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Drazin spectrum of operator matrices on the Banach space. (English) Zbl 1157.47004
An operator $T:X\rightarrow X$ on a complex Banach space $X$ is called Drazin invertible if there exists another operator $S$ on $X$ such that $ST=TS$, $STS=S$ and $T^{k+1}S=T^k$ for some nonnegative integer $k$. The Drazin spectrum of $T$ is the set $\sigma_D(T) :=\{\lambda\in \mathbb{C} : \lambda - T\text{ is not Drazin invertible}\}$. Given two complex Banach spaces $X$ and $Y$, the authors consider operators on the product space $X \times Y$ defined by a $2\times 2$ upper triangular matrix $M_C$ with $A$ and $B$ in the diagonal and $C$ in the upper right entry. They show that $\sigma_D(A)\cup \sigma_D(B)=\sigma_D(M_C) \cup W$, where $W$ is the union of certain holes in $\sigma_D(M_C)$ contained in $\sigma_D(A)\cap \sigma_D(B)$. Moreover, they study the set $\bigcap_{C\in B(Y,X)}\sigma_D(M_C)$.

47A10Spectrum and resolvent of linear operators
47A55Perturbation theory of linear operators
Full Text: DOI
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