Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces

TL;DR

This paper develops a comprehensive framework for classifying and decomposing finite systems of closed densely defined operators in Hilbert spaces, introducing new theorems, ideals, and a prime decomposition approach.

Contribution

It introduces new decomposition theorems, ideals, and a prime decomposition for $N$-tuples of operators, advancing the understanding of their structure and classification.

Findings

Established algebraic and order properties of $CDD_N$

Proposed a prime decomposition with uniqueness

Introduced a new partial order for $N$-tuples

Abstract

An \textit{ideal} of $N$-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for $N$-tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class $CDD_N$ of all unitary equivalence classes of such $N$-tuples are established and certain ideals in $CDD_N$ are distinguished. It is proved that infinite operations in $CDD_N$ may be reconstructed from the direct sum operation of a pair. \textit{Prime decomposition} in $CDD_N$ is proposed and its (in a sense) uniqueness is established. The issue of classification of ideals in $CDD_N$ (up to isomorphism) is discussed. A model for $CDD_N$ is described and its concrete realization is presented. A new partial order of $N$-tuples of operators is introduced and its fundamental properties are established. Extremal importance of unitary disjointness of $N$-tuples and the way how it `tidies up' the structure of $CDD_N$ are emphasized.