Strongly irreducible factorization of quaternionic operators and Riesz decomposition theorem

Abstract

Let $\mathcal {H}$ be a right quaternionic Hilbert space and let T be a bounded quaternionic normal operator on $\mathcal {H}$. In this article, 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 the norm $\Vert K\Vert < \delta$, a partial isometry W and a strongly irreducible operator S on $\mathcal {H}$ such that

$$\begin{aligned} T = (W+K)S. \end{aligned}$$

We illustrate our result with an example. In addition, we discuss the quaternionic version of the Riesz decomposition theorem and obtain a consequence that if the S-spectrum of a bounded (need not be normal) quaternionic operator is disconnected by a pair of disjoint axially symmetric closed subsets, then the operator is strongly reducible.