The Density Problem for Self–Commutators of Unbounded Bergman Operators

We give two sufficient conditions for the self–commutator of an unbounded Bergman operator to be densely defined. In conjunction with known results this leads to a strong Berger–Shaw type theorem for unbounded Bergman operators. Finally we present results regarding the density problem for unbounded sets in the plane of infinite area.     

Bergman Space Structure,Commutative Algebras of Toeplitz Operators,and Hyperbolic Geometry

We exhibit a surprising but natural connection among the Bergman space structure, commutative algebras of Toeplitz operators and pencils of hyperbolic straight lines. The commutative C*-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines the set of symbols consisting of functions which are constant on corresponding cycles, the orthogonal trajectories to lines forming a pencil. It turns out that the C*-algebra generated by Toeplitz operators with this class of symbols is commutative. Submitted: January 15, 2001?Revised: February 7, 2002     

Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains

In this paper we study the problem of the membership of H _ϕ in the Hilbert-Schmidt class, when and Ω is a planar domain. We find a necessary and sufficient condition.We apply this result to the problem of joint membership of H _φ and in the Hilbert-Schmidt class. Using the notion of Berezin Transform and a result of K. Zhu we are able to give a necessary and sufficient condition. Finally, we recover a result of Arazy, Fisher and Peetre on the case with φ holomorphic.     

Schatten-von Neumann Hankel Operators on the Bergman Space of Planar Domains

In this paper we study the problem of the joint membership of Hφ and in the Schatten-von Neumann p-class when φ ∈ L∞(Ω) and Ω is a planar domain. We use a result of K. Zhu and the localization near the boundary to solve the problem. Finally, we recover a result of Arazy, Fisher and Peetre on the case with φ holomorphic.      

Algebraic Properties of Toeplitz Operators with Radial Symbols on the Bergman Space of the Unit Ball

In this paper, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball in . We first determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. Next, we investigate the zero-product problem for several Toeplitz operators with radial symbols. Also, the corresponding commuting problem of Toeplitz operators whose symbols are of the form is studied, where and φ is a radial function. Ze-Hua Zhou: supported in part by the National Natural Science Foundation of China (Grand Nos.10671141, 10371091).     

Unbounded Toeplitz Operators

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H ², with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H ² are also studied. In memory of Paul R. Halmos     

Schatten Class Hankel Operators on the Harmonic Bergman Space of the Unit Ball

We give a necessary and sufficient condition for Hankel operators H_f on the harmonic Bergman space of the unit ball to be in the Schatten p-class for 2 ≤ p < ∞. A special case when symbol f is a harmonic function is also considered.     

On Some Product of Two Unbounded Self-Adjoint Operators

We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]). The author was supported in part by CNEPRU: B01820070020 (Ministry of Higher Education, Algeria).     

Hankel Operators on Bergman Spaces of Tube Domains over Symmetric Cones

We present here some criteria for Schatten-Von Neumann class membership for the small Hankel operator on Bergman space A ²(T _Ω), when T _Ω is the tube over the symmetric cone Ω. The author would like to thank professor Aline Bonami for helpful advices.     

The Completeness of the Bergman Distance of Planar Domains has a Local Character

It is shown that the property of completeness of the Bergman metric of a planar domain whose complement is not a polar set, is a local one.     

Weighted Composition Operators between Different Hardy Spaces

Let and be two analytic functions defined on such that. The operator given by is called a weighted composition operator. In this paper we deal with the boundedness, compactness, weak compactness, and complete continuity of weighted composition operators from a Hardy space H _p into another Hardy space H _q . We apply these results to study composition operators on Hardy spaces of a half-plane. Submitted: November 20, 2001.     

On the Algebra Generated by the Bergman Projection and a Shift Operator I

Let be a domain with smooth boundary and let α be a C ²- diffeomorphism on satisfying the Carleman condition .We denote by the C*-algebra generated by the Bergman projection of G, all multiplication operators aI and the operator where is the Jacobian of α. A symbol algebra of is determined and Fredholm conditions are given. We prove that the C*-algebra generated by the Bergman projection of the upper half-plane and the operator is isomorphic and isometric to . Submitted: February 11, 2001?Revised: January 27, 2002     

A Note on Wiener-Hopf Determinants and the Borodin-Okounkov Identity

The continuous analogue of a Toeplitz determinant identity for Wiener-Hopf operators is proved. An example which arises from random matrix theory is studied and an error term for the asymptotics of the determinant is computed. Submitted: October 15, 2001? Revised: January 3, 2002.     

Products of Toeplitz Operators on the Polydisk

This paper studies products of Toeplitz operators on the Hardy space of the polydisk. We show that T _f T _g = 0 if and only if T _f T _g is a finite rank if and only if T _f or T _g is zero. The product T _f T _g is still a Toeplitz operator if and only if there is a h $ \in $ L $ \infty $ (T ⁿ ) such that T _f T _g - T _h is a finite rank operator. We also show that there are no compact simi-commutators with symbols pluriharmonic on the polydisk. Submitted: October 5, 2000     

Continuity of the Norm of a Composition Operator

We explore the continuity of the map which, given an analytic selfmap of the disk, takes as its value the norm of the associated composition operator on the Hardy space . We also examine the continuity the functions which assign to a self-map of the disk the Hilbert-Schmidt norm or the essential norm of the associated composition operator and show these to be discontinuous. Additionally, we characterize when the norm of a composition operator is minimal. Submitted: January 3, 2002? Revised: March 1, 2002.     

The Matrix Multidisk Problem

The solutions of a class of matrix optimization problems (including the Nehari problem and its multidisk generalization) can be identified with the solutions of an abstract operator equation of the form T(., ., .) = 0. This equation can be solved numerically by Newton's method if the differential T' of T is invertible at the points of interest. This is typically too difficult to verify. However, it turns out that under reasonably broad conditions we can identify T' as the sum of a block Toeplitz operator and a compact block Hankel operator. Moreover, we can show that the block Toeplitz operator is a Fredholm operator and and in some cases can calculate its Fredholm index. Thus, T' will also be a Fredholm operator of the same index. In a number of cases that have been checked todate, numerical methods perform well when the Fredholm index is equal to zero and poorly otherwise. The main focus of this paper is on the multidisk problem alluded to above. However, a number of analogies with existing work on matrix optimization have been worked out and incorporated. Submitted: April 23, 2002.     

Toeplitz Operators Associated to Unimodular Algebras

We introduce a class of function algebras, that we call unimodular, and study Toeplitz operators on the Hardy spaces associated to representing measures on these algebras.We show that our class of function algebras is very extensive and that a number of important results for Toeplitz operators and their associated C*-algebras extend to the very general setting we consider. Submitted: July 1. 2001.     

The Iterated Aluthge Transform of an Operator

The Aluthge transform (defined below) of an operator T on Hilbert space has been studied extensively, most often in connection with p-hyponormal operators. In [6] the present authors initiated a study of various relations between an arbitrary operator T and its associated , and this study was continued in [7], in which relations between the spectral pictures of T and were obtained. This article is a continuation of [6] and [7]. Here we pursue the study of the sequence of Aluthge iterates { ⁽ⁿ⁾} associated with an arbitrary operator T. In particular, we verify that in certain cases the sequence { ⁽ⁿ⁾} converges to a normal operator, which partially answers Conjecture 1.11 in [6] and its modified version below (Conjecture 5.6). Submitted: December 5, 2000? Revised: August 30, 2001.