Extended Cesaro Operator from A φ ∞ to Bloch Space

Mingzhu Yang School of Tianjin University of Finance and Economics 25 Zhu Jiang Street, Tianjin 300222, China E-mail: mzhyang@mail.nankai.edu.cn Abstract Let g be a holomorphic function of the unit ball B in several complex variables, and denote by g T the induced extended Cesaro operator. This paper discussed the boundedness and compactness of g T acting from Aφ ∞ to Bloch space in the unit ball.


Introduction
Let B be the unit ball of n C , and ( ) H B denotes the class of analytic functions in B. Let p H be the standard Hardy space on the unit disc D. For 0 ( ) H , the classical Cesaro operator acting on f is given by the formula The study of Cesaro operator has become a major driving force in the development of modern complex analysis.The recent papers are good sources for information on much of the developments in the theory of Cesaro operators up to the middle of last decade.In the recent years, boundedness and compactness of extended Cesaro operator between several spaces of holomorphic functions have been studied by many mathematicians.It is well known that the operator C is bounded on the usual Hardy spaces p H and Bergman space, as well as the Dirichlet space.Basic results facts on Hardy spaces can be found in Durn(1970).For 0 p < < ∞ , Siskakis (1987) studied the spectrum of C , as a by-product he obtained that C is bounded on ( ) p H D .For 1 p = , the boundedness of C was given also by Siskakis (1990) by a particularly elegant method, independent of spectrum theory, a different proof of the result can be found in Giang and Morricz(1995)}.After that, for 0 1 p < < , Miao(1992) proved C is also bounded.For p = ∞ , the boundedness of C was given by Danikas and Sisakis(1993).

A little calculation shows
, it is natural to consider the extended Cesaro operator defined by It is easy to see that g T take ( ) H B into itself.In general, there is no easy way to determine when an extended Cesaro operator is bounded or compact.
The boundedness and compactness of this operator on weighted Bergman, mixed norm , Bloch, and Dirichlet spaces in the unit ball have been studied by Xiao and Hu.In this paper, we continue this line of research.Now we introduce some spaces first.We define Bloch space Bloch as the space of holomorphic functions ( ) Let φ denote a strictly decreasing continuous function : 1 φ = , it becomes the classical bounded function space.

Some Lemmas
In the following, we will use the symbol C or M to denote a finite positive number which does not depend on variable z and may depend on some norms and parameters f , not necessarily the same at each occurrence.
By Montel theorem and the definition of compact operator, the following lemma follows.

Assume that ( ) g H B ∈
. Then : T is bounded and for any bounded sequence ( ) Proof.Assume that g T is compact and suppose ( ) uniformly on compact sets of B .Since K is a compact subset of B , by the hypothesis and the definition of g T , ( ) T f z converges to zero uniformly on K .It follows from the arbitrary of K that the limit function h is equal to 0 .Since it's true for arbitrary subsequence of k f , we see that , where (0, ) therefore there is a subsequence km f which converges uniformly to ≤ <∞ and converges to 0 on compact subsets of B , by the hypothesis of this lemma, we have that T K is relatively compact, so g T is compact, finishing the proof.

Suppose ( ) g H B ∈
, then : Proof: We proof the sufficiency frist.Since (0) 0 Now we turn to the necessity.Setting the test function , for any

Since
B w ∈ is arbitrary, we get the necessity.
Remark: note that by take the test function 1 f = ,we can get g Bloch ∈ .
3.2 Suppose , and , then : , that is Rg belongs to the class of bounded holomorphic functions.
Proof: It is obvious from the 3.1.

Suppose ( ) g H B ∈
, then : Proof: We consider the sufficiency first.Assume the condition holds, then for any given 0 ε > , there exists a (0 1) ≤ , and for any sequence and k f converges to 0 uniformly on compact subsets of B .Notice that (0) 0 Now we turn to the necessity.For the necessity ,we choose the test functions as follows.For any sequence } and j h uniformly converges to 0 in any compact subset of B .That is to say j h satisfy the condition of lemma 2.1, then we have The conclusion follows by the arbitrary of the sequence } { j z ., we must have g is a constant.