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Drazin spectrum of operator matrices on the Banach space. (English) Zbl 1157.47004

An operator T:XX 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 σ D (T):={λ:λ-TisnotDrazininvertible}.

Given two complex Banach spaces X and Y, the authors consider operators on the product space X×Y defined by a 2×2 upper triangular matrix M C with A and B in the diagonal and C in the upper right entry. They show that σ D (A)σ D (B)=σ D (M C )W, where W is the union of certain holes in σ D (M C ) contained in σ D (A)σ D (B). Moreover, they study the set CB(Y,X) σ D (M C ).


MSC:
47A10Spectrum and resolvent of linear operators
47A55Perturbation theory of linear operators
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