Gregory F. Bachelis and John E. Gilbert. 1. Introduction. Let G be a compact group and X, Y right Banach La(G)-modules, with Y reflexive. This paper considers ...
Math. Z. 125, 285-297 (1972) 9 by Springer-Verlag 1972
Banach Spaces of Compact Multipliers and Their Dual Spaces Gregory F. Bachelis and John E. Gilbert 1. Introduction
Let G be a compact group and X, Y right Banach La(G)-modules, with Y reflexive. This paper considers the various dual spaces of the Banach space S(X, Y) of compact multipliers from X into Y. If X and Y satisfy? weak additional conditions it is proved that
S(X, Y)* = X** |
Y*,
and hence that the space Hom(X**, Y) of all multipliers from X** into Y is the double dual of the space S(X, Y) of compact multipliers for reflexive Y In specific familiar examples the following results are obtained: (1.1) If 1 < p < oo and 1 < q < oo then A~(G) is the dual space of the compact multipliers from LP(G) into Lq(G); hence the space Hom(LP(G), Lq(G)) of all multipliers is the double dual of the space of compact multipliers. (1.2) Suppose l < p < o o and G is abelian. Then the space SP(G) of Junctions in LP(G) with unconditionally convergent Fourier series is a reflexive Banach space. (Theorem 4.9.) After some preliminary ideas have been disposed of in w2, we prove in w3 the main results in a completely general setting using techniques from the theory of tensor products (cf. for instance [12-1 pp. 361-376). Section 4 contains illustrative examples including (1.1) and (1.2). The authors wish to thank Professor A. FigS-Talamanca for helpful suggestions.
2. Preliminaries
When B is a Banach algebra we say that a Banach space X is a right Banach B-module if (i) X is a right B-module in the algebraic sense, (ii) ]]x'bllx
