ABSTRACT
Theorem 11.4. Let P be a partial integral operator which acts between two ideal spaces X and Y. Then the associate operator P' of P is given by
(11.15)
lor any yE Y' with P'y E e. In particular, if the operator (11.1) is regular, then (11.15) holds for all y E Y'. o As was shown in the proof of Theorem 11.1, the operator ]P[ defined by (11.6) acts from X into 6 and is continuous. Consequently, the image N of the unit ball Ilzllx S 1 of X under ]P[ is a bounded subset of 5, and, hence, so is the set
(11.16) N =UHv: -]P[z S vS ]P[z}: IIzllx S I} (closure in 5). Denote by Y the ideal space whose unit ball coincides with the set (11.16); such a space may be easily constructed. By construction, the operator ]P[ acts from X into Y and is continuous.
