Some topologies related to nonseparable Hilbert spaces and their applications

Abstract

In this paper, we define some new topologies related to a nonseparable Hilbert space H, which are generalizations of the well-known strong operator topology (SOT), weak operator topology (WOT) on B(H) and the weak topology on H. We study properties of these new topologies, and obtain some results including that the generalized compact operators are dense (in the generalized SOT and WOT) in B(H). We also prove that a linear functional is continuous in the generalized strong operator topology if and only if it is continuous in the generalized weak operator topology. Finally, as an application of the generalized weak topology, we characterize continuous linear functionals which annihilate the generalized compact operators.