On reflexive norms for the direct product of Banach spaces

Abstract

Introduction. In a previous paper [7](1), for two Banach spaces E1, E2, the Banach spaces E1?E2, E; X E2, E G E2 ' [7, p. 205 ] are constructed. If the norm [7, Definition 3.1] is defined on E1?E2, then the associate norm N' [7, Definition 3.2 and Lemma 3.11 is defined on E; X E2. Similarly N denotes the norm on EP X E2'. 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 (E10 E2)'=E; XE2' holds? B. rs 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 sets up the relation (E1l0E2)' E' 0EX if, and only if, N= N. B*. For reflexive Banach spaces (that is, such that EP = E1, E' =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 (E1l E2)'= E( XE' holds, N = N. Since in general (for any norm N) all we can state is (E1l?E2)'DE; ?El [7, p. 205], we have no basis for assuming that N represents the norm in (E1l E2), or N =N for expressions in (E1, E2)C2[(E', E2') [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.