The Cross-Space of Linear Transformations
Robert Schatten · Annals of Mathematics · 1946
In a previous paper2 [7] for two Banach spaces 231, 32, we construct a linear set 03 0 02 Of (that is, finite sums) =lfi 0 gi ( E 01, gi E 32). These expressions are abstract algebraic elements for which the X operator possesses the distributive property. Introducing a norm a in 1 0 (D 0we obtain the normed linear 0 0a 02. Completing it in the usual fashion by adding as new elements fundamental (Cauchy) sequences of expressions in 9i 0i a 932, we obtain a Banach 3iB 0) a932 . The last one naturally depends on a. Together with -B1 0 02 we construct a linear set Q3*D 0 * of expressions il Fi ?& Gi (Fi E -, Gi E ). For a fixed expression Z71 Fj X (El=1 Fj X Gj) i g) = 2 Fj(fi)Gj(gi) represents an additive functional on 0, 0. -92Under certain conditions for every expression Z1 Fj 0 Gj, the bound a'(E'7.. Fj 0 Gj) of the additive functional which it represents, is finite. In such a case the bound also represents a norm a' in t X. For any crossnorm a ? A, a'(E1 Fj 0 Gj) is finite for every expression in Q31* 0 t3 Thus, (31 53a T2)* D 31 Ga' 932 . The last inclusion, depending on the particular Banach spaces and crossnorm. a, may in certain cases be a proper one. Thus, a given cross-space 31 Ga 02 uniquely determines an space Q1 Ga' 03'2 and a space (01 Ga 32)*. It is difficult to state the precise conditions imposed upon a crossnorm. a, such that for the resulting cross-space, the conjugate coincides with its associate space. Furthermore, it does not appear to be a simpler task to characterize the conjugate (i1 G3a Q32)* for a given cross-space 01 Ga 02 . At this point it seems advisable to mention.