C∗-algebras of multiplication operators on Bergman spaces

2011

In this paper, we develop a machinery to study multiplication operators on the Bergman space via the Hardy space of the bidisk. We show that only a multiplication operator by a finite Blaschke product has a unique reducing subspace on which its restriction is unitarily equivalent to the Bergman shift. Using the machinery we study the structure of reducing subspaces unitary equivalence of a multiplication operator on the Bergman space. As a consequence, we completely classify reducing subspaces of the multiplication operator by a Blaschke product φ with order three on the Bergman space to solve a conjecture of Zhu and obtain that the number of minimal reducing subspaces of the multiplication operator equals the number of connected components of the Riemann surface of φ −1 • φ over D.

Publicationes Mathematicae Debrecen

In this study, we present the necessary and sufficient conditions for the algebra generated by unilateral weighted shifts to be isometrically isomorphic to the polydisc algebra.

Pacific Journal of Mathematics, 1991

By using a natural localization method, one describes the finite codimensional invariant subspaces of the Bergman //-tuple of operators associated to some bounded pseudoconvex domains in C" , with a sufficiently nice boundary. with the structure and classification of the invariant subspaces of the Bergman n-tuple of operators, cf. Agrawal-Salinas [2], Axler-Bourdon [4], Bercovici [5], Douglas [7], Douglas-Paulsen . Due to the richness of this lattice of invariant subspaces, the additional assumption on finite codimension was naturally adopted by the above mentioned authors as a first step towards a better understanding of its properties. The present note arose from the observation that, when the L 2bounded evaluation points of a pseudoconvex domain lie in the Fredholm resolvent set of the associated Bergman rc-tuple, then the description of finite codimensional invariant subspaces is, at least conceptually, a fairly simple algebraic matter. This simplification requires only the basic properties of the sheaf model for systems of commuting operators introduced in [11]. The main result below is also available by some other recent methods. First is the quite similar technique of localizing Hubert modules over function algebras, due to Douglas and Douglas and Paulsen , and secondly is the study of the so-called canonical subspaces of some Hubert spaces with reproducing kernels, developed by Agrawal and Salinas . Both points of view will be discussed in §2 of this note. In fact the Bergman space of a pseudoconvex domain is only an example within a class of abstract Banach ^(C^-modules, whose finite codimensional submodules turn out to have a similar structure. The precise formulation of this remark ends the note. We would like to thank the referee, whose observations pointed out some bibliographical omissions in a first version of the manuscript. Let Ω be a bounded pseudoconvex domain in C n , n > 1, and let L%(Q) denote the corresponding Bergman

TURKISH JOURNAL OF MATHEMATICS

In this paper, we introduce a class of unitary operators defined on the Bergman space L 2 a (C+) of the right half plane C+ and study certain algebraic properties of these operators. Using these results, we then show that a bounded linear operator S from L 2 a (C+) into itself commutes with all the weighted composition operators Wa, a ∈ D if and only if S(w) = ⟨Sbw, bw⟩, w ∈ C+ satisfies a certain averaging condition. Here for a = c + id ∈ D, f ∈ L 2 a (C+), Waf = (f • ta) M ′ M ′ •ta , M s = 1−s 1+s , ta(s) = −ids+(1−c) (1+c)s+id , and bw(s) = 1 √ π 1+w 1+w 2Rew (s+w) 2 , w = M a, s ∈ C+. Some applications of these results are also discussed.

Theta, 2004, 2004

Journal of Operator Theory, 2016

Following upon results of Putinar, Sun, Wang, Zheng and the first author, we provide models for the restrictions of the multiplication by a finite Balschke product on the Bergman space in the unit disc to its reducing subspaces. The models involve a generalization of the notion of bundle shift on the Hardy space introduced by Abrahamse and the first author to the Bergman space. We develop generalized bundle shifts on more general domains. While the characterization of the bundle shift is rather explicit, we have not been able to obtain all the earlier results appeared, in particular, the facts that the number of the minimal reducing subspaces equals the number of connected components of the Riemann surface B(z) = B(w) and the algebra of commutant of T B is commutative, are not proved. Moreover, the role of the Riemann surface is not made clear also.

Proceedings of The Edinburgh Mathematical Society, 2002

We solve a joint similarity problem for pairs of operators of Foias-Williams/Peller type on weighted Bergman spaces. We show that for the single operator, the Hardy space theory established by Bourgain/Aleksandrov-Peller carries over to weighted Bergman spaces, by establishing the relevant weak factorizations. We then use this fact, together with a recent dilation result due to the first author and R. Rochberg to show that a commuting pair of such operators is jointly polynomially bounded if and only if it is jointly completely polynomially bounded. In this case, the pair is jointly similar to a pair of contractions by Paulsen's similarity theorem.

Complex Analysis and Operator Theory, 2023

In this paper, we study linear dynamical properties of shift operators on some classes of Segal algebras. Moreover, we characterize hypercyclic generalized bilateral shift operators on the standard Hilbert module.

Integral Equations and Operator Theory, 2013

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.

A Thesis Presented for the Doctor of Philosophy Degree in Pure Mathematics The University of Tennessee, Knoxville

Rocky Mountain Journal of Mathematics, 2014

Indiana University Mathematics Journal, 2002

A theorem of Aleman, Richter, and Sundberg asserts that every z-invariant subspace M of the Bergman space A 2 is generated by M zM, the orthocomplement of zM within M. The purpose of this paper is investigate the extent to which that property generalizes to g-invariant subspaces for a function g ∈ H ∞. Such a function g is said to have the wandering property in A 2 if every g-invariant subspace M of A 2 is generated by M gM. In the Hardy space H 2 , every inner function has the wandering property, while every function with this property must be the composition of an inner function with a conformal mapping. In this paper it is shown that the only functions that can have the wandering property in A 2 are essentially the classical inner functions. On the other hand, a large class of inner functions for which this property fails is exhibited. The wandering property is equivalent to the cyclicity of certain reproducing kernels; thus the proofs involve the approximation of noncyclic kernels by kernels corresponding to pullback measures.

Integral Equations and Operator Theory, 2013

2015

Let φ(z) = z m , z ∈ U, for some positive integer m, and Cφ be the composition operator on the Bergman space A 2 induced by φ. In this article, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators C * φ Cφ, CφC * φ as well as self-commutator and anti-self-commutators of Cφ. We also find the eigenfunctions of these operators.

Integral Equations and Operator Theory, 1993

has defined the (/4 +/(:)-orbit of an operator T acting on an nilbert space as (U+/C)(T) = {R-1TR : R invertible of the form unitary plus compact}. In this paper we consider the (U +/C)-orbit and the closure thereof for bilateral and unilateral weighted shifts. In particular, we determine which shifts are in the (//+/C)-orbits of injective weighted shifts and which shifts are in the closure of the (/d +/(:)-orbit of periodic injective shifts. Also, the closure of the (U +/(:)-orbit of injective essentially normal shifts is determined.