Product and Commutativity ofkth-Order Slant Toeplitz Operators

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.

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

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02602-1 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Sumin Kim ◽

Jongrak Lee

Keyword(s):

Toeplitz Operator ◽

Bergman Space ◽

Toeplitz Operators ◽

Bergman Spaces ◽

Sufficient Conditions ◽

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.

Toeplitz operators and Wiener-Hopf factorisation: an introduction

Concrete Operators ◽

10.1515/conop-2017-0010 ◽

2017 ◽

Vol 4 (1) ◽

pp. 130-145 ◽

Cited By ~ 2

Author(s):

M. Cristina Câmara

Keyword(s):

Integral Equations ◽

Half Plane ◽

Singular Integral ◽

Hardy Spaces ◽

Toeplitz Operators ◽

Sufficient Conditions ◽

Singular Integral Equations ◽

Upper Half Plane ◽

Abstract Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces Hp of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in L−1(ℝ)

Algebraic Properties of Toeplitz Operators on the Pluriharmonic Bergman Space

Journal of Function Spaces and Applications ◽

10.1155/2013/578436 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 2

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.

Multiplication Operator with BMO Symbols and Berezin Transform

Journal of Function Spaces ◽

10.1155/2015/754646 ◽

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 ◽

Special Symbol ◽

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.

Isometries of the product of composition operators on weighted Bergman space

Asian-European Journal of Mathematics ◽

10.1142/s1793557122501753 ◽

2021 ◽

Author(s):

Anuradha Gupta ◽

Geeta Yadav

Keyword(s):

Bergman Space ◽

Composition Operator ◽

Composition Operators ◽

Sufficient Conditions ◽

Weighted Bergman Space ◽

Counter Example ◽

Multiplicative Operator ◽

In this paper, the necessary and sufficient conditions for the product of composition operators to be isometry are obtained on weighted Bergman space. With the help of a counter example we also proved that unlike on [Formula: see text] and [Formula: see text] the composition operator on [Formula: see text] induced by an analytic self-map on [Formula: see text] with fixed origin need not be of norm one. We have generalized the Schwartz’s [Composition operators on [Formula: see text], thesis, University of Toledo (1969)] well-known result on [Formula: see text] which characterizes the almost multiplicative operator on [Formula: see text]

Commuting Quasihomogeneous Toeplitz Operator and Hankel Operator on Weighted Bergman Space

Abstract and Applied Analysis ◽

10.1155/2013/408168 ◽

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 ◽

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.

Schatten Class Toeplitz Operators on the Bergman Space

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2009/921324 ◽

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 ◽

We have shown that if the Toeplitz operatorTϕon the Bergman spaceLa2(𝔻)belongs to the Schatten classSp,1≤p<∞, thenϕ˜∈Lp(𝔻,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 disk𝔻. Further, ifϕ∈Lp(𝔻,dλ)thenϕ˜∈Lp(𝔻,dλ)andTϕ∈Sp. For certain subclasses ofL∞(𝔻), necessary and sufficient conditions characterizing Schatten class Toeplitz operators are also obtained.

On Superrecurrence

Canadian Mathematical Bulletin ◽

10.4153/cmb-1991-008-7 ◽

1991 ◽

Vol 34 (1) ◽

pp. 48-57

Author(s):

Karma Dajani

Keyword(s):

Lebesgue Space ◽

Sufficient Conditions ◽

Skew Products ◽

Ergodic Automorphism ◽

AbstractLet T be a non-singular, conservative, ergodic automorphism of a Lebesgue space. We study a kind of weighted cocycles called H-cocycles. We introduce the notions of H-superrecurrence and H-supertransience. We use skew products to give necessary and sufficient conditions for H-superrecurrence.

Necessary and sufficient conditions for boundedness of the Hardy-type operator from a weighted Lebesgue space to a Morrey-type space

Mathematical Inequalities & Applications ◽

10.7153/mia-16-01 ◽

2013 ◽

pp. 1-19 ◽

Cited By ~ 4

Author(s):

Victor Burenkov ◽

Ryskul Oinarov

Keyword(s):

Lebesgue Space ◽

Sufficient Conditions ◽

Type Operator ◽

Weighted Lebesgue Space ◽

Type Space ◽

Hardy Type ◽

Hardy Type Operator ◽

The extinction time of a general birth and death process with catastrophes

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Cited By ~ 1

Author(s):

P. J. Brockwell

Keyword(s):

Laplace Transform ◽

Size Distribution ◽

Sufficient Conditions ◽

Death Process ◽

Extinction Time ◽

Birth And Death Process ◽

Different Types ◽

The Laplace Transform ◽

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.