On Reflexive Norms for the Direct Product of Banach Spaces

Introduction. In a previous paper [7]('), for two Banach spaces Eu E2, the Banach spaces Ei®E2, E{ ®E2, E(' ®E2' [7, p. 205] are constructed. If the norm N [7, Definition 3.1] is defined on Ei®E2, then the associate norm N' [7, Definition 3.2 and Lemma 3.1] is defined on E{ ®E2. Similarly N" denotes the norm on E{' ®E". Among the unsolved problems (mentioned in [7, §6]), are listed the following two: A. What are the exact conditions imposed upon a crossnorm [7, Definition 3.3] under which (Ei®E2)'=E{ ®E2 holds? B. Is the associate with every crossnorm also a crossnorm, or do there exist crossnorms whose associates are not crossnorms? In the present paper we present a "partial" answer to problem A (which we denote by A*), and a "partial" answer to problem B (which we denote by B*). A*. A uniformly convex crossnorm N sets up the relation (Ei®E2)' =E{ ®El if, and only if, N" = N. B*. For reflexive Banach spaces (that is, such that El' =E\, E2 =E2) the associate with every crossnorm is also a crossnorm. In this paper we also show that the values of a crossnorm for all expressions of rank not greater than 2 do not necessarily determine the crossnorm. The following should be mentioned in immediate connection with problem A: It is evident that for norms for which (Ex®E2)'=E{ ®E2 holds, N" = N. Since in general (for any norm N) all we can state is (Ei®E2)'~Z)E{ ®E2 [7, p. 205], we have no basis for assuming that N" represents the norm in (Ei®Et)", or N" = N for expressions in 2l(£i, E2)C%(E{', El') [7, Definition 1.3]. Therefore, N"^N [7, Lemma 3.2] is the best that can be stated in the general case. In the present paper we present some results on reflexive norms, that is, such that N" = N.