Generalized Drazin spectrum of upper triangular matrices in Banach algebras

Abstract

Let \mathcal{A} be a Banach algebra with unit e and a,b,c\in \mathcal{A},M_c=\left(\begin{array}{cc}a\hfill & c\hfill \\ 0\hfill & b\hfill \end{array}\right)\in M_2\left(\mathcal{A}\right). The concepts of left and right generalized Drazin invertible of elements in a Banach algebra are proposed. A generalized Drazin spectrum of \alpha is defined by {\sigma }_{\mathrm{g}\mathrm{D}}\left(\alpha \right)=\{\lambda \in \mathbb{C}:\alpha -\lambda e\mathrm{i}\mathrm{s}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d}\mathrm{D}\mathrm{r}\mathrm{a}\mathrm{z}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}\}. It is shown that

　　　　　　　　　　 {\sigma }_{\mathrm{g}\mathrm{D}}\left(a\right)\cup {\sigma }_{\mathrm{g}\mathrm{D}}\left(b\right)={\sigma }_{\mathrm{g}\mathrm{D}}\left(M_c\right)\cup W_2,

where W_g is a union of certain holes {\sigma }_{\mathrm{g}\mathrm{D}} and W_g\subseteq {\sigma }_{\mathrm{g}\mathrm{D}}\left(a\right)\cap {\sigma }_{\mathrm{g}\mathrm{D}}\left(b\right), or more finely W_g\subseteq {\sigma }_{\mathrm{r}\mathrm{g}\mathrm{D}}\left(a\right)\cap {\sigma }_{\mathrm{l}\mathrm{g}\mathrm{D}}\left(b\right). In addition, some properties of generalized Drazin spectrum of elements in a Banach algebra are studied.

Key words： Banach algebra; generalized Drazin inverse; generalized Drazin spectrum; upper; triangular matrices

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