De Branges spaces and Fock spaces

Abstract

Relations between two classes of Hilbert spaces of entire functions, de Branges spaces and Fock-type spaces with nonradial weights, are studied. It is shown that any de Branges space can be realized as a Fock-type space with equivalent area norm, and several constructions of a representing weight are suggested. For some special classes of weights (e.g. weights depending on the imaginary part only) the corresponding de Branges spaces are explicitly described.     

Decomposition ofq-deformed Fock spaces

A decomposition of the level-oneq-deformed Fock space representations ofU _q(sl _n ) is given. It is found that the action ofU _q(sl _n ) on these Fock spaces is centralized by a Heisenberg algebra, which arises from the center of the affine Hecke algebra _N in the limitN . Theq-deformed Fock space is shown to be isomorphic as aU _q(sl _n )-Heisenberg-bimodule to the tensor product of a level-one irreducible highest weight representation ofU _q(sl _n ) and the Fock representation of the Heisenberg algebra. The isomorphism is used to decompose theq-wedging operators, which are intertwiners between theq-deformed Fock spaces, into constituents coming fromU _q(sl _n ) and from the Heisenberg algebra.     

Frames in generalized Fock spaces

Let A denote a real linear transformation on Cn which is symmetric and positive-definite relative to the real inner product Re〈z,w〉, z,w∈Cn. Let FA(Cn) denote the Fock space consisting of holomorphic functions on Cn which are square integrable with respect to the Gaussian measure . For w∈Cn, let , z∈Cn, where KA is the reproducing kernel for FA(Cn). The main aim of this paper is to show that there exist a,b>0 such that the set of functions forms a frame in FA.     

Factorization and reflexivity on Fock spaces

The framework of the paper is that of the full Fock space and the Banach algebraF which can be viewed as non-commutative analogues of the Hardy spacesH ² andH respectively.An inner-outer factorization for any element in as well as characterization of invertible elements inF are obtained. We also give a complete characterization of invariant subspaces for the left creation operatorsS ₁ ,..., S _n of . This enables us to show that every weakly (strongly) closed unital subalgebra of {(S ₁ ,..., S _n ) F } is reflexive, extending in this way the classical result of Sarason [S]. Some properties of inner and outer functions and many examples are also considered.The first author was supported in part by NSF DMS 93-21369 1991Mathematics Subject Classification. Primary 47D25, Secondary 32A35, 47A67.     

Taylor coefficients of functions in Fock spaces

Analogues of the Hausdorff-Young and Hardy-Littlewood theorems are proved for functions in Fock spaces. Estimates of Taylor coefficients are shown to be sharp in the sense of Duren-Taylor.     

Bergman kernel asymptotics for generalized Fock spaces

Letm(t)dt be a positive measure onR ⁺. We investigate the relations among the growth ofm, the growth of its moment sequence {y_n}, the growth of its Bergman kernel functionk x=Σγ{_n/^-1}x ⁿ, and the growth of the kernel function associated to the measureK(t) ⁻¹m(t) dt. For a large class of measures, we find that these quantities satisfy asymptotic relations similar to the simple exact relations which hold in the model casem(t)=e ⁻¹. Supported in part by a grant from the National Science Foundation.     

Perfect codes as isomorphic spaces

Linear equivalence between perfect codes is defined. This definition gives the concept of general perfect 1-error correcting binary codes. These are defined as 1-error correcting perfect binary codes, with the difference that the set of errors is not the set of weight one words, instead any set with cardinality n and full rank is allowed. The side class structure defines the restrictions on the subspace of any general 1-error correcting perfect binary code. Every linear equivalence class will contain all codes with the same length, rank and dimension of kernel and all codes in the linear equivalence class will have isomorphic side class structures.     

Schatten class Toeplitz operators on generalized Fock spaces

In this paper we characterize the Schatten p   class membership of Toeplitz operators with positive measure symbols acting on generalized Fock spaces for the full range 0<p<∞$0<p<\infty$.     

Zero sets and interpolating sets in Fock spaces

An example is constructed to show that interpolating sets for Fock spaces are not necessarily zero sets.

     

Schatten class weighted composition operators on weighted Fock spaces

We describe the Schatten class weighted composition operators on Fock–Sobolev spaces and a large class of weighted Fock spaces, where the weights defining such spaces are radial, decay at least as fast as the classical Gaussian weight, and satisfy certain mild smoothness condition. To prove our main results, we characterize the Schatten class membership of the Toeplitz operators T _μ induced by nonnegative measures μ on the complex space ${\mathbb{C}^n}$ .     

Sampling and interpolation in large Bergman and Fock spaces

We obtain sampling and interpolation theorems in weighted spaces of analytic functions for radial weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted Bergman spaces.     

Duality of Fock spaces with respect to the minimal norm

In this note we prove a duality theorem for generalized Fock spaces.     

Toeplitz operators with BMO and IMO symbols between Fock spaces

In this paper, given $$f \in BMO$$, for all possible $$0< p< q<\infty $$, we characterize the boundedness (or compactness) of the Toeplitz operators $$T_{f}$$ from the Fock space $$F^{p}$$ to $$F^{q}$$. With $$f\in IMO$$ (the space of integrable mean oscillation functions), for all possible $$0< q< p<\infty $$, we characterize those symbols f for which the Toeplitz operators $$T_{f}$$ are bounded (or compact) from $$F^{p}$$ to $$F^{q}$$.