ESSENTIAL NORMS OF PRODUCTS OF WEIGHTED COMPOSITION OPERATORS AND DIFFERENTIATION OPERATORS BETWEEN BANACH SPACES OF ANALYTIC FUNCTIONS

We obtain several estimates of the essential norms of the products of differen- tiation operators and weighted composition operators between weighted Banach spaces of analytic functions with general weights. As applications, we also give estimates of the es- sential norms of weighted composition operators between weighted Banach space of analytic functions and Bloch-type spaces.

Essential Norms of Composition Operators Between Hardy Spaces of the Unit Disc

The authors express the essential norms of composition operators between Hardy spaces of the unit disc in terms of the natural Nevanlinna counting function.

Essential Norms of Composition Operators between Weighted Bergman Spaces of the Unit Disc

In this paper, we express the essential norms of composition operators between weighted Bergman spaces of the unit disc in terms of the generalized Nevanlinna counting function.

Essential Norm of Extended Ceshro Operators between Bloch-type Spaces

The Bloch-type space Bω consists of all functions f ∈ H（B） for which｜｜f｜｜Bω =sup z∈Bω（z）｜△f（z）｜〈∞Let Tφ be the extended Cesaro operator with holomorphic symbol φ. The essential norm of Tφ as an operator from Bω to Bμ is denoted by ｜｜Tφ｜｜e,Bω→Bμ. The purpose of this paper is to prove that, for w, ω normal and φ ∈ H（B）｜｜Tφ｜｜e,Bω→Bμ≈lim sup｜z｜→1μ（z）｜Rφ（z）｜∫0^｜z｜dt/ω（t）.

AN UPPER BOUND OF THE ESSENTIAL NORM OF COMPOSITION OPERATORS BETWEEN WEIGHTED BERGMAN SPACES

In this paper, we define the generalized counting functions in the higher dimensional case and give an upper bound of the essential norms of composition operators between the weighted Bergman spaces on the unit ball in terms of these counting functions. The sufficient condition for such operators to be bounded or compact is also given.

MULTIPLICATION OPERATORS ON INVARIANT SUBSPACES OF FUNCTION SPACES

Let Mφ be the operator of multiplication by φ on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator Mz, we derive some spectral properties of the multiplication operator Mφ ： F→F. We characterize norm, spectrum, essential norm and essential spectrum of such operators when F has the codimension n property with n∈{1,2,...,＋∞}.

THE PRODUCT OPERATOR BETWEEN BLOCH-TYPE SPACES OF SLICE REGULAR FUNCTIONS

There is little work concerning the properties of quaternionic operators acting on slice regular function spaces defined on quaternions.In this paper,we present an equivalent characterization for the boundedness of the product operator C_(φ)D^(m) acting on Bloch-type spaces of slice regular functions.After that,an equivalent estimation for its essential norm is established,which can imply several existing results on holomorphic spaces.

Linear Combination of Derivative Weighted Composition Operators

The relation between composition operators on the Dirichlet spaces in the open unit disk and derivative weighted composition operators on the Bergman spaces in the open unit disk is investigated firstly,and for a combination of several derivative weighted composition operators which acts on classic Bergman space,the lower bound of its essential norm is estimated in terms of the boundary data of the symbols of d-composition operators.Some similar results about composition operators on the Dirichlet space are also presented.A necessary condition is given to determine the compactness of the combination of several derivative weighted composition operators on Bergman spaces.

A Note on the Essential Norm of Composition Operators from H^p(B_N ) to H^q(B_N )

The authors give an upper bound of the essential norms of composition opera- tors between Hardy spaces of the unit ball in terms of the counting function in the higher dimensional value distribution theory defined by Professor S. S. Chern. The sufficient condition for such operators to be bounded or compact is also given.

An Upper Bound of Essential Norm of Composition Operator on H^2(B_n)

The authors give an upper bound of the essential norm of a composition operator on H2(Bn),which involves the counting function in the higher dimensional value distribution theory defined by S.S.Chern.A criterion is also given to assure that the composition operator on H2(Bn) is bounded or compact.

Integral operators on analytic Morrey spaces

In this note,we characterize the boundedness of the Volterra type operator Tg and its related integral operator Ig on analytic Morrey spaces.Furthermore,the norm and essential norm of those operators are given.As a corollary,we get the compactness of those operators.

Toeplitz Products on Bergman–Sobolev Spaces over the Unit Polydisk

In this paper,we characterize the compactness and Fredholmness of Toeplitz operators and Toeplitz products on Bergman-Sobolev spaces over the unit polydisk.We also calculate the essential norm of finite sums of finite Toeplitz products on these spaces.

On the Compactness of Grunsky Differential Operators

Grunsky operators play an important role in classical geometric function theory and in the study of Teichmüller spaces.The Grunsky map is known to be holomorphic on the universal Teichmüller space.In this paper the authors deal with the compactness of a Grunsky differential operator.They will give upper and lower estimates of the essential norm of a Grunsky differential operator and discuss when a Grunsky differential operator is a p-Schatten class operator.