Topologies and bornologies determined by operator ideals, II

Let be an operator ideal on LCS's. A continuous seminorm p of a LCS X is said to be - continuous if $Q̃_p ∈ ^{inj}(X,X̃_p)$, where $X̃_p$ is the completion of the normed space $X_p = X/p^{-1}(0)$ and $Q̃_p$ is the canonical map. p is said to be a Groth()- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map $Q̃_{pq} : X̃_q → X̃_p$ belongs to $(X̃_q,X̃_p)$. It is well known that when is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous seminorm p of X is -continuous if and only if every continuous seminorm p of X is a Groth()-seminorm. In this paper, we extend this equivalence to arbitrary operator ideals and discuss several aspects of these constructions which were initiated by A. Grothendieck and D. Randtke, respectively. A bornological version of the theory is also obtained.