Zero products of Toeplitz operators with harmonic symbols

On the Bergman space of the unit ball in Cⁿ, we solve the zero-product problem for two Toeplitz operators with harmonic symbols that have continuous extensions to (some part of) the boundary. In the case where symbols have Lipschitz continuous extensions to the boundary, we solve the zero-product problem for multiple products with the number of factors depending on the dimension n of the underlying space; the number of factors is n+3. We also prove a local version of this result but with loss of a factor.     

Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball

We calculate the operator norm of the weighted composition operator from a weighted Bergman space to a weighted-type space on the unit ball of Cⁿ. We also characterize the compactness of the operator.     

On exposed functions in Bernstein spaces

For σ > 0, the Bernstein space {ie427-01} consists of those L ¹(ℝ) functions whose Fourier transforms are supported by [−σ, σ]. Since {ie427-02} is separable and dual to some Banach space, the closed unit ball {ie427-03} of {ie427-04} has sufficiently large sets of both exposed and strongly exposed points: {ie427-05} coincides with the closed convex hull of its strongly exposed points. We investigate some properties of exposed points, construct several examples, and obtain as corollaries relations between the sets of exposed, strongly exposed, weak* exposed, and weak* strongly exposed points of {ie427-06}.     

Reproducing kernels for harmonic Bergman spaces of the unit ball

We study reproducing kernels for harmonic Bergman spaces of the unit ball inR ⁿ . We establish some new properties for the reproducing kernels and give some applications of these properties.     

Zero Products of Toeplitz Operators with <Emphasis Type="Italic">n</Emphasis>-Harmonic Symbols

On the Bergman space of the unit polydisk in the complex n-space, we solve the zero-product problem for two Toeplitz operators with n-harmonic symbols that have local continuous extension property up to the distinguished boundary. In the case where symbols have additional Lipschitz continuity up to the whole distinguished boundary, we solve the zero-product problem for products with four factors. We also prove a local version of this result for products with three factors.     

高维加权Bergman空间上的Toeplitz算子

本文研究了高维加权Bergman空间Ap(Bn,dVpφ)(1＜p＜∞).上的Toeplitz算子.利用Toeplitz算子的Berezin变换,获得了Ap(Bn,dVp)(1＜p＜∞)上具有L∞(Bn)符号的Toeplitz算子的有限乘积的有限和是紧算子的一些等价刻画,推广了加权Bergman空间Ap(D,dmpφ)上的Toeplitz算子的有限乘积的有限和是紧的当且仅当它的Berezin变换在单位圆盘的边界消失为0的结论     

Continuity of Bergman and Szegö projections on weighted-sup function spaces on pseudoconvex domains

We study regularity of Bergman and Szeg? projections on Sobolev type weighted-sup spaces. The paper covers the case of strongly pseudoconvex domains with C⁴ boundary and, partially, domains of finite type in the sense of D’Angelo. Received: 6 October 2005     

Quasi-Radial Quasi-Homogeneous Symbols and Commutative Banach Algebras of Toeplitz Operators

We present here a quite unexpected result: Apart from already known commutative C*-algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C*-algebra, and for n = 1 all of them collapse to the algebra generated by Toeplitz operators with radial symbols.     

Positive Toeplitz operators between the pluriharmonic Bergman spaces

We study Toeplitz operators between the pluriharmonic Bergman spaces for positive symbols on the ball. We give characterizations of bounded and compact Toeplitz operators taking a pluriharmonic Bergman space b ^p into another b ^q for 1 < p, q < ∞ in terms of certain Carleson and vanishing Carleson measures. This research was supported by KOSEF (R01-2003-000-10243-0) and Korea University Grant.     

On the Algebra Generated by the Bergman Projection and a Shift Operator I

Let be a domain with smooth boundary and let α be a C ²- diffeomorphism on satisfying the Carleman condition .We denote by the C*-algebra generated by the Bergman projection of G, all multiplication operators aI and the operator where is the Jacobian of α. A symbol algebra of is determined and Fredholm conditions are given. We prove that the C*-algebra generated by the Bergman projection of the upper half-plane and the operator is isomorphic and isometric to . Submitted: February 11, 2001?Revised: January 27, 2002     

Strongly exposed points, Helson-Szegö weights and Toeplitz operators

We investigate the strongly exposed points of the unit ball ofH ¹ and show that these functions are characterized by the equality of two De Branges' spaces that appeared in earlier results on exposed points. It then follows that the strongly exposed points are induced by Helson-Szegö weights, leading to several interesting properties of this class of functions.Dedicated to my father, Albert Jan Beneker.     

Schatten Class Hankel Operators on the Harmonic Bergman Space of the Unit Ball

We give a necessary and sufficient condition for Hankel operators H_f on the harmonic Bergman space of the unit ball to be in the Schatten p-class for 2 ≤ p < ∞. A special case when symbol f is a harmonic function is also considered.     

Finite rank Toeplitz products with harmonic symbols

On the Bergman space of the unit ball of Cn, we study the finite rank problem for Toeplitz products with harmonic symbols. We first solve the problem with two factors in case symbols have local continuous extension property up to the boundary. Also, in case symbols have additional Lipschitz continuity up to (some part of) the boundary, we solve the problem for multiple products with number of factors depending on the dimension n. Analogous theorems on the polydisk are also obtained.     

Denting and strongly extreme points in the unit ball of spaces of operators

For 1 ≤p ≤ ∞ we show that there are no denting points in the unit ball of ℓ(l^p). This extends a result recently proved by Grząślewicz and Scherwentke whenp = 2 [GS1]. We also show that for any Banach spaceX and for any measure space (Ω, A, μ), the unit ball of ℓ(L ¹ (μ), X) has denting points iffL ¹(μ) is finite dimensional and the unit ball ofX has a denting point. We also exhibit other classes of Banach spacesX andY for which the unit ball of ℓ(X, Y) has no denting points. When X* has the extreme point intersection property, we show that all ‘nice’ operators in the unit ball of ℓ(X, Y) are strongly extreme points.     

Commutative Toeplitz Banach Algebras on the Ball and Quasi-Nilpotent Group Action

Studying commutative C*-algebras generated by Toeplitz operators on the unit ball it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C*-algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each standard weighted Bergman space. There are five different pairwise non-conjugate model classes of such subgroups: quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent and quasi-nilpotent. Recently it was observed in Vasilevski (Integr Equ Oper Theory. 66:141–152, 2010) that there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were subordinated to the quasi-elliptic group, the corresponding commutative operator algebras were Banach, and being extended to C*-algebras they became non-commutative. These results were extended then to the classes of symbols, subordinated to the quasi-hyperbolic and quasi-parabolic groups. In this paper we prove the analogous commutativity result for Toeplitz operators whose symbols are subordinated to the quasi-nilpotent group. At the same time we conjecture that apart from the known C*-algebra cases there are no more new Banach algebras generated by Toeplitz operators whose symbols are subordinated to the nilpotent group and which are commutative on each weighted Bergman space.     

Backward iteration in strongly convex domains

We prove that a backward orbit with bounded Kobayashi step for a hyperbolic, parabolic or strongly elliptic holomorphic self-map of a bounded strongly convex C² domain in Cd necessarily converges to a repelling or parabolic boundary fixed point, generalizing previous results obtained by Poggi-Corradini in the unit disk and by Ostapyuk in the unit ball of Cd.     

Commutant Lifting Theorem for the Bergman Space

The central question of this paper is the one of finding the right analogue of the Commutant Lifting Theorem for the Bergman space L_a². We also analyze the analogous problem for weighted Bergman spaces L_a,α², − 1 < α < ∞.     

Boundary behavior of Gleason''''s problem in hyperbolic harmonic Bergman spaces

Necessary and sufficient conditions are obtained for the boundedness of Berezin transformation on Lebesgue space Lp(B, dVβ) in the real unit ball B in Rn. As an application, we prove that Gleason type problem is solvable in hyperbolic harmonic Bergman spaces. Furthermore we investigate the boundary behavior of the solutions of Gleason type problem.     

Exposed 2-homogeneous polynomials on Hilbert spaces

We show that every extreme point of the unit ball of 2-homogene- ous polynomials on a separable real Hilbert space is its exposed point and that the unit ball of 2-homogeneous polynomials on a non-separable real Hilbert space contains no exposed points. We also show that the unit ball of 2-homogeneous polynomials on any infinite dimensional real Hilbert space contains no strongly exposed points.

     

Multipliers between $ BMO $ Spaces on Open Unit Ball

and spaces consist of all those locally integrable functions on the open unit ball such that their mean oscillations on any Bergman ball are, respectively, bounded and vanishing to zero as the Bergman ball approaches the boundary of the unit ball. The harmonic (or analytic) versions of the and spaces are the Bloch space and the little Bloch space, respectively. The study of these spaces plays an important role in modern analysis, especially in the the area of operator theory and holomorphic spaces. In this paper, we characterize systematically the multipliers of and spaces. Submitted: January 9, 2001?Revised: October 16, 2002