Topologies and bornologies determined by operator ideals, II

Abstract

The notion of topologies, introduced by Stephani[10], is useful for studying the injective hull of an operator ideal. Using Randtke’s idea(see [8,p.90] or [12,(3.2.1)and(3.2.7)]), we can characterize generating topologies in terms of seminorms which satisfy some expected properties(see Lemma3.3 and Theorems3.4 and 3.9). By a well-known and useful ideal of Grothendieck, the dual notions of generating topologies and ideal-topologies, the so-called generating bornologies, are given and studied in Sect.4. In terms of ideal-bornogies, the surjective hull of an operator ideal on Banach spaces is given(see Lemma4.8 and Theorem4.10). In terms of these two dual concepts, we are able to classify locally convex spaces, and to study their dual results. For instance, we show that if A is a symmetric(resp.completely symmetric) operator ideal on Bnanch spaces then a Banach space E is an A-topological space(A-bornological space) if and only if its Banach dual space E ′ is A-bornological(resp.A-topological)(Theorem 5.9). Also we are able to define the most natural and the most applicable type of operator ideals on LCS ′ s, namely the G − B-operators. This is an extension of the notions of quasiSchwartz operators defined by Randtke[8,p.91] and of cone-prenuclear maps defined by Wong[12,p.142]. Sufficient conditions are given to ensure that the G−B-operators from an injective(resp.surjective) operator ideal(see Propositions 6.3 and 6.4). Finally, we point out that a formula concerning with the injective hull of a bounded operator ideal, given by Franco and Pineiro[3, Theorem 1 in Sect.2], is not true(see Example 6.6)