scholarly journals Schatten Class Toeplitz Operators on the Bergman Space

2009 ◽  
Vol 2009 ◽  
pp. 1-16
Author(s):  
Namita Das
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Berezin Transform ◽  
Open Unit ◽  
Open Unit Disk ◽  
Area Measure ◽  
Schatten Class ◽  
Necessary And Sufficient

We have shown that if the Toeplitz operatorTϕon the Bergman spaceLa2(&#x1D53B;)belongs to the Schatten classSp,1≤p<∞, thenϕ˜∈Lp(&#x1D53B;,dλ), whereϕ˜is the Berezin transform ofϕ,dλ(z)=dA(z)/(1−|z|2)2, anddA(z)is the normalized area measure on the open unit disk&#x1D53B;. Further, ifϕ∈Lp(&#x1D53B;,dλ)thenϕ˜∈Lp(&#x1D53B;,dλ)andTϕ∈Sp. For certain subclasses ofL∞(&#x1D53B;), necessary and sufficient conditions characterizing Schatten class Toeplitz operators are also obtained.

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 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 Toeplitz Operators on the Bergman Space of Planar Domains with Essentially Radial Symbols

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Roberto C. Raimondo
Keyword(s):  
Bergman Space ◽  
Toeplitz Operators ◽  
Boundary Behavior ◽  
Berezin Transform ◽  
Planar Domain ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Planar Domains ◽  
Necessary And Sufficient ◽  
The Relationship

We study the problem of the boundedness and compactness of when and is a planar domain. We find a necessary and sufficient condition while imposing a condition that generalizes the notion of radial symbol on the disk. We also analyze the relationship between the boundary behavior of the Berezin transform and the compactness of

scholarly journals Toeplitz operators with BMO symbols and the Berezin transform

2003 ◽  
Vol 2003 (46) ◽  
pp. 2929-2945 ◽  
Author(s):  
Nina Zorboska
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Berezin Transform ◽  
Boundary Behaviour

We prove that the boundedness and compactness of the Toeplitz operator on the Bergman space with aBMO1symbol is completely determined by the boundary behaviour of its Berezin transform. This result extends the known results in the cases when the symbol is either a positiveL1-function or anL∞function.

scholarly journals On Fully-Convex Harmonic Functions and their Extension

2018 ◽  
Vol 38 (2) ◽  
pp. 51-60
Author(s):  
Shahpour Nosrati ◽  
Ahmad Zireh
Keyword(s):  
Harmonic Functions ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Uniformly Convex ◽  
Circular Arc ◽  
Coefficient Condition ◽  
Necessary And Sufficient ◽  
Convex Univalent Functions

‎Uniformly convex univalent functions that introduced by Goodman‎, ‎maps every circular arc contained in the open unit disk with center in it into a convex curve‎. ‎On the other hand‎, ‎a fully-convex harmonic function‎, ‎maps each subdisk $|z|=r<1$ onto a convex curve‎. ‎Here we synthesis these two ideas and introduce a family of univalent harmonic functions which are fully-convex and uniformly convex also‎. ‎In the following we will mention some examples of this subclass and obtain a necessary and sufficient conditions and finally a coefficient condition will attain with convolution‎.

scholarly journals Applications of Borel Distribution Series on Analytic Functions

2020 ◽  
pp. 71-82
Author(s):  
Abbas Kareem Wanas ◽  
Jubran Abdulameer Khuttar
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates ◽  
Sigma Delta ◽  
The Family ◽  
Necessary And Sufficient

The purpose of the present paper is to determine the necessary and sufficient conditions for the power series B_{\mu} whose coefficients are probabilities of the Borel distribution to be in the family H(\lambda, \sigma, \delta, \mu) of analytic functions which defined in the open unit disk. We derive a number of important geometric properties, such as, coefficient estimates, integral representation, radii of starlikeness and convexity. Also we discuss the extreme points and neighborhood property for functions belongs to this family.

scholarly journals Multiplication Operator with BMO Symbols and Berezin Transform

2015 ◽  
Vol 2015 ◽  
pp. 1-4
Author(s):  
Xue Feng ◽  
Kan Zhang ◽  
Jianguo Dong ◽  
Xianmin Liu ◽  
Chi Guan
Keyword(s):  
Unit Ball ◽  
Bergman Space ◽  
Sufficient Conditions ◽  
Multiplication Operator ◽  
Berezin Transform ◽  
Weighted Bergman Space ◽  
Necessary And Sufficient Conditions ◽  
Special Symbol ◽  
Necessary And Sufficient

We discuss multiplication operator with a special symbol on the weighted Bergman space of the unit ball. We give the necessary and sufficient conditions for the compactness of multiplication operator on the weighted Bergman space of the unit ball.

scholarly journals Product and Commutativity ofkth-Order Slant Toeplitz Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Chaomei Liu ◽  
Yufeng Lu
Keyword(s):  
Bergman Space ◽  
Lebesgue Space ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Harmonic Polynomial ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

The commutativity ofkth-order slant Toeplitz operators with harmonic polynomial symbols, analytic symbols, and coanalytic symbols is discussed. We show that, on the Lebesgue space and Bergman space, necessary and sufficient conditions for the commutativity ofkth-order slant Toeplitz operators are that their symbol functions are linearly dependent. Also, we study the product of twokth-order slant Toeplitz operators and give some necessary and sufficient conditions.

scholarly journals Commuting Quasihomogeneous Toeplitz Operator and Hankel Operator on Weighted Bergman Space

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Jun Yang
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Hankel Operator ◽  
Sufficient Conditions ◽  
Weighted Bergman Space ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Quasihomogeneous Symbols

We characterize the commuting Toeplitz operator and Hankel operator with quasihomogeneous symbols. Also, we use it to show the necessary and sufficient conditions for commuting Toeplitz operator and Hankel operator with ordinary functions.

scholarly journals A Subclass of q-Starlike Functions Defined by Using a Symmetric q-Derivative Operator and Related with Generalized Symmetric Conic Domains

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 917
Author(s):  
Shahid Khan ◽  
Saqib Hussain ◽  
Muhammad Naeem ◽  
Maslina Darus ◽  
Akhter Rasheed
Keyword(s):  
Unit Disk ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Necessary And Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Derivative Operator ◽  
Structural Formulas ◽  
Necessary And Sufficient

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain Ωk,q,α˜, which generalizes the symmetric conic domains. By using the domain Ωk,q,α˜, we define a new subclass of analytic and q-starlike functions in the open unit disk U and establish some new results for functions of this class. We also investigate a number of useful properties and characteristics of this subclass, such as coefficients estimates, structural formulas, distortion inequalities, necessary and sufficient conditions, closure and subordination results. The proposed approach is also compared with some existing methods to show the reliability and effectiveness of the proposed methods.

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