Direct integrals of matrices

TL;DR

This paper demonstrates that certain linear operators on separable Hilbert spaces can be uniquely represented as direct integrals of irreducible matrices, simplifying their decomposition and multiplicity analysis.

Contribution

It introduces a unique representation of operators as direct integrals of matrices, enhancing the understanding of their structure and decomposition.

Findings

Unique representation of operators as direct integrals of matrices

Simplification of prime and central decomposition

Development of multiplicity theory for such operators

Abstract

It is shown that each linear operator on a separable Hilbert space which generates a finite type I von Neumann algebra has, up to unitary equivalence, a unique representation as a direct integral of inflations of mutually unitary inequivalent irreducible matrices. This leads to a simplification of the so-called prime (or central) decomposition and the multiplicity theory for such operators. The concept of so-called p-isomorphisms between special classes of such operators is discussed. All results are formulated in more general settings; that is, for tuples of closed densely defined operators affiliated with finite type I von Neumann algebras.