Two application of nets

TL;DR

This paper explores two applications of nets: extending the Bochner integral to broader spaces and characterizing trace class operators via nets of principal minors, advancing integration theory and operator classification.

Contribution

It introduces a generalized integration theory for vector-valued functions and characterizes determinant class operators using nets of minors, linking to trace class operators.

Findings

Extended Bochner integral to locally convex spaces

Characterization of trace class operators via nets of minors

Normal operators' determinant class condition coincides with trace class condition

Abstract

Two applications of nets are given. The first is an extension of the Bochner integral to arbitrary locally convex spaces, leading to an integration theorye of more general vector valued functions then in the classical approach by Gelfand and Pettis. The second application starts with the observation that an operator on a Hilbert space is trace class if and only if the net of "principal trace minors" converges. The notion of a "determinant class operator" then is defined as one for which the net of determinantal principal minors converges. It is shown that for a normal operator A this condition coincides with 1-A being trace class.