The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces

The characterization of thosef for which the Hankel operatorsH _f belongs to various trace ideals over Bergman spaces on pseudoconvex domains of finite type in complex dimension two is given. In particular, we determine how the cutoff values are affected by the boundary geometry.All three authors supported by grants from the National Science Foundation     

Hankel operators on generalizedH 2 spaces

In the paper, we give characterizations of Hankel operators on generalizedH ² spaces and obtain some properties for corresponding Hankel algebras.Supported in part by NSF of China     

Compact products of Hankel operators

In this paper we completely characterize compact products of three Hankel operators on the Hardy space. We obtain a necessary and sufficient condition for that , and are simultaneously compact.This work was partly supported by NSF grants. The second author was also partly supported by the Research Council of Vanderbilt University.     

Compact Hankel operators on harmonic Bergman spaces

We study Hankel operators on the harmonic Bergman spaceb ²(B), whereB is the open unit ball inR ⁿ,n2. We show that iff is in then the Hankel operator with symbolf is compact. For the proof we have to extend the definition of Hankel operators to the spacesb ^p(B), 1<p<, and use an interpolation theorem. We also use the explicit formula for the orthogonal projection ofL ²(B, dV) ontob ²(B). This result implies that the commutator and semi-commutator of Toeplitz operators with symbols in are compact.     

Maximal inner spaces and Hankel operators on the Bergman space

The paper deals with two closely related questions about the Bergman space of the unit disk. First, we investigate a special class of invariant subspaces of the Bergman space, namely, invariant subspaces induced by certain Hankel operators. We show that such spaces always have the co-dimension 1 or 2 property; and we determine exactly when such a space has the co-dimension 1 property. Second, we introduce the notion of inner spaces in the Bergman space and give several characterizations of when an inner space is maximal.Research supported by the National Science Foundation     

Toeplitz and Hankel type operators on the upper half-plane

An orthogonal decomposition of admissible wavelets is constructed via the Laguerre polynomials, it turns to give a complete decomposition of the space of square integrable functions on the upper half-plane with the measurey dxdy. The first subspace is just the weighted Bergman (or Dzhrbashyan) space. Three types of Ha-plitz operators are defined, they are the generalization of classical Toeplitz, small and big Hankel operators respectively. Their boundedness, compactness and Schatten-von Neumann properties are studied.Research was supported by the National Natural Science Foundation of China.     

Paraproducts and Hankel operators of Schatten class via p-John-Nirenberg Theorem

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of Schatten-von Neumann class S_p, if and only if its symbol is in the dyadic Besov space B_p^d. Our main tools are a product formula for paraproducts and a “p-John-Nirenberg-Theorem” due to Rochberg and Semmes.We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols.Using an averaging technique by Petermichl, we retrieve Peller's characterizations of scalar and vector Hankel operators of Schatten-von Neumann class S_p for 1<p<∞. We then employ vector techniques to characterise little Hankel operators of Schatten-von Neumann class, answering a question by Bonami and Peloso.Furthermore, using a bilinear version of our product formula, we obtain characterizations for boundedness, compactness and Schatten class membership of products of dyadic paraproducts.     

Composition operators on Orlicz spaces

In this paper we characterize the composition operators on Orlicz spaces and study some of their properties.Research supported by NBHM DDF No. 40/11/95 (R&D-II)/1429     

Hankel operators on the Bergman spaces of strongly pseudoconvex domains

We characterize functions fL ²(D) such that the Hankel operators H_f are, respectively, bounded and compact on the Bergman spaces of bounded strongly pseudoconvex domains.Research partially supported by a grant of the National Science Foundation.     

Two distinguished subspaces of product BMO and Nehari-AAK theory for Hankel operators on the torus

In this paper we show that the theory of Hankel operators in the torus ^d , ford>1, presents striking differences with that on the circle , starting with bounded Hankel operators with no bounded symbols. Such differences are circumvented here by replacing the space of symbolsL ( ) by BMOr( ^d ), a subspace of product BMO, and the singular numbers of Hankel operators by so-called sigma numbers. This leads to versions of the Nehari-AAK and Kronecker theorems, and provides conditions for the existence of solutions of product Pick problems through finite Picktype matrices. We give geometric and duality characterizations of BMOr, and of a subspace of it, bmo, closely linked withA ₂ weights. This completes some aspects of the theory of BMO in product spaces.Sadosky was partially supported by NSF grants DMS-9205926, INT-9204043 and GER-9550373, and her visit to MSRI is supported by NSF grant DMS-9022140 to MSRI.     

Compact and finite rank operators satisfying a Hankel type equationT 2X=XT 1 *

In 1997, V. Pták defined the notion of generalized Hankel operator as follows: Given two contractions and , an operatorX: is said to be a generalized Hankel operator ifT ₂ X=XT ₁ ^* andX satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations ofT ₁ andT ₂. The purpose behind this kind of generalization is to study which properties of classical Hankel operators depend on their characteristic intertwining relation rather than on the theory of analytic functions. Following this spirit, we give appropriate versions of a number of results about compact and finite rank Hankel operators that hold within Pták's generalized framework. Namely, we extend Adamyan, Arov and Krein's estimates of the essential norm of a Hankel operator, Hartman's characterization of compact Hankel operators and Kronecker's characterization of finite rank Hankel operators.Dedicated to the memory of our master and friend Vlastimil Pták     

Zero or compact products of several Hankel operators

In this paper we give a necessary and sufficient condition for the product of several Hankel operators on the Hardy space of the unit disk to be zero or compact. This extends results of Brown and Halmos [2], Axler, Chang and Sarason [1], Volberg [8] and Xia and Zheng [9] on zero or compact products of two or three Hankel operators.     

Products of Hankel operators

This paper studies products of Hankel operators on the Hardy space. We show thatH _f ^(1)* H _f(2) H _f ^(3)* =0 for all permutation if and only if eitherH _f1 orH _f2 orH _f3 is zero. Using Douglas' localization theorem and Izuchi's theorem on Sarason's three functions problem, we show that

On operators commuting with Toeplitz operators modulo the finite rank operators

Let X be a bounded linear operator on the Hardy space H² of the unit disk. We show that if is of finite rank for every inner function θ, then X=T_?+F for some Toeplitz operator T_? and some finite rank operator F on H². This solves a variant of an open question where the compactness replaces the finite rank conditions.     

Hankel and Toeplitz operators on Dirichlet spaces

In this paper we study Hankel and Toeplitz operators on Dirichlet type spaces D. We obtain necessary and sufficient condition on the symbols for these operators to be bounded and to belong to the Schatten ideal S^p for certain and p.     

Multiplication operators between Orlicz spaces

The invertible, compact and Fredholm multiplication operators on Orlicz spaces are characterized in this paper.     

Kernels of Hankel operators and hyponormality of Toeplitz operators

We give a formula for and describe when . We explore the hyponormality of Toeplitz operators whose symbols are of circulant type and some more general types. In addition, we discuss formulas for and estimates of the rank of the self-commutator of a hyponormal Toeplitz operator. Received September 17, 1999 / Revised May 25, 2000 / Published online December 8, 2000     

A band limited and Besov class functional calculus for Tadmor-Ritt operators

For Banach space operators T satisfying the Tadmor-Ritt condition a band limited H^∞ calculus is established, where and a is at most of the order C(T)⁵. It follows that such a T allows a bounded Besov algebra B_{∞ 1}⁰ functional calculus, These estimates are sharp in a convenient sense. Relevant embedding theorems for B_{∞ 1}⁰ are derived. Received: 25 October 2004; revised: 31 January 2005