On The Uniqueness of The Strongly Irreducible Decompositions of Operators up to Similarity

TL;DR

This paper generalizes the Jordan canonical form theorem to a class of bounded linear operators on complex separable Hilbert spaces, focusing on the uniqueness of strongly irreducible decompositions up to similarity.

Contribution

It introduces a new framework for understanding the uniqueness of strongly irreducible decompositions using direct integrals.

Findings

Established conditions for the uniqueness of decompositions

Extended Jordan form concepts to broader operator classes

Provided a new perspective on operator similarity

Abstract

We give a generalization of the Jordan canonical form theorem for a class of bounded linear operators on complex separable Hilbert spaces in terms of direct integrals. Precisely, we study the uniqueness of strongly irreducible decompositions of the operators on the Hilbert spaces up to similarity.