Dechao Zheng

Compact composition operators on BMOA and the Bloch space

We give a new and simple compactness criterion for composition operators C ϕ on BMOA and the Bloch space in terms of the norms of ϕ n in the respective spaces.

Isolated points and essential components of composition operators on H^c^h^i

We consider the topological space of all composition operators on the Banach algebra of bounded analytic functions on the unit disk. We obtain a function theoretic characterization of isolated points and show that each isolated composition operator is essentially isolated.

Toeplitz and Hankel operators on Bergman spaces

In this paper we consider Toeplitz and Hankel operators on the Bergman spaces of the unit ball and the polydisk in C n whose symbols are bounded measurable functions. We giv e necessary and sufficient conditions on the symbols for these operators to be compact. We study the Fredholm theory of Topelitz operators for which the corresponding Hankel operator is compact. For these Toeplitz operators the essential spectrum is computed and shown to be connected. We also consider symbols that extend to continuous functions on the maximal ideal space of H ∞(Ω); for these symbols we describe when the Toeplitz or Hankel operators are compact

Commuting Toeplitz operators with pluriharmonic symbols

Let dA(z) denote the Lebesgue volume measure on the open unit ball Bn °f cn, normalized so that the measure of Bn equals 1. The 13ergman space L2(Bn) is the EIilbert space consisting of holomorphic functions on Bn that are also in L2(Bn) dA). For f in L90(Bn), the Toeplitz operator Tf with symbol f is deSned by T(h) = P(fh): where P is the orthogonal projection from L2(BnndA) onto ol2tBesdA)n called the Bergman projection. Let dsr be the surface area measure on the unit sphere Sn For 1 < p < 007 H(S4) is the Banach space of holomorphic functions on Bn with the norm defined by

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].

Hankel operators and Toeplitz operators on the Bergman space

Abstract We consider in this paper the question of when the semi-commutator T fg − T f T g on the Bergman space with bounded harmonic symbols is compact. Several conditions equivalent to compactness of T fg − T f T g are given. As a consequence we prove a conjecture of Axler that for bounded analytic functions f and g on the unit disk, T fg ∗ − T g T f ∗ is compact iff either f or g is constant on each Gleason part P ( m ) except D .

Sampling sets and closed range composition operators on the Bloch space

We give a necessary and sufficient condition for a composition operator C Φ on the Bloch space to have closed range. We show that when Φ is univalent, it is sufficient to consider the action of C Φ on the set of Mobius transforms. In this case the closed range property is equivalent to a specific sampling set satisfying the reverse Carleson condition.

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.

Classification of Reducing Subspaces of a Class of Multiplication Operators on the Bergman Space via the Hardy Space of the Bidisk

In this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk. Let D be the open unit disk in C. Let dA denote Lebesgue area measure on the unit disk D, normalized so that the measure of D equals 1. The Bergman space La is the Hilbert space consisting of the analytic functions on D that are also in the space L(D, dA) of square integrable functions on D. For a bounded analytic function φ on the unit disk, the multiplication operator Mφ with symbol φ is defined on the Bergman space La given by Mφh = φh for h ∈ La. On the basis {en}n=0, where en is equal to √ n+ 1z, the multiplication operator Mz by z is a weighted shift operator, said to be the Bergman shift: Mzen = √ n+ 1 n+ 2 en+1. A reducing subspace M for an operator T on a Hilbert space H is a subspace M of H such that TM ⊂ M and T ∗M ⊂ M . A reducing subspace M of T is called minimal if M does not have any nontrivial subspaces which are reducing subspaces. The goal of this paper is to classify reducing subspaces ofMφ for the Blaschke product φwith four zeros by identifying its minimal reducing subspaces. Our main idea is to lift the Bergman shift up as a compression of a commuting pair of isometries on a nice subspace of the Hardy space of the bidisk. This idea was used in studying the Hilbert modules by R. Douglas and V. Paulsen [5], operator theory in the Hardy space over the bidisk by R. Dougals and R. Yang [6], [18], [19] and [20]; the higher-order Hankel forms by S. Ferguson and R. Rochberg [7] and [8] and and the lattice of the invariant subspaces of the Bergman shift by S. Richter [12]. On the Hardy space of the unit disk, for an inner function φ, the multiplication operator by φ is a pure isometry. So its reducing subspaces are in one-to-one correspondence with the closed subspaces of H φH [4], [10]. Therefore, it has infinitely many reducing subspaces provided that φ is any inner function other than a Mobius function. Many people have studied the problem of determining reducing subspaces of a multiplication operator on the Hardy space of the unit circle [1], [2] and [11]. The multiplication operators on the Bergman space possess a very rich structure theory. Even the lattice of the invariant subspaces of the Bergman shift Mz is huge [3]. But the lattice of reducing subspaces of the multiplication operator by a finite Blaschke on the The first author was supported in part by the National Natural Science Foundation-10471041 of China. The second author was partially supported by the National Science Foundation. 1 2 SUN, ZHENG AND ZHONG Bergman space seems to be simple. On the Bergman space, Zhu [21] showed that for a Blaschke product φ with two zeros, the multiplication operatorMφ has exact two nontrivial reducing subspacesM0 andM0 . In fact, the restriction of the multiplication operator on M0 is unitarily equivalent to the Bergman shift. Using the Hardy space of bidisk in [9], we show that the multiplication operator with a finite Blaschke product φ has a unique reducing subspaceM0(φ), on which the restriction ofMφ is unitarily equivalent to the Bergman shift and if a multiplication operator has a such reducing subspace, then its symbol must be a finite Blaschke product. The spaceM0(φ) is called the distinguished reducing subspace of Mφ and is equal to ∨ {φ′φn : n = 0, 1, · · · ,m, · · · } if φ vanishes at 0 in [15], i.e,

Algebraic and spectral properties of dual Toeplitz operators

Dual Toeplitz operators on the orthogonal complement of the Bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement. In this paper we study algebraic and spectral properties of dual Toeplitz operators.