In this paper, we introduce and study the class of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$g_z$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation>-invertible linear relations in Banach spaces. We develop this concept as a natural extension of generalized Drazin invertibility introduced in [<CitationRef CitationID="CR15">15</CitationRef>]. A linear relation T is called <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$g_z$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation>-invertible if there exists <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(M, N) \in \operatorname {Red}(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo>Red</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$T_M$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>M</mi> </msub> </math></EquationSource> </InlineEquation> is an invertible linear relation and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$T_N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> is a zeroloid operator. Among other characterizations, we show that T is <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$g_z$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation>-invertible if and only if 0 does not belong to the set of accumulation points of accumulation points of its spectrum. Using tools from local spectral theory, we investigate, as an application, the stability of the generalized Drazin-zeroloid spectrum for a closed, everywhere defined linear relation T under perturbations by power finite rank and quasinilpotent operators.

On $g_z$-invertible Multivalued Linear Operators and Their Perturbations

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发表日期：

2026年6月29日

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Melik Lajnef,
Teresa Álvarez

摘要

In this paper, we introduce and study the class of $g_z$ $g_{z}$ -invertible linear relations in Banach spaces. We develop this concept as a natural extension of generalized Drazin invertibility introduced in [15]. A linear relation T is called $g_z$ $g_{z}$ -invertible if there exists $(M, N) \in \operatorname {Red}(T)$ $(M, N) \in Red (T)$ such that $T_M$ $T_{M}$ is an invertible linear relation and $T_N$ $T_{N}$ is a zeroloid operator. Among other characterizations, we show that T is $g_z$ $g_{z}$ -invertible if and only if 0 does not belong to the set of accumulation points of accumulation points of its spectrum. Using tools from local spectral theory, we investigate, as an application, the stability of the generalized Drazin-zeroloid spectrum for a closed, everywhere defined linear relation T under perturbations by power finite rank and quasinilpotent operators.

On \(g_z\)-invertible Multivalued Linear Operators and Their Perturbations

摘要