ABSTRACT
Since the limit, as n -+ 00, of the left-hand side of (11.18) exists for any z E X, the sequence (P#U"),, is X-weakly Cauchy in X'. But the space X' is X -weakly complete (see ZABREJKO [1966,1968]), and hence p#U" -+ Ply for some y E Y'. Since this function yobviously satisfies
(P:r,y) = (z,P'y) (z E X,y E Y'), we have shown that the associate operator P' of P also exists if we consider P as an operator from X into Y. Now let y E Y' and P#,I E 6. Then JP#[ 1111 E 6 and, by Lebesgue's theorem, the sequence (P#U"),, converges (in 6) to P#U. But, by what has been proved before, the sequence (P#Yn)n converges as well X-weakly to P'y. We conclude that P'fI = p#y, and so we are done.•
11.4. Algebras of partial integral operators Given two ideal spaces X and Y, denote by .c(X, Y) the space of all bounded linear operators and by .c"(X, Y) the space of all regular bounded linear operators between X and Y. Similarly, by £p(X, Y) and .c;(X, Y) (where the subscript p stands for "partial integral operator") we denote the space of all operators and regular operators, respectively, of the form (11.1). Theorems 11.1 and 11.2 state that £p(X, Y) ~ .c(X, Y) and .c;(X, Y) ~ .c"(X,Y). H .c(X, Y) and .c"(X, Y) are equipped with the usual operator norm, the subspaces £p(X, Y) and .c;(X, Y) are not closed. However, if we consider .c"(X,Y) with the norm (11.19) I!PII.crex,Y) := II IPI lI.c(x,Y) , where IPI is the module of P as in Subsection 11.2, then .c"(X, Y) becomes a Banach space in the norm (11.19), and .c;(X,Y) is closed in .c"(X,Y). In order to state more precise results, some auxiliary definitions are in order. These definitions are the same as in § 3 and § 4, but now for functions of two variables. Recall that, given two ideal spaces X and Y over T x 5, by Y/X we denote the multiplicator space of all functions e such that ez E Y for
all x EX. This is an ideal space with norm
(11.20) IIclly/x =sup {llczlly : IIxllx ~ 1}. Further, by !'t,(X,Y),mm(X,Y) and !'tn(X,Y) we denote the sets of all measurable functions I = 'et, 8, r) on T x S x T, m =met, 8,0') on T X S X S, and n = net, 8, r, 0') on T X S X T X S, respectively, such that 'et, 8, r) =0 for 8 E Sri. and met, 8, 0') =0 for t E TrI.. All these three sets are ideal spaces equipped with the norms
(11.21)
(11.22)
and
(11.23)
1111I!Jt,(X,Y) = sup 11'/I(.,.,r)x(r,·)ldp(r)/1 '1I~lIxSl IT Y
IlnlltJtn(x.Y) = sup II' In(.,.,r,O')z(r,O')ld(p X JI)(r,O')I/ '1I~lIxSl lTxS Y
respectively. Of course, these definitions are formulated just in such a way that the kernel norms (11.21), (11.22) and (11.23) coincide with the operator norms of the corresponding (regular) integral operators, i.e.
