A Short-type Decomposition Of Forms

TL;DR

This paper introduces a decomposition theorem for nonnegative sesquilinear forms, generalizing the operator short, and explores applications to operators, set functions, and positive functionals.

Contribution

It presents a novel short-type decomposition theorem for sesquilinear forms, extending the concept of operator short to a broader mathematical context.

Findings

Decomposition of forms into shorted and singular parts.

Analogous results for bounded positive operators.

Applications to additive set functions and positive functionals.

Abstract

The main purpose of this paper is to present a decomposition theorem for nonnegative sesquilinear forms. The key notion is the short of a form to a linear subspace. This is a generalization of the well-known operator short defined by M. G. Krein. A decomposition of a form into a shorted part and a singular part (with respect to an other form) will be called short-type decomposition. As applications, we present some analogous results for bounded positive operators acting on a Hilbert space; for additive set functions on a ring of sets; and for representable positive functionals on a $\ast$-algebra.