De Branges Spaces of Entire Functions Symmetric About the Origin

We define and investigate the class of symmetric and the class of semibounded de Branges spaces of entire functions. A construction is made which assigns to each symmetric de Branges space a semibounded one. By employing operator theoretic tools it is shown that every semibounded de Branges space can be obtained in this way, and which symmetric spaces give rise to the same semibounded space. Those subclasses of Hermite-Biehler functions are determined which correspond to symmetric or semibounded, respectively, nondegenerated de Branges spaces. The above assignment is determined in terms of the respective generating Hermite-Biehler functions.     

On Lagrange-Type Interpolation Series and Analytic Kramer Kernels

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. A challenging problem is to characterize the situations when these sampling formulas can be written as Lagrange-type interpolation series. This article gives a necessary and sufficient condition to ensure that when the sampling formula is associated with an analytic Kramer kernel, then it can be expressed as a quasi Lagrange-type interpolation series; this latter form is a minor but significant modification of a Lagrange-type interpolation series. Finally, a link with the theory of de Branges spaces is established. Received: October 8, 2007. Revised: December 13, 2007.     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

Compact Operators on Bergman Spaces

We prove that a bounded operator S on L _a ^p for p > 1 is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disk if S satisfies some integrable conditions. Some estimates about the norm and essential norm of Toeplitz operators with symbols in BT are obtained.     

Various Slicing Indices on Banach Spaces

We give a short and direct proof for the computation of the Szlenk index of the C(K) spaces, when K is a countable compact space and determine their Lavrientiev indices. We also compute the Szlenk index of certain C(α) spaces, where α is an uncountable ordinal. Finally, we show that if the Szlenk index of a Banach space is ω (first infinite ordinal), then its weak*-dentability index is at most ω² and that this estimate is optimal. The first author was supported by the grants: Institutional Research Plan AV0Z10190503, A100190502, GA ČR 201/04/0090.     

For which reproducing kernel Hilbert spaces is Pick's theorem true?

Pick's theorem tells us that there exists a function inH , which is bounded by 1 and takes given values at given points, if and only if a certain matrix is positive.H is the space of multipliers ofH ², and this theorem has a natural generalisation whenH is replaced by the space of multipliers of a general reproducing kernel Hilbert spaceH(K) (whereK is the reproducing kernel). J. Agler has shown that this generalised theorem is true whenH(K) is a certain Sobolev space or the Dirichlet space, so it is natural to ask for which reproducing kernel Hilbert spaces this generalised theorem is true. This paper widens Agler's approach to cover reproducing kernel Hilbert spaces in general, replacing Agler's use of the deep theory of co-analytic models by a relatively elementary, and more general, matrix argument. The resulting theorem gives sufficient (and usable) conditions on the kernelK, for the generalised Pick's theorem to be true forH(K), and these are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces and for a functional Hilbert space introduced by Saitoh.     

Operator ranges and non-cyclic vectors for the backward shift

In this paper we look at operators on the Hardy spaceH ²(D) with range containing all of the non-cyclic vectors of the backward shift. We show several classes of such operators must be surjective, including Toeplitz, Hankel and composition operators.     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

Quasi-Free Resolutions Of Hilbert Modules

The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra $\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and $\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.     

Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence

Let G be a connected linear semisimple Lie group with Lie algebra , and let be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in and is the nilpotent -orbit in associated to Ω by the Kostant-Sekiguchi correspondence. We show that the corank of the Hamiltonian K-space Ω is twice the complexity of the variety .     

Schrödinger operators and de Branges spaces

We present an approach to de Branges's theory of Hilbert spaces of entire functions that emphasizes the connections to the spectral theory of differential operators. The theory is used to discuss the spectral representation of one-dimensional Schrödinger operators and to solve the inverse spectral problem.     

Overcompleteness of Sequences of Reproducing Kernels in Model Spaces

We give necessary conditions and sufficient conditions for sequences of reproducing kernels (k_Θ(·, λ_n))_{n ≥ 1} to be overcomplete in a given model space K_Θ^p where Θ is an inner function in H^∞, p ∈ (1, ∞), and where (λ_n)_{n ≥ 1} is an infinite sequence of pairwise distinct points of Under certain conditions on Θ we obtain an exact characterization of overcompleteness. As a consequence we are able to describe the overcomplete exponential systems in L² (0, a).     

De Branges spaces of entire functions closed under forming difference quotients

With a de Branges spaceH(E) of entire functions a functionq, analytic in ⁺ and satisfying there Imq(z)0, is associated. In this note we give necessary and sufficient conditions forH(E) to be closed under forming certain difference quotients in terms of the poles and zeros ofq. Moreover, we obtain a criterion whether a functionq possessing the above mentioned properties can be written as the quotient of the right upper and right lower entry of an entire matrix functionW (z) satisfying a certain kernel condition.     

Subspaces of L p , p > 2, with unconditional bases have equivalent partition and weight norms

In this note we give a simple proof that every subspace of L_p, 2 < p < ∞, with an unconditional basis has an equivalent norm determined by partitions and weights. Consequently L_p has a norm determined by partitions and weights. Received: 31 January 2005     

Nearly Invariant Subspaces Related to Multiplication Operators in Hilbert Spaces of Analytic Functions

We consider Hilbert spaces of analytic functions defined on an open subset of , stable under the operator M_u of multiplication by some function u. Given a subspace of which is nearly invariant under division by u, we provide a factorization linking each element of to elements of on the inverse image under u of a certain complex disc, for which we give a relatively simple formula. By applying these results to and u(z) = z, we obtain interesting results involving a H²-norm control. In particular, we deduce a factorization for the kernel of Toeplitz operators on Dirichlet spaces. Finally, we give a localization for the problem of extraneous zeros.Submitted: January 18, 2003 Revised: December 20, 2003     

On Two Variable Jordan Block (II)

On the Hardy space over the bidisk H²(D²), the Toeplitz operators and are unilateral shifts of infinite multiplicity. A closed subspace M is called a submodule if it is invariant for both and . The two variable Jordan block (S₁, S₂) is the compression of the pair to the quotient H²(D²) ⊖M. This paper defines and studies its defect operators. A number of examples are given, and the Hilbert-Schmidtness is proved with good generality. Applications include an extension of a Douglas-Foias uniqueness theorem to general domains, and a study of the essential Taylor spectrum of the pair (S₁, S₂). The paper also estabishes a clean numerical estimate for the commutator [S₁*, S₂] by some spectral data of S₁ or S₂. The newly-discovered core operator plays a key role in this study.     

Positive Functionals on some Spaces of Fractions

The aim of this work is to study norm preserving extensions of positive functionals on some spaces of fractions. The main result is stated in the case of unitary complex Banach algebras with involution. Moreover, we deal with C*-algebras and the commutative case as well. An application is also given.