Products of Bergman space Toeplitz operators on the polydisk (original) (raw)

Products of Toeplitz and Hankel operators on the Bergman space in the polydisk

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica, 2018

In this paper we obtain a condition for analytic square integrable functions \(f,g\) which guarantees the boundedness of products of the Toeplitz operators \(T_fT_{\bar g}\) densely defined on the Bergman space in the polydisk. An analogous condition for the products of the Hankel operators \(H_fH^*_g\) is also given.

Bounded Toeplitz products on the Bergman space of the polydisk

Journal of Mathematical Analysis and Applications, 2003

We consider the question for which square integrable analytic functions f and g on the polydisk the densely defined products T f Tḡ are bounded on the Bergman space. We prove results analogous to those we obtained in the setting of the unit disk [K.

Two Questions on Products of Toeplitz Operators on the Bergman Space

2009

The zero product problem and the commuting problem for Toeplitz operators on the Bergman space over the unit disk are some of the most interesting unsolved problems. For bounded harmonic symbols these are solved but for general bounded symbols it is still far from being complete. This paper shows that the zero product problem holds for a special case where one of the symbols has certain polar decomposition and the other is a general bounded symbol. We also prove that the commutant of Tz+z is sum of powers of itself.

Products of Toeplitz Operators on the Bergman Space

Integral Equations and Operator Theory, 2005

In 1962 Brown and Halmos gave simple conditions for the product of two Toeplitz operators on Hardy space to be equal to a Toeplitz operator. Recently, Ahern andCucković showed that a similar result holds for Toeplitz operators with bounded harmonic symbols on Bergman space. For general symbols, the situation is much more complicated. We give necessary and sufficient conditions for the product to be a Toeplitz operator (Theorem 6.1), an explicit formula for the symbol of the product in certain cases (Theorem 6.4), and then show that almost anything can happen (Theorem 6.7).

Toeplitz operators in polyanalytic Bergman type spaces

Functional Analysis and Geometry, 2019

We consider Toeplitz operators in Bergman and Fock type spaces of polyanalytic L2textup−L^2\textup{-}L2textup−functions on the disk or on the half-plane with respect to the Lebesgue measure (resp., on mathbbC\mathbb{C}mathbbC with the plane Gaussian measure). The structure involving creation and annihilation operators, similar to the classical one present for the Landau Hamiltonian, enables us to reduce Toeplitz operators in true polyanalytic spaces to the ones in the usual Bergman type spaces, however with distributional symbols. This reduction leads to describing a number of properties of the operators in the title, which may differ from the properties of the usual Bergman-Toeplitz operators.

Toeplitz and Hankel Operators on Bergman Spaces

Transactions of the American Mathematical Society, 1992

In this paper we consider Toeplitz and Hankel operators on the Bergman spaces of the unit ball and the polydisk in C whose symbols are bounded measurable functions. We give necessary and sufficient conditions on the symbols for these operators to be compact. We study the Fredholm theory of Toeplitz operators for which the corresponding Hankel operator is compact. For these Toeplitz operators the essential spectrum is computed and shown to be connected. We also consider symbols that extend to continuous functions on the maximal ideal space of H°°(Ci) ; for these symbols we describe when the Toeplitz or Hankel operators are compact.

Finite Sums of Toeplitz Products on the Polydisk

Potential Analysis, 2009

On the Bergman space of the unit polydisk, we study a class of operators which contains sums of finitely many Toeplitz products with pluriharmonic symbols. We give characterizations of when an operator in that class has finite rank or is compact. As one of applications we show that sums of a certain number, depending on and increasing with the dimension, of semicommutators of Toeplitz operators with pluriharmonic symbols cannot be compact without being the zero operator.

Algebraic Properties of Toeplitz Operators on the Polydisk

Abstract and Applied Analysis, 2011

We discuss some algebraic properties of Toeplitz operators on the Bergman space of the polydiskDn. Firstly, we introduce Toeplitz operators with quasihomogeneous symbols and property (P). Secondly, we study commutativity of certain quasihomogeneous Toeplitz operators and commutators of diagonal Toeplitz operators. Thirdly, we discuss finite rank semicommutators and commutators of Toeplitz operators with quasihomogeneous symbols. Finally, we solve the finite rank product problem for Toeplitz operators on the polydisk.