OPERATORS ACTING ON CERTAIN BANACH SPACES OF ANALYTIC FUNCTIONS

. Let be reflexive Banach space of functions analytic plane domain 12 such that for every A in i2 the functional of evaluation at A is bounded Assume further that X contains the constants and Mz multiplication by the independent variable ., is bounded operator on ’. We give sufficient conditions for Mz to be reflexive. In particular, we prove that the operators Mz on EP{i} and certain H{} reflexive. We also prove that the algebra of multiplication operators on Bergman spaces is reflexive, giving simpler proof of result of Eschmeier.


INTRODUCTION.
Let f be a bounded domain in the complex plane C. Suppose Z is a reflexive Banach space consisting of functions that are analytic on fl such that X, for each A in the functional e(A)" Z C of evaluation at A given by e(A)(f)-< f, e(A) >-f(A) is bounded, and if f X then zf X. Note that the last condition allows us to define Mz X X by Mzf zf, f X. It is easy to see that Mz is actually abounded operator on X. If Z is a Hilbert space, the operator Mz and many of its properties have been studied in Shields and Wallen [1]; Bourdon and Shapiro [2]. We would like to give some sufficient conditions so that the operator Mz becomes reflexive.
Let f be a bounded open set in C and let p be a real number with _< p < oo. We denote by /.2(1) the LP-space of the 2-dimensional Lebesgue measure restricted to The space of analytic functions on fl is denoted by H(fl) and as usual g (fl) is the Banach space of all bounded functions analytic on fl equipped with the supremum norm. In this article we shall prove that the algebra S (Mf[ f H(fl)} is reflexive. We give a shorter proof of a result of J. Eschmeier [3] in case fl is a plane domain.

PRELIMINARIES.
In this section we make a few definitions and set our notation straight. If G is a bounded domain in the plane, the Carathodory hull (C-hull) of G is the complement For the algebra (.') of all bounded operators on a Banach space X, the weak operator topology (WOT) is the one in which a net As converges to A if Aax Ax weakly, x X. A complex valued function on for which Cf " for every " is called a multgher of " and the collection of all these multipliers IS denoted by 34(2") Because Mz is a bounded operator on Z, the adjoInt M; r-X'. satisfies Mz'e(,) ,e(,).
In general each multiplier of Z determines a multiplication operator M e defined by Me Cf, ,r Also Me'e( (,)e() It is well known that each multiplier is a bounded analytic function, Shields and Wallen [1]    We shall use the formal notation I(z) ,o ](n)z" for e D the umt disc in C (See Shields [5]   EXAMPLE 8. In Example 7 assume (n) for all n. In this case, it follows from(l) that Ilf g c fl for any compact K cD, where C depends on K. We present an example of a Banach space satisfying the hypothesis of Theorem 10. EXAMPLE 11. Let be a circular domain and < p < . Since L() is closed in (), () is reflexive. By Lemma 3.7 of Garnett [7] every point of is a bounded point evaluation for (). It is also clear that l]Mp] N lip n for every polynomial p. By Theorem 4 the multiplication operator Mz on L() is reflexive.
ACKNOWLEDGEMENT. Research of the first author was partially supported by a grant (no. 67-SC-520-276) from Shiraz University Research Council.