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 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 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 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 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 On the Commutativity of a Certain Class of Toeplitz Operators

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Issam Louhichi ◽  
Fanilo Randriamahaleo ◽  
Lova Zakariasy
Keyword(s):  
Toeplitz Operator ◽  
Unit Disk ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Harmonic Bergman Space

AbstractOne of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex place C is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators defined on the harmonic Bergman space.

scholarly journals Schatten Class Operators in ℒ(La2(ℂ+)) \msbm=MTMIB${\cal L}\left( {L_a^2 \left( {{\msbm C}_+ } \right)} \right)$

2017 ◽  
Vol 55 (2) ◽  
pp. 91-117
Author(s):  
Namita Das ◽  
Jitendra Kumar Behera
Keyword(s):  
Bergman Space ◽  
Half Plane ◽  
Toeplitz Operators ◽  
Hankel Operators ◽  
Bounded Operators ◽  
Area Measure ◽  
Schatten Class ◽  
The Right

AbstractIn this paper, we consider Toeplitz operators defined on the Bergman space\msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$of the right half plane and obtain Schatten class characterization of these operators. We have shown that if the Toeplitz operators 𝕿φon\msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$belongs to the Schatten classSp, 1 ≤p < ∞,then\msbm=MTMIB$\tilde \phi \in L^p \left( {{\msbm C}_+ ,d\nu } \right)$, where$\tilde \phi \left( w \right) = \left\langle {\phi b_{\bar w} ,b_{\bar w} } \right\rangle $w ∈ℂ+and$b_{\bar w} (s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + \bar w}}{{2 Rew} \over {\left( {s + w} \right)^2 }}$. Here$d\nu (w) = \left| {B(\bar w,w)} \right|d\mu (w)$, wheredμ(w) is the area measure on ℂ+and$B(\bar w,w) = \left( {b_{\bar w} (\bar w)} \right)^2 $: Furthermore, we show that ifφ ∈ Lp(ℂ+,dv),then\msbm=MTMIB$\tilde \phi \in L^p ({\msbm C}_+ ,d\nu )$and 𝕿φ∈Sp. We also use these results to obtain Schatten class characterizations of little Hankel operators and bounded operators defined on the Bergman space\msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$

scholarly journals Normal Toeplitz Operators on the Fock Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1615
Author(s):  
Jongrak Lee
Keyword(s):  
Toeplitz Operator ◽  
Toeplitz Operators ◽  
Fock Spaces ◽  
Basic Properties

We characterize normal Toeplitz operator on the Fock spaces F2(C). First, we state basic properties for Toeplitz operator Tφ on F2(C). Next, we study the normal Toeplitz operator Tφ on F2(C) in terms of harmonic symbols φ. Finally, we characterize the normal Toeplitz operators Tφ with non-harmonic symbols acting on F2(C).

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.

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