A , which is equivalent to asserting that there exists afunction g in H ∞ such that || g || ∞ = 1 /δ and(3.4) φ = B g. Since Id H = ( M ∗ φ ) | H = ( M ∗ B ) | H ( M ∗ g ) | H , equation (3.4) says that ( M ∗ B ) | H has an inverse bounded by /δ , as || M ∗ g || = || g || ∞ = 1 /δ .Conversely, if M ∗ B admits an inverse T bounded by /δ on H , the Pick property of theSzegö kernel says that T extends isometrically to some M ∗ g , and thanks to (3.3) B ( A ) g ( A ) = Id, as M ∗ B g acts like the identity on H . (cid:3) The author extended in [6] Carleson’s characterizations, [4] and [3], of interpolating se-quences to sequences of square matrices, together with the characterization of interpolatingsequences in terms of Riesz systems conditions from [12]. Nevertheless, the analogous ofTheorem 1.2 was proven with the additional assumption that the dimensions of the matri-ces in A are uniformly bounded, and used the solution of the Feichtinger conjecture: Theorem 3.3.
Let A = ( A n ) n ∈ N be a sequence of matrices with spectra in the unit disc,and let H = ( H n ) n ∈ N be the associated sequence of model spaces defined in (3.1) . Thefollowing are equivalent: (i): A is interpolating for H ∞ ; (ii): A is strongly separated; (iii): H is a Riesz system.Moreover, if the dimensions of the matrices in A are uniformly bounded, (i) (and hencealso all the conditions above) is equivalent to (iv): A is weakly separated and H is a Bessel system. Thanks to equivalence between conditions (i), (ii) and (iii) in Theorem 3.3, Theorem 1.3applied to the sequence of model spaces ( H n ) n ∈ N says, together with Theorem 1.1, thatthe extra assumption on the dimensions of the matrices in A in condition (iv) of Theorem3.3 can be dropped. Therefore that Theorem 1.2 extends to sequences of matrices of anysizes: Theorem 3.4. A is interpolating if and only if it is weakly separated and H is a Besselsystem. EAKLY SEPARATED BESSEL SYSTEMS OF MODEL SPACES 11
In particular, Theorem 1.4 follows from Theorem 1.3.We present below two examples of sequences of matrices having dyadic roots of unity astheir eigenvalues. Section 3.1 will give an example of an interpolating sequence of matrices,together with some useful tool to estimate the angle between model spaces arising fromBlaschke products. Section 3.2 will use a similar construction in order to exhibit a Besselsystem of model spaces which is not the disjoint union of finitely many weakly separatedsequences, and hence not the finite union of finitely many Riesz systems. This will give anegative answer to a question posed by the author in [6], where it was asked whether thepositive answer to the Feichtinger conjecture can be extended to multi-dimensional modelsub-spaces of the Hardy space.3.1.
Interpolating Matrices with Equidistributed Eigenvalues.
Let, for any posi-tive integer n , ω n := e πi n be a primitive n -root of unity, and let W n := diag(1 , ω n , . . . , ω n − n ) be a n × n diagonal matrix having n equi-distributed points on the unit circle as itseigenvalues. Let ( r n ) n ∈ N be a sequence in (0 , that re-scales the sequence ( W n ) n ∈ N sothat its spectra belong to D :(3.5) A n := r n W n , n ∈ N . We will discuss here how fast must ( r n ) n ∈ N go to in order for A := ( A n ) n ∈ N to beinterpolating. In particular, we will show that A is interpolating if and only if it is a zerosequence , that is, if and only if there exists bounded analytic function on D that vanisheson A and that doesn’t vanish outside the spectra of the matrices in A : Theorem 3.5.
The sequence of matrices defined in (3.5) is interpolating if and only if itis a zero sequence.
The proof of Theorem 3.5 requires that we are able to estimate (from below) the anglebetween two model spaces arising from Blaschke products. Such a tool is the content ofLemma 3.6 below. Let ( B n ) n ∈ N be a sequence of Blaschke products such that B = Q n ∈ N B n converges uniformly on any compact subset of D to a non zero inner function, and let ( H n ) n ∈ N be the associated sequence of model spaces in H . For any subset σ of N we willdefine, for the sake of brevity, H σ := span i ∈ σ { H i } and B σ = Y i ∈ σ B i . Lemma 3.6.
Let σ and τ be two disjoint subsets of N , and suppose that ( H i ) i ∈ σ is a Rieszsystem with Riesz bound γ . Then sin( H σ , H τ ) ≥ γ inf i ∈ σ sin( H i , H τ ) . Proof.
For any i in σ let δ i := sin( H i , H τ ) , and let δ := inf i ∈ σ δ i . It suffices to show that T : H σ ∪ τ → H σ ∪ τ such that T | H σ = δ Id | H σ T | H τ = 0 is bounded by γ . Let T i be, for any i in σ , the restriction to H i of M ∗ B τ . Thanks toLemma 3.2, each T i is bounded below by δ i . Fix then a vector x = u + v in H σ ∪ τ , where u is in H σ and v is in H τ . There exists a sequence ( h i ) i ∈ σ of unit vectors such that h i isin H i for any i in σ so that u can be written as a linear combination of ( h i ) i ∈ σ : u = X i ∈ σ α i h i . Since each T i is a contraction and it is bounded below by δ , the sequence ( T i ( h i )) i ∈ N isbounded above and below and therefore || T ( x ) || = || T ( u ) || = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ∈ σ δα i h i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ γ δ X i ∈ σ α i ≤ γ X i ∈ σ δ i α i ≤ γ X i ∈ σ α i || T i ( h i ) || ≤ γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ∈ σ α i T i ( h i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = γ || M ∗ B τ ( u ) || = γ || M ∗ B τ ( x ) || . Since M ∗ B τ is a contraction, this shows that the norm of T doesn’t exceed γ , as weclaimed. (cid:3) Remark 3.7.
Suppose that also ( H j ) j ∈ τ is a Riesz system. Then a double application ofLemma . implies that the distance between H σ and H τ is comparable with the minimaldistance attained by a model space labeled by an index in σ and one with a label in τ .If the sequence ( H n ) n ∈ N is weakly separated, this says roughly speaking that the set of sparse subsequences of ( H n ) n ∈ N is a separated set as well.We are now ready to prove Theorem 3.5. Here ( B n ) n ∈ N and ( H n ) n ∈ N arise from thesequence of matrices A defined in (3.5). Observe that if A is not a zero sequence theneach H n is contained in the closure of the linear span of all other model spaces, and hence ( H n ) n ∈ N is not strongly separated. What is left to show then is that A is strongly separated(and hence interpolating) provided that it is a zero sequence: Proof of Theorem 3.5.
Let, for any positive integer n ,(3.6) r n = 1 − α n − n . Since A is a zero sequence, then the spectra of the matrices in A form a zero sequence andtherefore X n ∈ N α n < ∞ . Let γ n be the Riesz bound of the basis { ˆ s r n , . . . , ˆ s r n ω n − n } of H n . Then ( γ n ) n ∈ N is uniformlybounded if and only if the strong separation constants n − Y l =1 | b r n ω ln ( r n ) | EAKLY SEPARATED BESSEL SYSTEMS OF MODEL SPACES 13 are uniformly bounded below. Let, for any j and n in N , M j ( n ) := j X l =1 | h ˆ s r j ω lj , ˆ s r n i | = j X l =1 (1 − r n )(1 − r j ) | − r n r j ω lj | (3.7)Since | b w ( z ) | = 1 − | h ˆ s w , ˆ s z i | w, z ∈ D , then ( γ n ) n ∈ N is bounded if and only if ( M n ( n )) n ∈ N is, which is the case, thanks to Lemma3.8 below, since ( α n ) n ∈ N is bounded. Therefore by Lemma 3.6 in order to show that A isstrongly separated it suffices to show that inf n ∈ N Y j = n | B j ( r n ) | > , which is true if and only if each term on the product is uniformly bounded below and sup n ∈ N X j ∈ N M j ( n ) < ∞ . Thanks to Lemma 3.8, this is true since ( α n ) n ∈ N is summable and the sequence ( r n ) n ∈ N isweakly separated, together with the fact that thanks to Remark 3.7 the sine of the anglebetween H n and H j is, for any n and j , comparable with the pseudo-hyperbolic distancebetween r n and r j . (cid:3) A technical tool for the proof of Theorem 3.5 is the following computation, which relatesthe quantity M j ( n ) to the parameters ( α n ) n ∈ N defined in (3.6): Lemma 3.8.
Let r n := 1 − α n − n n ∈ N be a sequence in (0 , and let M j ( n ) be defined as in (3.7) . Then, for any j and n positiveintegers, M j ( n ) ≃ α n α j α n + α j n − j − α n α j − j . Proof.
Let, for any j in N and for any l = 1 , . . . , j , θ jl := arg ω lj = 2 πl j . Then − cos( θ jl ) ≃ ( θ jl ) ≃ l j +1 and therefore M j ( n ) ≃ (1 − r j )(1 − r n ) j X l =1 | − r n r j ω lj | = (1 − r j )(1 − r n ) j X l =1 − r n r j ) + 2((1 − cos( θ jl ))) r n r j ≃ (1 − r j )(1 − r n )(1 − r j r n ) j X l =1
11 + (cid:16) √ r n r j j (1 − r n r j ) l (cid:17) ≃ (1 − r j )(1 − r n )(1 − r j r n ) Z j
11 + (cid:16) √ r n r j j (1 − r n r j ) x (cid:17) dx = 2 j (1 − r j )(1 − r n )(1 − r j r n ) √ r j r n Z √ rjrn (1 − rjrn ) √ rjrn j (1 − rjrn )
11 + x dx ≃ j (1 − r j )(1 − r n )1 − r j r n = α n α j α n + α j n − j − α n α j − j . (cid:3) Bessel Systems of Model Spaces.
Thanks to the positive answer to the Feichtingerconjecture, any Bessel system of lines in a Hilbert space is the disjoint union of finitelymany Riesz systems. We show here that this is not the case for multi-dimensional modelspaces in H , as we will construct a sequence of matrices A which can not be written as thedisjoint union of finitely many weakly separated sequences and whose associated sequenceof model spaces is a Bessel system. This implies that [1, Th. 9.11] doesn’t extend to multi-dimensional model spaces, and since any Riesz system is weakly separated it will showthat the positive answer to the Feichtinger conjecture doesn’t extend to multi-dimensionalmodel spaces whose dimensions are not uniformly bounded.The starting point of our construction is the sequence A defined in (3.5). Let, for any n in N , t n be the positive number greater than r n such that(3.8) | b r n ( t n ) | = 12 n n ∈ N , and let Λ = ( λ n,l ) be the sequence in D consisting of the union of all n equi-distributedpoints at distance − t n from the unit circle: λ n,l := t n ω ln , n ∈ N , l = 1 , . . . , n . The reader can think of Λ as a sequence whose elements tends to approach each eigenvalueof the matrices in A . In particular, thanks to (3.8), the sequence of matrices B := A ∪ Λ can not be written as the finite union of weakly separated sequences, as for any positive ε and for any n in N each of the n eigenvalues of A n has a point of Λ at pseudo-hyperbolicdistance less than ε . Therefore what is left to show is that, if ( r n ) n ∈ N is chosen to beconverging to adequately fast, the sequence of model spaces associated with the sequence EAKLY SEPARATED BESSEL SYSTEMS OF MODEL SPACES 15 B is a Bessel system. Let A ′ = ( r n ω ln ) be the (scalar) sequence of all the eigenvalues ofthe matrices in A , and let B ′ := A ′ ∪ Λ = ( z n ) n ∈ N . We can then recursively choose the sequence ( r n ) n ∈ N to approach fast enough so that sup n ∈ N X n ∈ N − | b z n ( z ) | < ∞ , and therefore thanks to Remark 2.6 the sequence of lines spanned by the Szegö kernels atthe point of B ′ is a Bessel system. This, together with an extra separation condition onthe eigenvalues of the matrices in A , implies that the model spaces associated with thesequence B forms a Bessel system: Lemma 3.9.
Let ( H n ) n ∈ N be a sequence of closed sub-spaces of a Hilbert space H , and let,for any n in N , { x n , . . . , x m n n } be a basis of H n of unit vectors such that (3.9) m n X l =1 | c l | ≤ C n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m n X l =1 c l x ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , c , . . . , c m n ∈ C . If (3.10) C := sup n ∈ N C n < ∞ and ( x ln ) is a Bessel system with bound M , then ( H n ) n ∈ N is a Bessel system with bound CM .Proof. Let ( h n ) n ∈ N be a sequence of unit vectors in H such that h n belongs to H n for any n in N , and let ( a n ) n ∈ N be an l sequence. Write h n = P m n l =1 b ln x ln , and observe that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ∈ N a n h n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ∈ N m n X l =1 a n b ln x ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M X n ∈ N | a n | m n X l =1 | b ln | ≤ C M X n ∈ N | a n | , thanks to (3.9) and (3.10). (cid:3) As we showed during the proof of Theorem 3.5, by eventually increasing the rate ofconvergence of ( r n ) to we can assume that the Reisz bound of the basis { ˆ s r n , . . . , ˆ s r n ω n − n } is uniformly bounded in n , thus in particular condition (3.9) holds. Therefore thanks toLemma 3.9 the sequence of model spaces associated to the sequence B is a Bessel system. References [1]
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