In the context of a Banach space X endowed with a right quaternionic structure, we examine a bounded right linear operator T defined on X. We introduce and investigate the spherical local spectrum of T, which serves as the quaternionic counterpart to the local spectrum of bounded operators on complex Banach spaces. To generalize the concept of the single-valued extension property originally formulated in the complex setting to the quaternionic framework, we adopt the definition provided by Colombo et al. (Spectral theory on the S-spectrum for quaternionic operators. Birkhäuser, Basel, 2018). This definition posits the existence of a unique right slice hyperholomorphic extension of the pseudo-resolvent <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R_{s}(T)=\mathcal {Q}_{s}(T)(x\bar{s}-Tx)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>s</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi mathvariant="script">Q</mi> <mi>s</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mover accent="true"> <mrow> <mi>s</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>-</mo> <mi>T</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for any axially symmetric open neighborhood <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {U} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">U</mi> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho _{s}(T)\subseteq \mathcal {U}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mi>s</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊆</mo> <mi mathvariant="script">U</mi> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, leveraging the properties of the quaternionic right Banach space, we establish and prove several fundamental results analogous to those in the complex setting, extending them to the quaternionic context. This systematic approach enables the development of local spectral theory within quaternionic right Banach spaces. Additionally, we introduce and analyze key subsets of X that are pivotal to quaternionic local spectral theory. These subsets encompass the spherical local spectral subspace of T, the spherical global spectral subspace of T, and the spherical quasi-nilpotent part of T.

Spherical Local Spectrum in Quaternionic Banach Spaces

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发表日期：

2025年11月16日

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Abdelkhalek El Amrani,
Aziz Blali,
Mohammed Drissi-Alami

摘要

In the context of a Banach space X endowed with a right quaternionic structure, we examine a bounded right linear operator T defined on X. We introduce and investigate the spherical local spectrum of T, which serves as the quaternionic counterpart to the local spectrum of bounded operators on complex Banach spaces. To generalize the concept of the single-valued extension property originally formulated in the complex setting to the quaternionic framework, we adopt the definition provided by Colombo et al. (Spectral theory on the S-spectrum for quaternionic operators. Birkhäuser, Basel, 2018). This definition posits the existence of a unique right slice hyperholomorphic extension of the pseudo-resolvent $R_{s}(T)=\mathcal {Q}_{s}(T)(x\bar{s}-Tx)$ $R_{s} (T) = Q_{s} (T) (x \bar{s} - T x)$ for any axially symmetric open neighborhood $\mathcal {U} $ $U$ such that $\rho _{s}(T)\subseteq \mathcal {U}$ $ρ_{s} (T) \subseteq U$ . Furthermore, leveraging the properties of the quaternionic right Banach space, we establish and prove several fundamental results analogous to those in the complex setting, extending them to the quaternionic context. This systematic approach enables the development of local spectral theory within quaternionic right Banach spaces. Additionally, we introduce and analyze key subsets of X that are pivotal to quaternionic local spectral theory. These subsets encompass the spherical local spectral subspace of T, the spherical global spectral subspace of T, and the spherical quasi-nilpotent part of T.