Characterization of the condition S-spectrum of a compact operator in a right quaternionic Hilber space

We define a new type of spectrum, called the condition pseudo S-spectra of linear operators in a right quaternionic Hilbert space as σεS(T):={q∈H:‖Qq(T)‖‖Qq(T)-1‖>1ε}⋃{q∈H:‖Qq(T)‖‖Qq(T)-1‖=1ε}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _\varepsilon ^S(T):=\Big \{{\mathbf {q}}\in {\mathbb {H}}: \Vert Q_{{\mathbf {q}}}(T)\Vert \Vert Q_{{\mathbf {q}}}(T)^{-1}\Vert >\frac{1}{\varepsilon }\Big \}\bigcup \Big \{{\mathbf {q}}\in {\mathbb {H}}:\Vert Q_{{\mathbf {q}}}(T)\Vert \Vert Q_{{\mathbf {q}}}(T)^{-1}\Vert =\frac{1}{\varepsilon }\Big \}.$$\end{document} This is expected to be useful in solving operator equations. The goal of this paper consists of establishing a necessary and sufficient condition for the characterization of the condition spectrum of a compact operator in a right quaternionic Hilbert space.