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 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.

BOUNDED TOEPLITZ AND HANKEL PRODUCTS ON THE WEIGHTED BERGMAN SPACES OF THE UNIT BALL

2015 ◽  
Vol 99 (2) ◽  
pp. 237-249
Author(s):  
MAŁGORZATA MICHALSKA ◽  
PAWEŁ SOBOLEWSKI
Keyword(s):  
Unit Ball ◽  
Toeplitz Operators ◽  
Hankel Operator ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Berezin Transform ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Necessary Condition ◽  
Sufficient Condition

Let $A_{{\it\alpha}}^{p}$ be the weighted Bergman space of the unit ball in ${\mathcal{C}}^{n}$, $n\geq 2$. Recently, Miao studied products of two Toeplitz operators defined on $A_{{\it\alpha}}^{p}$. He proved a necessary condition and a sufficient condition for boundedness of such products in terms of the Berezin transform. We modify the Berezin transform and improve his sufficient condition for products of Toeplitz operators. We also investigate products of two Hankel operators defined on $A_{{\it\alpha}}^{p}$, and products of the Hankel operator and the Toeplitz operator. In particular, in both cases, we prove sufficient conditions for boundedness of the products.

Convex Subordination Chains in Several Complex Variables

2009 ◽  
Vol 61 (3) ◽  
pp. 566-582 ◽  
Author(s):  
Ian Graham ◽  
Hidetaka Hamada ◽  
Gabriela Kohr ◽  
John A. Pfaltzgraff
Keyword(s):  
Unit Ball ◽  
Sufficient Conditions ◽  
Complex Variables ◽  
Necessary And Sufficient Conditions ◽  
Several Complex Variables ◽  
Sufficient Condition ◽  
Subordination Chain ◽  
Euclidean Unit Ball ◽  
Necessary And Sufficient

Abstract.In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in ℂn. We also obtain a sufficient condition for injectivity off(z/‖z‖, ‖z‖) onBn\ ﹛0﹜, wheref(z,t) is a convex subordination chain over (0, 1).

scholarly journals COMPACT DIFFERENCES OF COMPOSITION OPERATORS ON BERGMAN SPACES IN THE BALL

2010 ◽  
Vol 89 (3) ◽  
pp. 407-418 ◽  
Author(s):  
XIANG DONG YANG ◽  
LE HAI KHOI
Keyword(s):  
Unit Ball ◽  
Composition Operator ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Compact Operators ◽  
Necessary And Sufficient Conditions ◽  
Weighted Bergman Spaces ◽  
Finite Sum ◽  
Necessary And Sufficient

AbstractWe obtain necessary and sufficient conditions for the compactness of differences of composition operators acting on the weighted Bergman spaces in the unit ball. A representation of a composition operator as a finite sum of composition operators modulo compact operators is also studied.

scholarly journals Some properties of Glisson distances in the upper half-plane

2021 ◽  
Vol 2131 (3) ◽  
pp. 032039
Author(s):  
M Ovchintsev
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Half Plane ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ordinate Axis ◽  
Upper Half Plane ◽  
Necessary And Sufficient

Abstract The author compares the Gleason distance with the distance of Euclid in the unit disk in the upper half plane. The concept of “the Gleason distance” was formulated in the work of H.S. Bear [1] The Gleason distance is defined as follows (see [1]): d = sup |f(z2)-f(z1)|, f(Z)εB1(K) where B 1 (K) is the unit ball in the space of bounded analytic in K functions. The author of the article proves that in the circle K the distances of Gleason and Euclid are equal only when the points are opposite. He found necessary and sufficient conditions, when the distances are equal for the two given points which are symmetrical about the ordinate axis.

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 Analytic functions over valued fields

1990 ◽  
Vol 13 (2) ◽  
pp. 247-252
Author(s):  
R. Bhaskaran ◽  
V. Karunakaran
Keyword(s):  
Unit Ball ◽  
Fixed Points ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Ball ◽  
Valued Fields ◽  
Complete Field ◽  
Necessary And Sufficient

LetKbe a non-archimedean, non-trivially (rank 1) valued complete field.B,B0denote the closed and open unit ball ofKrespectively. Necessary and sufficient conditions for analytic functions defined onB,B0with values inKto be injective, necessary and sufficient conditions for fixed points, the problem of subordination are studied in this paper.

scholarly journals Deformations and equitopological deformations of strongly pseudoconvex manifolds

1981 ◽  
Vol 82 ◽  
pp. 113-129 ◽  
Author(s):  
Stephen S.-T. Yau
Keyword(s):  
Unit Ball ◽  
Deformation Theory ◽  
Complex Analysis ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Order Perturbation ◽  
First Order ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
First Order Perturbation

One of the main problems in complex analysis has been to determine when two open sets D1, D2 in Cn are biholomorphically equivalent. In [26] Poincaré studied perturbations of the unit ball B2 in C2 of a particular kind, and found necessary and sufficient conditions on a first order perturbation that the perturbed domain be biholomorphically equivalent to B2. Recently Burns, Shnider and Wells [7] (cf. also Chern-Moser [9]) have studied the deformations of strongly pseudoconvex manifolds. They proved that there is no finite-dimensional deformation theory for M if one keeps track of the boundary.

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