Kunyu Guo

Analytic Hilbert modules

The seminal 1989 work of Douglas and Paulsen in the theory of analytic Hilbert modules precipitated a number of major research efforts. This in turn led to some intriguing and valuable results, particularly in the areas of operator theory and functional analysis. With the field now beginning to blossom, the time has come to collect those results under one cover. Written by two of the most active and often-cited researchers in the field, Analytic Hilbert Modules reports on the progress made by the authors and others, including the characteristic space theory, rigidity, the equivalence problem, the Arveson modules, extension theory, and reproducing Hilbert spaces on n-dimensional complex space.

Multiplication operators on the Bergman space via the Hardy space of the bidisk

Abstract In this paper, we develop a machinery to study multiplication operators on the Bergman space via the Hardy space of the bidisk. Using the machinery we study the structure of reducing subspaces of a multiplication operator on the Bergman space. As a consequence, we completely classify reducing subspaces of the multiplication operator by a Blaschke product φ with order three on the Bergman space to solve a conjecture of Zhu [J. London Math. Soc. 62: 553–568, 2000].

Toeplitz algebra and Hankel algebra on the harmonic Bergman space

In this paper we completely characterize compact Toeplitz operators on the harmonic Bergman space. By using this result we establish the short exact sequences associated with the Toeplitz algebra and the Hankel algebra. We show that the Fredholm index of each Fredholm operator in the Toeplitz algebra or the Hankel algebra is zero.

Essentially commuting Hankel and Toeplitz operators

Abstract We characterize when a Hankel operator and a Toeplitz operator have a compact commutator.

On operators which commute with analytic Toeplitz operators modulo the finite rank operators

It is shown that an operator S on the Hardy space H 2 (D n ) (or H 2 (B n )) commutes with all analytic Toeplitz operators modulo the finite rank operators if and only if S = T 9 + F. Here F is a finite rank operator, and in the case n = 1, g is a sum of a rational function and a bounded analytic function, and in the case n >2, g is a bounded analytic function.

Homogeneous quasi-invariant subspaces of the fock space

In this paper, we prove that two homogeneous quasi-invariant subspaces are similar only if they are equal. Moreover, we exhibit an example to show how to determine the similarity orbits of quasi-invariant subspaces.

Indices, characteristic numbers and essential commutants of Toeplitz operators

For an essentially normal operatorT, it is shown that there exists a unilateral shift of multiplicitym inC*(T) if and only if γ(T)≠0 and γ(T)/m. As application, we prove that the essential commutant of a unilateral shift and that of a bilateral shift are not isomorphic asC*-algebras. Finally, we construct a naturalC*-algebra ε + ε* on the Bergman spaceLa2(Bn), and show that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators.

Essentially commutative C *-algebras with essential spectrum homeomorphic to S 2 n −1

This paper gives a complete classification of essentially commutative C∗–algebras whose essential spectrum is homeomorphic to S2n−1 by their characteristic numbers. Let A1, A2 be such two C∗–algebras, then they are C∗–isomorphic if and only if they have the same n-th characteristic number. Furthermore, let γn(A) = m, then A is C∗– isomorphic to C(Mz1 , · · · ,Mzn) if m = 0, A is C∗–isomorphic to C(Tz1 , · · · , Tzn−1 , Tzm n ) if m 6= 0. Some examples are given to show applications of the classification theorem. We finally remark that the proof of the theorem depends on a construction of a complete system of representatives of Ext(S2n−1). 2000 Mathematics subject classification (Amer.Math.Soc.): 47A53; 47B35; 47B37.

Compact Perturbations of Hankel Operators

Abstract We give some necessary and sufficient conditions of when the product of two Hankel operators is a compact perturbation of a Hankel operator on the Hardy space.

EXTENSIONS OF HARDY MODULES OVER THE POLYDISK ALGEBRA

In this paper, the Ext groups of Hardy modules over the polydisk algebra A(Dn) are calculated. Of particular importance is that the calculating of Ext-groups is closely related to harmonic analysis of polydisks. Finally, the authors point out that Ext-groups reveal rigidity of Hardy submodules over A(Dn) for n > 1.