Operator theory in the Hardy space over the bidisk (I)

In this paper, we identify the vector valued Hardy space with the Hardy space over the bidisk and construct a universal model for thecontractive analytic functions. We will also study some elementary properties of the submodules and show, in some cases, how the operator theoretical properties are related to the module theoretical properties. The last part focus on the study of double commutativity of compression operators.     

Operator theory in the Hardy space over the bidisk (II)

This paper is a continuation of a project of developing a systematic operator theory inH ²(D ²). A large part of it is devoted to a study ofevaluation operator which is a very useful tool in the theory. A number of elementary properties of the evaluation operator are exhibited, and these properties are used to derive results in other topics such as interpretation of characteristic opertor function inH ²(D ²), spectral equivalence, compactness, compressions of shift operators, etc., Even though some results reflect the two variable nature ofH ²(D ²), the goal of this paper is to manifest a close tie between the operator theory inH ²(D ²) and classical single operator theory. The unilateral shift of a finite multiplicity and the Bergman shift will be used as examples to illustrate some of the results.Research in this paper is partially supported by a grant from the national science foundation DMS 9970932.     

A reproducing kernel thesis for operators on Bergman-type function spaces

In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, the unit polydisc and more generally to weighted Fock spaces.     

Oscillation of linear difference equations in Banach spaces

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

     

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     

OnJ-conservative scattering system realizations in several variables

We prove that an arbitrary function, which is holomorphic on some neighbourhood ofz=0 in ^N and vanishes atz=0, and whose values are bounded linear operators mapping one separable Hilbert space into another one, can be represented as the transfer function of some multi-parameter discrete time-invariant conservative scattering linear system whose state space is a Krein space.The author is thankful to Prof. D.Z. Arov for suggesting this problem. He wishes also to thank Leeds University, where the revised version of this paper was prepared, for its hospitality, and Dr. V.V. Kisil who organized his visit there under the International Short Visits Scheme of LMS (grant no. 5620).     

Geometric Arveson–Douglas conjecture

We prove the Arveson–Douglas essential normality conjecture for graded Hilbert submodules that consist of functions vanishing on a given homogeneous subvariety of the ball, smooth away from the origin. Our main tool is the theory of generalized Toeplitz operators of Boutet de Monvel and Guillemin.     

A joint norm control Nehari type theorem forN-tuples of Hardy spaces

If N ∈ ℕ, 0 < p ≤ 1, and(X_k) _k=1 ^N are r.i.p-spaces, it is shown that there is C(= C(p, N)) > 0, such that for every ƒ ∈ ∩ _k=1 ^N X_k, there exists with , for every 1 ≤ k ≤ N. Also, if ⊓ is a convex polygon in ℝ², it is proved that the N-tuple (H(X₁),…, H(X_n)) is K_⊓-closed with respect to (X₁,…, X_N) in the sense of Pisier. Everything follows from Theorem 2.1, which is a general analytic partition of unity type result.     

One-sided tangential interpolation for operator-valued Hardy functions on polydisks

All solutions of one-sided tangential interpolation problems with Hilbert norm constraints for operator-valued Hardy functions on the polydisk are described. The minimal norm solution is explicitly expressed in terms of the interpolation data.The research of this author is partially supported by NSF grant DMS 9800704, and by the Faculty Research Assignment grant from the College of William and Mary.     

Domination spaces and factorization of linear and multilinear summing operators

It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, σ)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p1,?. . .?, pn)-dominated multilinear operators and dominated (p1,?. . .?, pn; σ)-continuous multilinear operators.     

A characterization of the Morrey-Campanato spaces

In this paper, we give a new characterization of the Morrey–Campanato spaces by using the convolution _tB*f(x) to replace the minimizing polynomial P_Bf of a function f in the Morrey-Campanato norm, where is an appropriate Schwartz function.D.G. Deng and L.X. Yan are partially supported by NSF of China and the Foundation of Advanced Research Center, Zhongshan University. X.T. Duong and L.X. Yan are supported by a Discovery grant from Australia Research Council.     

Continuity properties for modulation spaces, with applications to pseudo-differential calculus—I

Let M^p,q denote the modulation space with parameters p,q∈[1,∞]. If 1/p₁+1/p₂=1+1/p₀ and 1/q₁+1/q₂=1/q₀, then it is proved that . The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of Ψdo (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and Ψdo in the framework of modulation spaces.     

On the continuity of the spectrum in certain Banach algebras

In this paper we describe some classes of linear operatorsTL(H) (mainly Toeplitz, Wiener-Hopf and singular integral) on a Hilbert spacesH such that the spectrum (T, L(H)) is continuous at the pointsT from these classes. We also describe some subalgebras of the algebras for which the spectrum (x,) becomes continuous at the pointsx when (x,) is restricted to the subalgebra . In particular, we show that the spectrum (x,) is continuous in Banach algebras with polynomial identities. Examples of such algebras are given.This research was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

A Hardy inequality in the half-space

Here we prove a Hardy-type inequality in the upper half-space which generalize an inequality originally proved by Maz’ya (Sobolev Spaces, Springer, Berlin, 1985, p. 99). Here we present a different proof, which enable us to improve the constant in front of the remainder term. We will also generalize the inequality to the L^p case.     

Symmetric relations and module spaces

We study module spaces for linear relations (multi-valued operators) in a Hilbert space. The defect spaces are not required to be finite-dimensional. In particular we pay attention to module spaces for symmetric relations.     

Nonlinear mean ergodic theorems for nonexpansive semigroups in Banach spaces

We prove two nonlinear ergodic theorems for noncommutative semigroups of nonexpansive mappings in Banach spaces. Using these results, we obtain some nonlinear ergodic theorems for discrete and one-parameter semigroups of nonexpansive mappings. Dedicated to Professors Albrecht Dold and Ed Fadell