THE COMPACT COMPOSITION OPERATOR ON THE μ-BERGMAN SPACE IN THE UNIT BALL

Let p 〉 0 and μ be a normal function on [0, 1）, u（r） = （1 - r2）1＋n^pμ（r） for r ∈ [0, 1）. In this article, the bounded or compact weighted composition operator Tφ,ψ from the μ-Bergman space AP（p） to the normal weight Bloch type space β （r）in the unit ball is characterized. The briefly sufficient and necessary condition that the composition operator Cφ is compact from A^p（μ） to βv, is given. At the same time, the authors give the briefly sufficient and necessary condition that Cv is compact on βμ, for a 〉 1.

COMPOSITION OPERATOR ON BERS-TYPE SPACES

Necessary and sufficient conditions are established for a composition operator C(phi)f = f o phi to be bounded or compact on the Bers-type space H-alpha(infinity) and the little Bers-type space H-alpha(infinity). The boundedness and compactness of the composition operator C-phi on A(infinity)(phi) are characterized, which generalize the case of C-phi on H-alpha(infinity).

COMPACT DIFFERENCES OF COMPOSITION OPERATORS ON HOLOMORPHIC FUNCTION SPACES IN THE UNIT BALL

We find a lower bound for the essential norm of the difference of two composition operators acting on H 2（BN ） or As2 （BN ） （s 1）. This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.

COMPOSITION OPERATORS FROM B^a TO F(p,q,s)

Necessary and sufficient conditions are given for a composition operator Cof = f o o to be bounded and compact from Ba to F(p, q, s).

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.

Weighted Composition Operators between Hardy Spaces on the Unit Ball

We consider the weighted composition operators between Hardy spaces on the unit ball, and obtain some sufficient and necessary conditions of bounded or compact weighted composition operators. We also prove that the operator from H^1 to H^1 is compact if and only if it is weakly compact. Meanwhile, we get the analogue on the Bergman spaces.

COMPOSITION OPERATORS FROM p-BLOCH SPACES TO LITTLE q-BLOCH SPACES ON THE UNIT BALL OF C^n

For all 0 〈 p, q 〈 ∞, let Cφ denote the composition operator from q-Bloch spaces βp to little p-Bloch spaces β0q on the unit ball of C^n. In this article, necessary and sufficient conditions for Cφ to be a bounded or compact operator are given.

COMPOSITION OPERATORS BETWEEN BLOCH TYPE SPACES IN THE UNIT BALL

In this paper, we obtain some new necessary and sufficient conditions for the boundedness and compactness of composition operators Cφ between Bloch type spaces in the unit ball Bn.

WEIGHTED COMPOSITION OPERATORS ON WEIGHTED DIRICHLET SPACES

We characterize the boundedness and compactness of weighted composition operators on weighted Dirichlet spaces in terms of Nevanlinna counting functions and Caxleson measure.

COLLECTIVELY COMPACT COMPOSITION OPERATOR SEQUENCES BETWEEN BERGMAN AND HARDY SPACES

This paper studies the collective compactness of composition operator sequences between the Bergman and Hardy spaces. Some sufficient and necessary conditions involving the generalized Nevanlinna counting functions for composition operator sequences to be collectively compact between weighted Bergman spaces are given

Weighted composition operators from F(p,q,s) spaces to Bers-type spaces in the unit ball

This paper deals with the boundedness and compactness of the weighted composition operators from the F（p, q, s） spaces, including Hardy space, Bergman space, Qp space, BMOA space, Besov space and α-Bloch space, to Bers-type spaces Hv^∞（ or little Bers-type spaces Hv,o∞ ）, where v is normal.

Composition Operators Mapping into the E(p,q) Spaces

Composition operators are used to study the E(p,q) spaces. The boundedness of these operators is also considered. The criteria for these operators to be bounded are given in terms of the Carleson measure.

Composition Operators on Compact Riemann Surfaces

In the present paper, the characterization of invertible composition oper-ators on compact Riemann surfaces is obtained, which is very different from that in the case of Hardy or Bergman spaces on a disk of the complex plane C.

Composition Operators Between the Weighted Dirichlet Spaces on Bounded Symmetric Domains of C^n

In this paper, we define the weighted Dirichlet space D p α(Ω) on bounded symmetric domains Ω of C n. Using η-α Carleson measure,we study the boundedness and compactmess of the composition operators between the weighted Dirichlet spaces.

WEIGHTED COMPOSITION OPERATORS ON THE HILBERT SPACE OF DIRICHLET SERIES

In this paper, we study weighted composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The Hermitianness, Fredholmness and invertibility of such operators are characterized, and the spectra of compact and invertible weighted composition operators are also described.

COMPACT COMPOSITION OPERATORS ON GENERALIZED LIPSCHITZ SPACES

In this paper, boundedness and compactness of the composition operator on the generalized Lipschitz spaces Λα （α 〉 1） of holomorphic functions in the unit disk are characterized.

TWO PROBLEMS ABOUT COMPOSITION OPERATORS ON HARDY SPACE

In this paper, two problems about composition operator on Hardy space are considered. Firstly, a new estimation of the norm of a class of composition operators is given. Secondly, the cyclic behavior of the adjoint operator of a composition operator is discussed.

BOUNDEDNESS OF COMPOSITION OPERATORS ON D_T SPACES FOR T > 1

In this note, some conditions of composition operators on DT spaces to be bounded are given by means of Carleson measures and pointwise multipliers, for some ranges of T. The authors prove that (i) Let 1 < T n + 2 and 2k < T 2k + 1 (or 2k - 1 < T 2k) for some positive integer k. Suppose = ( 1,… , n) be a univalent mapping from B into itself, denote dμj(l) (z) = R(l) j (z) 2(k-l+2) (1 - z 2)2k-t+1dv(z) for l= 1, 2,… , k + 1. If μj(l)-1 are (T - 2k + 2l-4)-Carleson measures for all l, then the composition operator C on DT is bounded; (ii) Let 1 < T n + 2, = ( 1,… , n) be univalent and the Frechet derivative of -1 be bounded on (B). If R j ∈ M(DT-2) for all j, then the composition operator C on DT is bounded; (iii) Let T > n + 2 and as in (ii). If j ∈ DT for all j, then the composition operator C on DT is bounded.

Composition Operators with Linear Fractional Symbols on Vector-Valued Bergman Spaces

Let φ and ψ be linear fractional self\|maps of the unit disk D and X a separable Hilbert space. In this paper we completely characterize the weak compactness of the product operators of a composition operation C φ with another one's adjoint C * ψ on the vector\|valued Bergman space B 1(X) for forms C φC * ψ and C * ψC φ.

A Note on Weighted Composition Operators on the Fock Space

Based on a new characterization of bounded and compact weighted compositionoperators on the Fock space obtained by Le T （Le T. Normal and isometricweighted composition operators on the Fock space. Bull. London. Math. Soc., 2014,46： 847-856）, this paper shows that a bounded weighted composition operator onthe Fock space is a Fredholm operator if and only if it is an invertible operator, andif and only if it is a nonzero constant multiple of a unitary operator. The result isvery different from the corresponding results on the Hardy space and the Bergmanspace.