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 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 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 Isometries of the product of composition operators on weighted Bergman space

2021 ◽  
Author(s):  
Anuradha Gupta ◽  
Geeta Yadav
Keyword(s):  
Bergman Space ◽  
Composition Operator ◽  
Composition Operators ◽  
Sufficient Conditions ◽  
Weighted Bergman Space ◽  
Necessary And Sufficient Conditions ◽  
Counter Example ◽  
Multiplicative Operator ◽  
Necessary And Sufficient

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]

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 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.

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

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

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&gt; 0 for eachj&gt; 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.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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