Bounded analytic functions of two complex variables

Abstract

Introduction The purpose of this paper is to study the Banach space Ho~ (G x G) of bounded analytic functions in a region G x G in C 2 by representing it as a space of bounded linear operators. By considering the compact operators we are able to relate H~ (G x G) and its subspace H~ (G)Q~H~ (G). (Definitions are given below.) In general, this subspace can be proper thereby providing an answer to a question raised by RossI during the Tulane Symposium on Function Algebras in 1965. We obtain a representation of H~o (G x G) as the dual space of a 7-tensor product of measures which is identified with M 1 (G x G) and LI(G x G)/N2. Furthermore, the compact linear operators from MI(G) to H~o (G) are characterized in terms of a 2-tensor product of L| spaces. By a constructive method we give a large class of functions on the unit polydisc D x D which are in H~ (D x D), but which are not in H~ (D) | (D). Finally, in Section 5, we summarize the relationships of all the Banach spaces considered in this paper. Obviously, we are strongly influenced by the papers of RtJBEL and SmELOS [5, 6]. They indicated the possibility of extending their results to higher dimensions. Such an extension to two complex dimensions is obtained when H~ (G x G) is exhibited as the dual of the separable Banach space M I(G x G). Using similar proofs and replacing arguments involving point masses by arguments based on approximate identities, many of our results can be proved independently of [5, 6] for the Banach space H~(TxT) of functions on the torus which are boundary values of bounded analytic functions on DxD. We wish to express our gratitude to Professor QtJIGLEY for many helpful discussions and to Professor STOUT for communicating some useful remarks.