This paper examines the pseudo S-spectrum of a bounded right quaternionic operator A defined on a right quaternionic Hilbert space. In this context, a generalized spectrum <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\left( \text{ defined } \text{ for } ~\varepsilon>0~ \text{ as }~\sigma _{\varepsilon }^S(A):=\sigma ^S(A)\bigcup \Big \{\textbf{q}\in \mathbb {H}:~\Vert R_\textbf{q}(A) ^{-1}\Vert >\frac{1}{\varepsilon }\Big \} \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mspace width="0.333333em" /> <mtext>defined</mtext> <mspace width="0.333333em" /> <mspace width="0.333333em" /> <mtext>for</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="3.33333pt" /> <mspace width="0.333333em" /> <mtext>as</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <msubsup> <mi>σ</mi> <mrow> <mi>ε</mi> </mrow> <mi>S</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <msup> <mi>σ</mi> <mi>S</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>⋃</mo> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">{</mo> </mrow> <mi mathvariant="bold">q</mi> <mo>∈</mo> <mi mathvariant="double-struck">H</mi> <mo>:</mo> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">‖</mo> <msub> <mi>R</mi> <mi mathvariant="bold">q</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">‖</mo> </mrow> <mo>></mo> <mfrac> <mn>1</mn> <mi>ε</mi> </mfrac> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">}</mo> </mrow> </mfenced> </math></EquationSource> </InlineEquation> is employed as the pseudo S-spectrum. We will concentrate on the case where A is compact to more clearly define the properties of this pseudo S-spectrum.

Findings on the pseudo S-spectra of quaternionic operators within a right quaternionic Hilbert space

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发表日期：

2026年5月8日

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Bilel Saadaoui

摘要

This paper examines the pseudo S-spectrum of a bounded right quaternionic operator A defined on a right quaternionic Hilbert space. In this context, a generalized spectrum $\left( \text{ defined } \text{ for } ~\varepsilon>0~ \text{ as }~\sigma _{\varepsilon }^S(A):=\sigma ^S(A)\bigcup \Big \{\textbf{q}\in \mathbb {H}:~\Vert R_\textbf{q}(A) ^{-1}\Vert >\frac{1}{\varepsilon }\Big \} \right) $ $(defined for ε > 0 as σ_{ε}^{S} (A) : = σ^{S} (A) ⋃ {q \in H : ‖ R_{q} {(A)}^{- 1} ‖ > \frac{1}{ε}})$ is employed as the pseudo S-spectrum. We will concentrate on the case where A is compact to more clearly define the properties of this pseudo S-spectrum.