𝔖 Bobbio Scriptorium
✦   LIBER   ✦

A Dichotomy for Linear Spaces of Toeplitz Operators

✍ Scribed by Edward A Azoff; Marek Ptak

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
293 KB
Volume
156
Category
Article
ISSN
0022-1236

No coin nor oath required. For personal study only.

✦ Synopsis

Let S be a linear manifold of bounded Hilbert space operators. An operator A belongs to the reflexive closure of S if Af belongs to the closure of S f for each vector f in the underlying Hilbert space. Two extreme possibilities are (1) S is reflexive in the sense that ref S=S, and (2) S is transitive in the sense that ref S includes all bounded operators on the underlying space. We show that every linear space B of Toeplitz operators which is closed in the ultraweak operator topology is either transitive or reflexive. No intermediate behavior is possible. The full space of all Toeplitz operators is transitive, but if B is properly contained in this space and contains all analytic Toeplitz operators, then B must be reflexive. In particular, the space of Toeplitz operators whose matrices have zeros on a fixed superdiagonal is reflexive.

1998 Academic Press

1. Introduction

In [Sar], D. Sarason proved that the algebra A of analytic Toeplitz operators is reflexive. Moreover, it is elementary in the sense that every ultraweakly continuous linear functional on A is induced by an operator of rank at most one. It follows that A is hereditarily reflexive, i.e., that every ultraweakly closed subspace of A is also reflexive.

In this paper, we study the full space F of all Toeplitz operators and its various ultraweakly closed subspaces. It is fairly easy to see that F is transitive article no.

πŸ“œ SIMILAR VOLUMES

A Theorem of Brown–Halmos Type for Bergm
A Theorem of Brown–Halmos Type for Bergman Space Toeplitz Operators
✍ Patrick Ahern; Ε½eljko ČučkoviΔ‡ πŸ“‚ Article πŸ“… 2001 πŸ› Elsevier Science 🌐 English βš– 146 KB

We study the analogues of the Brown-Halmos theorem for Toeplitz operators on the Bergman space. We show that for f and g harmonic, T f T g =T h only in the trivial case, provided that h is of class C 2 with the invariant laplacian bounded. Here the trivial cases are f Β―or g holomorphic. From this w

Products of Hankel and Toeplitz Operator
Products of Hankel and Toeplitz Operators on the Bergman Space
✍ Karel Stroethoff; Dechao Zheng πŸ“‚ Article πŸ“… 1999 πŸ› Elsevier Science 🌐 English βš– 245 KB

We consider the question for which square integrable analytic functions f and g on the unit disk the densely defined products T f T gΓ„ are bounded on the Bergman space. We prove results analogous to those obtained by the second author [17] for such Toeplitz products on the Hardy space. We furthermor

Linear fractional relations for Hilbert
Linear fractional relations for Hilbert space operators
✍ V. A. Khatskevich; M. I. Ostrovskii; V. S. Shulman πŸ“‚ Article πŸ“… 2006 πŸ› John Wiley and Sons 🌐 English βš– 207 KB πŸ‘ 1 views

## Abstract In this paper we study linear fractional relations defined in the following way. Let ℋ︁~__i__~ and ℋ︁~__i__~ ^β€²^, __i__ = 1, 2, be Hilbert spaces. We denote the space of bounded linear operators acting from ℋ︁~__j__~ to ℋ︁~__i__~ ^β€²^ by __L__ (ℋ︁~__j__~ , ℋ︁~__i__~ ^β€²^). Let __T__ ∈

Toeplitz Operators and Hankel Operators
Toeplitz Operators and Hankel Operators on the Hardy Space of the Unit Sphere
✍ Dechao Zheng πŸ“‚ Article πŸ“… 1997 πŸ› Elsevier Science 🌐 English βš– 396 KB

A sufficient condition is found for the product of two Toeplitz operators on the Hardy space of the unit sphere to be a compact perturbation of a Toeplitz operator. The condition leads to a criterion for a Hankel operator to be compact. ## 1997 Academic Press The object of this present paper is to