Factoring positive operators on reproducing kernel Hilbert spaces

We characterize when positive operators can be factored by analytic Toeplitz type operators. As a corollary, we give an operator theory characterization of those invariant subspaces of doubly commuting unilateral shifts, which are generated by a single inner function on the bidisk. The last result extends to shifts of arbitrary (countable) multiplicity.     

On an intertwining lifting theorem for certain reproducing kernel Hilbert spaces

We provide an alternate approach to an intertwining lifting theorem obtained by Ball, Trent and Vinnikov. The results are an exact analogue of the classical Sz-Nagy-Foias theorem in the case of multipliers on a class of reproducing kernel spaces, which satisfy the Nevanlinna-Pick property.     

Products of Bergman space Toeplitz operators on the polydisk

Motivated by recent works of Ahern and uković on the disk, we study the generalized zero product problem for Toeplitz operators acting on the Bergman space of the polydisk. First, we extend the results to the polydisk. Next, we study the generalized compact product problem. Our results are new even on the disk. As a consequence on higher dimensional polydisks, we show that the generalized zero and compact product properties are the same for Toeplitz operators in a certain case.The first three authors were partially supported by KOSEF(R01-2003-000-10243-0) and the last author was partially supported by the National Science Foundation.     

Interpolation problems,extensions of symmetric operators and reproducing kernel spaces II

By an oversight on the part of the authors this section was not included in the paper previously published in Integral Equations Operator Theory, volume 14/4 (1991), 466–500. Present address:Department of Mathematics Ben-Gurion University of the Negev Beersheva Israel     

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.     

A continuum of unusual self-adjoint linear partial differential operators

In an earlier publication a linear operator _THar$T_{\mathrm{Har}}$ was defined as an unusual self-adjoint extension generated by each linear elliptic partial differential expression, satisfying suitable conditions on a bounded region Ω$\Omega$ of some Euclidean space. In this present work the authors define an extensive class of _THar$T_{\mathrm{Har}}$-like self-adjoint operators on the Hilbert function space _L2(Ω);$L_2(\Omega );$ but here for brevity we restrict the development to the classical Laplacian differential expression, with Ω$\Omega$ now the planar unit disk. It is demonstrated that there exists a non-denumerable set of such _THar$T_{\mathrm{Har}}$-like operators (each a self-adjoint extension generated by the Laplacian), each of which has a domain in _L2(Ω)$L_2(\Omega )$ that does not lie within the usual Sobolev Hilbert function space W²(Ω)$W^2(\Omega )$. These _THar$T_{\mathrm{Har}}$-like operators cannot be specified by conventional differential boundary conditions on the boundary of ∂Ω$\mathrm{\partial }\Omega$, and may have non-empty essential spectra.     

Toeplitz operators on Dirichlet spaces

In this paper we consider Toeplitz operators on Dirichlet spaces of the unit disk in whose symbols are nonnegative measures. We obtain necessry and sufficient conditions on the symbols for the operator to be bounded and compact. If the symbols are supported in a cone we also get necessary and sufficient conditions for the operators to belong to the Schatten p-class. Application to the Hankel operators are indicated.This work supported in part by NSF grant DMS 8701271     

Global bifurcation for equations involving nonhomogeneous operators in an unbounded domain

We study the structure of the set of solutions of a nonlinear equation involving nonhomogeneous operators:

     

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 operators on harmonic Bergman spaces

We study Toeplitz operators on the harmonic Bergman spaceb ^p (B), whereB is the open unit ball inR ⁿ(n2), for 1<p. We give characterizations for the Toeplitz operators with positive symbols to be bounded, compact, and in Schatten classes. We also obtain a compactness criteria for the Toeplitz operators with continuous symbols.     

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     

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     

A class of linear fractional maps of the ball and their composition operators

We study a class of linear fractional self-maps of the ball which seems to be a good generalization of parabolic non-automorphisms of the unit disk. We give a normal form of these maps and use it to compute the spectrum of the composition operators induced by them. We also show that these composition operators are never hypercyclic. Applications are given to the study of more general linear fractional transformations.     

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.     

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.     

Bases of reproducing kernels in de-Branges spaces

This paper deals with geometric properties of sequences of reproducing kernels related to de-Branges spaces. If b is a nonconstant function in the unit ball of H^∞, and T_b is the Toeplitz operator, with symbol b, then the de-Branges space, H(b), associated to b, is defined by , where H² is the Hardy space of the unit disk. It is equipped with the inner product such that is a partial isometry from H² onto H(b). First, following a work of Ahern-Clark, we study the problem of orthogonal basis of reproducing kernels in H(b). Then we give a criterion for sequences of reproducing kernels which form an unconditional basis in their closed linear span. As far as concerns the problem of complete unconditional basis in H(b), we show that there is a dichotomy between the case where b is an extreme point of the unit ball of H^∞ and the opposite case.     

Comparing different classes of absolutely summing multilinear operators

Let C denote the composition operator defined on the standard Hardy spaces H^p as where is an analytic self-map of the unit disk in the complex plane. In this paper we discuss those invariant subspaces of C in H^p which are invariant under the shift operator, We restrict our attention to the case where is an inner function. Our main result characterises these invariant subspaces. We also consider C when restricted to such an invariant subspace and we describe the structure of the operator and find a formula for the essential spectral radius.Received: 27 January 2004     

Oscillation of linear difference equations in Banach spaces

One can easily show that almost all solutions of the difference equation

     

Discretizations of Integral Operators and Atomic Decompositions in Vector-Valued Weighted Bergman Spaces

We prove a general atomic decomposition theorem for weighted vector-valued Bergman spaces, which applies to duality problems and to the study of compact Toeplitz type operators.