Topological Hilbert Nullstellensatz for Bergman spaces

We prove Bergman space analogues of a conjecture of Douglas and Paulsen related to the classification of invariant subspaces for multiplication operators in several variables.     

Weighted Bergman Spaces: Shift-Invariant Subspaces and Input/State/Output Linear Systems

It is well known that subspaces of the Hardy space over the unit disk which are invariant under the backward shift occur as the image of an observability operator associated with a discrete-time linear system with stable state-dynamics, as well as the functional-model space for a Hilbert space contraction operator, while forward shift-invariant subspaces have a representation in terms of an inner function. We discuss several variants of these statements in the context of weighted Bergman spaces on the unit disk.     

多圆盘的加权Bergman空间上的不变子空间和约化子空间

本文主要研究多圆盘的加权Bergman 空间上的不变子空间和约化子空间, 给出了某些解析Toeplitz 算子的极小约化子空间的完全刻画, 以及一类解析Toeplitz 算子Tzi (1≤i≤n) 的不变子空间的Beurling 型定理.     

A cross section from invariant subspaces to inner functions

Beurling's well known theorem connects the study of invariant subspaces to that of inner functions over the unit disc. In this paper, we will further explore this connection and, as a corollary of the result, show a one to one correspondence between the components of the invariant subspace lattice and the components of the space of inner functions.     

Norm Approximation by Polynomials in Some Weighted Bergman Spaces

The polynomials are shown to be dense in weighted Bergman spaces in the unit disk whose weights are superbiharmonic and vanish in an average sense at the boundary. This leads to an alternative proof of the Aleman-Richter-Sundberg Beurling-type theorem for zero-based invariant subspaces in the classical Bergman space. Additional consequences are deduced.     

Large Bergman spaces: invertibility, cyclicity, and subspaces of arbitrary index

In a wide class of weighted Bergman spaces, we construct invertible non-cyclic elements. These are then used to produce z-invariant subspaces of index higher than one. In addition, these elements generate non-trivial bilaterally invariant subspaces in anti-symmetrically weighted Hilbert spaces of sequences.     

A Note on Maximal Inner Spaces of the Bergman Space

For an invariant subspace I of the Bergman space on the unit disk D, the associated inner space I zI has been known to have nice properties K. Zhu has recently given, in terms of kernels of Hankel operators, several characterizations for an inner space to be maximal. We show that maximality of inner spaces can be understood alternatively by use of the adjoint operator of the Bergman shift operator on      

Some remarks on analytic continuation in weighted backward shift invariant subspaces

By a famous result of Douglas, Shapiro, and Shields, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation outside the closed unit disk. More can be said when the spectrum of the associated inner function has holes on \mathbb T{{\mathbb T}}. Then the functions of the invariant subspaces even extend analytically through these holes. Here we will be interested in weighted backward shift invariant subspaces which appear naturally in the context of kernels of Toeplitz operators. Note that such kernels are special cases of so-called nearly invariant subspaces. In our setting a result by Aleksandrov allows to deduce analytic continuation properties which we will then apply to consider embeddings of weighted invariant subspaces into their unweighted companions. We hope that this connection might shed some new light on known results. We will also establish a link between the spectrum of the inner function and the approximate point spectrum of the backward shift in the weighted situation in the spirit of results by Aleman, Richter, and Ross.     

On Beurling-type theorems in weighted and Bergman spaces

We prove that analytic operators satisfying certain series of operator inequalities possess the wandering subspace property. As a corollary, we obtain Beurling-type theorems for invariant subspaces in certain weighted and Bergman spaces.

     

Nevanlinna–Pick Interpolation and Factorization of Linear Functionals

If \mathfrakA{\mathfrak{A}} is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \mathbbA₁(1){\mathbb{A}_1(1)}, then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \mathbbA₁(1){\mathbb{A}_1(1)}, and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-* closed subalgebras of H ^∞ acting on Hardy space or on Bergman space.     

A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

An analogue of the Paley–Wiener theorem is developed forweighted Bergman spaces of analytic functions in the upper half-plane.The result is applied to show that the invariant subspaces ofthe shift operator on the standard Bergman space of the unitdisk can be identified with those of a convolution Volterraoperator on the space L²(⁺, (1/t)dt).     

Hermitian Weighted Composition Operators and Bergman Extremal Functions

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are ${(1 - \overline{w}z)^{-\kappa}}$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.     

Approximate Factorization in Generalized Hardy Spaces

In this paper we establish a connection between the approximate factorization property appearing in the theory of dual algebras and the spectral inclusion property for a class of Toeplitz operators on Hilbert spaces of vector valued square integrable functions. As an application, it follows that a wide range of dual algebras of subnormal Toeplitz operators on various Hardy spaces associated to function algebras have property (A ₁(1)). It is also proved that the dual algebra generated by a spherical isometry (with a possibly infinite number of components) has the same property. One particular application is given to the existence of unimodular functions sitting in cyclic invariant subspaces of weak* Dirichlet algebras. Moreover, by this method we provide a unified approach to several Toeplitz spectral inclusion theorems. Research partially supported by grant CNCSIS GR202/2006 (cod 813).     

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.     

Operators on harmonic Bergman spaces

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theL ^p boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

Invariant subspaces of finite codimension in Banach spaces of analytic functions

We give a characterization of invariant subspaces of finite codimension in Banach spaces of vector-valued analytic functions in several variables, where invariant refers to invariance under multiplication by any polynomial. We obtain very weak conditions under which our characterization applies, that unifies and improves a number of previous results. In the vector-valued case, the results are new even for one complex variable. As a concrete application in several variables, we consider the Bergman space on a strictly pseudo-convex domain, and we improve previous results (assuming C^∞-boundary) to the case of C²-boundary.     

Invariant Subspaces of Positive Quasinilpotent Operators on Ordered Banach Spaces

In this paper we find invariant subspaces of certain positive quasinilpotent operators on Krein spaces and, more generally, on ordered Banach spaces with closed generating cones. In the later case, we use the method of minimal vectors. We present applications to Sobolev spaces, spaces of differentiable functions, and C*-algebras.      

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.     

A sharp estimate for extremal functions

We prove a sharp pointwise estimate for extremal functions of invariant subspaces of some weighted Bergman spaces on the unit disk. The allowed weights include standard radial weights and logarithmically subharmonic weights.

     

Single-point extremal functions in weighted bergman spaces

For weighted Berman spaces in the unit disk the extremal functions for invariant subspaces formed by functions vanishing at a fixed point are studied. In the case where the weight is radial and logarithmically subharmonic, it is shown that such extremal functions can serve for the separation of single zeros. It is also proved that the reproducing kernels of the Bergman spaces are univalent functions. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15, 1995, pp. 241–252.