Drazin spectrum of operator matrices on the Banach space
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 \(\bigcap_{C\in B(Y,X)}\sigma_D(M_C)\).
scientific article; zbMATH DE number 5349171
scientific article