Strongly irreducible factorization of quaternionic operators and Riesz decomposition theorem

TL;DR

This paper establishes a new factorization method for quaternionic normal operators involving strongly irreducible components and extends the Riesz decomposition theorem to quaternionic operators, revealing conditions for strong reducibility.

Contribution

It introduces a strongly irreducible factorization for quaternionic normal operators and proves a quaternionic Riesz decomposition theorem, extending classical operator theory to quaternionic settings.

Findings

Any quaternionic normal operator can be approximated by a strongly irreducible operator plus a small compact perturbation.

Disconnection of the spherical spectrum implies strong reducibility of the operator.

The paper provides an example illustrating the factorization result.

Abstract

Let $T$ be a bounded quaternionic normal operator on a right quaternionic Hilbert space $\mathcal{H}$. We show that $T$ can be factorized in a strongly irreducible sense, that is, for any $\delta >0$ there exist a compact operator $K$ with $\|K\|< \delta$, a partial isometry $W$ and a strongly irreducible operator $S$ on $\mathcal{H}$ such that \begin{equation*} T = (W+K) S. \end{equation*} We illustrate our result with an example. We also prove a quaternionic version of the Riesz decomposition theorem and as a consequence, show that if the spherical spectrum of a bounded quaternionic operator (need not be normal) is disconnected by a pair of disjoint axially symmetric closed subsets, then it is strongly reducible.