The aim of this paper is to introduce and characterize generalized Drazin invertible operators relative to a regularity <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> (generalized Drazin-<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> invertible operators, for short) to the context of closed operators. Also, by considering <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(T(t))_{t \ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> a <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$C_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-semigroup of operators on a Banach space X and A its infinitesimal generator, we study the following spectral inclusions <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$e^{t\sigma _{\mathcal{D}\mathcal{R}}(A)} \subseteq \sigma _{\mathcal{D}\mathcal{R}}(T(t))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>e</mi> <mrow> <mi>t</mi> <msub> <mi>σ</mi> <mrow> <mi mathvariant="script">D</mi> <mi mathvariant="script">R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> <mo>⊆</mo> <msub> <mi>σ</mi> <mrow> <mi mathvariant="script">D</mi> <mi mathvariant="script">R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\sigma _{\mathcal{D}\mathcal{R}}(T(t)) \subseteq e^{t \sigma _{\mathcal{D}\mathcal{R}}(A)} \cup \{0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mrow> <mi mathvariant="script">D</mi> <mi mathvariant="script">R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>⊆</mo> <msup> <mi>e</mi> <mrow> <mi>t</mi> <msub> <mi>σ</mi> <mrow> <mi mathvariant="script">D</mi> <mi mathvariant="script">R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> <mo>∪</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\sigma _{\mathcal{D}\mathcal{R}}(.)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mrow> <mi mathvariant="script">D</mi> <mi mathvariant="script">R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> represents the spectrum relative to the class of generalized Drazin-<InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> invertible operators. Finally, we provide new applications of the concept of closed generalized Drazin-<InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> invertible operators in relation with delay differential equations and also with second-order partial differential equations.

Generalized drazin-$\mathcal {R}$ inverses, spectral inclusions for $C_{0}$-semigroups and applications on differential equations

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2026年3月23日

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Othman Abad,
Hamid Boua

摘要

The aim of this paper is to introduce and characterize generalized Drazin invertible operators relative to a regularity $\mathcal {R}$ $R$ (generalized Drazin- $\mathcal {R}$ $R$ invertible operators, for short) to the context of closed operators. Also, by considering $(T(t))_{t \ge 0}$ ${(T (t))}_{t \geq 0}$ a $C_{0}$ $C_{0}$ -semigroup of operators on a Banach space X and A its infinitesimal generator, we study the following spectral inclusions $e^{t\sigma _{\mathcal{D}\mathcal{R}}(A)} \subseteq \sigma _{\mathcal{D}\mathcal{R}}(T(t))$ $e^{t σ_{D R} (A)} \subseteq σ_{D R} (T (t))$ and $\sigma _{\mathcal{D}\mathcal{R}}(T(t)) \subseteq e^{t \sigma _{\mathcal{D}\mathcal{R}}(A)} \cup \{0\}$ $σ_{D R} (T (t)) \subseteq e^{t σ_{D R} (A)} \cup {0}$ , where $\sigma _{\mathcal{D}\mathcal{R}}(.)$ $σ_{D R} (.)$ represents the spectrum relative to the class of generalized Drazin- $\mathcal {R}$ $R$ invertible operators. Finally, we provide new applications of the concept of closed generalized Drazin- $\mathcal {R}$ $R$ invertible operators in relation with delay differential equations and also with second-order partial differential equations.

Generalized drazin-\(\mathcal {R}\) inverses, spectral inclusions for \(C_{0}\)-semigroups and applications on differential equations

摘要