scholarly journals Toeplitz Operators with Quasihomogeneous Symbols on the Bergman Space of the Unit Ball

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Bo Zhang ◽  
Yufeng Lu
Keyword(s):  
Unit Ball ◽  
Toeplitz Operator ◽  
Bergman Space ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Quasihomogeneous Symbols

We consider when the product of two Toeplitz operators with some quasihomogeneous symbols on the Bergman space of the unit ball equals a Toeplitz operator with quasihomogeneous symbols. We also characterize finite-rank semicommutators or commutators of two Toeplitz operators with quasihomogeneous symbols.

scholarly journals Algebraic Properties of Toeplitz Operators on the Pluriharmonic Bergman Space

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jingyu Yang ◽  
Liu Liu ◽  
Yufeng Lu
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Rank Product ◽  
Algebraic Properties ◽  
Necessary And Sufficient ◽  
Product Problem ◽  
Quasihomogeneous Symbols

We study some algebraic properties of Toeplitz operators with radial or quasihomogeneous symbols on the pluriharmonic Bergman space. We first give the necessary and sufficient conditions for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator and discuss the zero-product problem for several Toeplitz operators with radial symbols. Next, we study the finite-rank product problem of several Toeplitz operators with quasihomogeneous symbols. Finally, we also investigate finite rank commutators and semicommutators of two Toeplitz operators with quasihomogeneous symbols.

scholarly journals Algebraic Properties of Quasihomogeneous and Separately Quasihomogeneous Toeplitz Operators on the Pluriharmonic Bergman Space

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Hongyan Guan ◽  
Liu Liu ◽  
Yufeng Lu
Keyword(s):  
Unit Ball ◽  
Toeplitz Operator ◽  
Bergman Space ◽  
Trivial Solution ◽  
Toeplitz Operators ◽  
The Other ◽  
Homogeneous Functions ◽  
Algebraic Properties ◽  
Product Problem

We study some algebraic properties of Toeplitz operator with quasihomogeneous or separately quasihomogeneous symbol on the pluriharmonic Bergman space of the unit ball inℂn. We determine when the product of two Toeplitz operators with certain separately quasi-homogeneous symbols is a Toeplitz operator. Next, we discuss the zero-product problem for several Toeplitz operators, one of whose symbols is separately quasihomogeneous and the others are quasi-homogeneous functions, and show that the zero-product problem for two Toeplitz operators has only a trivial solution if one of the symbols is separately quasihomogeneous and the other is arbitrary. Finally, we also characterize the commutativity of certain quasihomogeneous or separately quasihomogeneous Toeplitz operators.

scholarly journals Algebraic Properties of Toeplitz Operators on the Polydisk

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Bo Zhang ◽  
Yanyue Shi ◽  
Yufeng Lu
Keyword(s):  
Bergman Space ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Rank Product ◽  
Algebraic Properties ◽  
Product Problem ◽  
Quasihomogeneous Symbols

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.

scholarly journals Commuting and Semi-commuting Monomial-type Toeplitz Operators on Some Weakly Pseudoconvex Domains

2019 ◽  
Vol 62 (02) ◽  
pp. 327-340
Author(s):  
Cao Jiang ◽  
Xing-Tang Dong ◽  
Ze-Hua Zhou
Keyword(s):  
Unit Ball ◽  
Bergman Space ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Pseudoconvex Domains ◽  
Weakly Pseudoconvex

AbstractIn this paper, we completely characterize the finite rank commutator and semi-commutator of two monomial-type Toeplitz operators on the Bergman space of certain weakly pseudoconvex domains. Somewhat surprisingly, there are not only plenty of commuting monomial-type Toeplitz operators but also non-trivial semi-commuting monomial-type Toeplitz operators. Our results are new even for the unit ball.

QUASIHOMOGENEOUS TOEPLITZ OPERATORS WITH INTEGRABLE SYMBOLS ON THE HARMONIC BERGMAN SPACE

2014 ◽  
Vol 90 (3) ◽  
pp. 494-503 ◽  
Author(s):  
XING-TANG DONG ◽  
CONGWEN LIU ◽  
ZE-HUA ZHOU
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Mellin Transform ◽  
Harmonic Bergman Space ◽  
Funct Anal ◽  
Quasihomogeneous Symbols

AbstractIn this paper, we completely determine the commutativity of two Toeplitz operators on the harmonic Bergman space with integrable quasihomogeneous symbols, one of which is of the form $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}e^{ik\theta }r^{\, {m}}$. As an application, the problem of when their product is again a Toeplitz operator is solved. In particular, Toeplitz operators with bounded symbols on the harmonic Bergman space commute with $T_{e^{ik\theta }r^{\, {m}}}$ only in trivial cases, which appears quite different from results on analytic Bergman space in Čučković and Rao [‘Mellin transform, monomial symbols, and commuting Toeplitz operators’, J. Funct. Anal.154 (1998), 195–214].

scholarly journals Hyponormality of Toeplitz operators with non-harmonic symbols on the Bergman spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sumin Kim ◽  
Jongrak Lee
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

AbstractIn this paper, we present some necessary and sufficient conditions for the hyponormality of Toeplitz operator $T_{\varphi }$ T φ on the Bergman space $A^{2}(\mathbb{D})$ A 2 ( D ) with non-harmonic symbols under certain assumptions.

scholarly journals Toeplitz Operators on the Weighted Pluriharmonic Bergman Space with Radial Symbols

2011 ◽  
Vol 2011 ◽  
pp. 1-15
Author(s):  
Zhi Ling Sun ◽  
Yu Feng Lu
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Spectral Decomposition ◽  
Toeplitz Operators ◽  
Complete Information ◽  
Isometric Isomorphism

We construct an operatorRwhose restriction onto weighted pluriharmonic Bergman Spacebμ2(Bn)is an isometric isomorphism betweenbμ2(Bn)andl2#. Furthermore, using the operatorRwe prove that each Toeplitz operatorTawith radial symbols is unitary to the multication operatorγa,μI. Meanwhile, the Wick function of a Toeplitz operator with radial symbol gives complete information about the operator, providing its spectral decomposition.

scholarly journals Normal Toeplitz Operators on the Bergman Space

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1463
Author(s):  
Sumin Kim ◽  
Jongrak Lee
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Basic Properties

In this paper, we give a characterization of normality of Toeplitz operator Tφ on the Bergman space A2(D). First, we state basic properties for Toeplitz operator Tφ on A2(D). Next, we consider the normal Toeplitz operator Tφ on A2(D) in terms of harmonic symbols φ. Finally, we characterize the normal Toeplitz operators Tφ with non-harmonic symbols acting on A2(D).

scholarly journals The Detectable Subspace for the Friedrichs Model: Applications of Toeplitz Operators and the Riesz–Nevanlinna Factorisation Theorem

2020 ◽  
Vol 21 (10) ◽  
pp. 3141-3156
Author(s):  
S. Naboko ◽  
I. Wood
Keyword(s):  
Toeplitz Operator ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Model Operator ◽  
Hardy Classes ◽  
Finite Rank Perturbation ◽  
Boundary Measurements ◽  
Independent Variable ◽  
Friedrichs Model

Abstract We discuss how much information on a Friedrichs model operator (a finite rank perturbation of the operator of multiplication by the independent variable) can be detected from ‘measurements on the boundary’. The framework of boundary triples is used to introduce the generalised Titchmarsh–Weyl M-function and the detectable subspaces which are associated with the part of the operator which is ‘accessible from boundary measurements’. In this paper, we choose functions arising as parameters in the Friedrichs model in certain Hardy classes. This allows us to determine the detectable subspace by using the canonical Riesz–Nevanlinna factorisation of the symbol of a related Toeplitz operator.

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