Pseudo $S$-spectra of special operators in quaternionic Hilbert spaces

TL;DR

This paper introduces and analyzes the pseudo $S$-spectrum for quaternionic operators, extending pseudospectrum concepts from complex to quaternionic Hilbert spaces, with explicit computations and characterizations for special operator classes.

Contribution

It defines the pseudo $S$-spectrum for quaternionic operators, explores its properties, computes it for specific classes, and characterizes certain operators via this spectrum.

Findings

Pseudo $S$-spectrum generalizes complex pseudospectrum to quaternionic operators.

Explicit pseudo $S$-spectra are computed for special classes like normal and self-adjoint operators.

Characterization of left multiplication operators via pseudo $S$-spectrum circularization.

Abstract

For a bounded quaternionic operator $T$ on a right quaternionic Hilbert space $\mathcal{H}$ and $\varepsilon >0$, the pseudo $S$-spectrum of $T$ is defined as \begin{align*} \Lambda_{\varepsilon}^{S}(T) := \sigma_S (T) \bigcup \left \{ q \in \mathbb{H}\setminus \sigma_S(T):\; \|\Delta_{q}(T)^{-1}\| \geq \frac{1}{\varepsilon} \right\}, \end{align*} where $\mathbb{H}$ denotes the division ring of quaternions, $\sigma_S(T)$ is the $S$-spectrum of $T$ and $\Delta_q(T)= T^2-2 \text{Re}(q)T+|q|^2I$. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo $S$-spectrum and explicitly compute the pseudo $S$-spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of $S$-functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number $r$ if and only if for every $\varepsilon>0$ the pseudo $S$-spectrum of the operator is the circularization of a closed disc in the complex plane centered at $r$ with the radius $\sqrt{\varepsilon}$. Further, we propose a $G_1$-condition for quaternionic operators and prove some results in this setting.