Diagonalization and representation results for nonpositive sesquilinear form measures

TL;DR

This paper investigates how to decompose sesquilinear form measures into positive parts and explores their diagonal vector expansions, connecting these decompositions to trace class measures and Naimark dilations.

Contribution

It introduces a novel approach to decompose sesquilinear form measures into positive components using an auxiliary Hilbert space, linking to dilation theory.

Findings

Decomposition of sesquilinear form measures into positive parts.

Representation of measures as trace class valued measures.

Relations established with Naimark dilations and direct integrals.

Abstract

We study decompositions of operator measures and more general sesquilinear form measures $E$ into linear combinations of positive parts, and their diagonal vector expansions. The underlying philosophy is to represent $E$ as a trace class valued measure of bounded variation on a new Hilbert space related to $E$. The choice of the auxiliary Hilbert space fixes a unique decomposition with certain properties, but this choice itself is not canonical. We present relations to Naimark type dilations and direct integrals.