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Variations of Weyl Type Theorems for Upper Triangular Operator Matrices

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Let \(\mathcal X\) be a Banach space and let T be a bounded linear operator on \(\mathcal {X}\). We denote by S(T) the set of all complex \(\lambda \in \mathcal {C}\) such that T does not have the single-valued extension property. In this paper it is shown that if MC is a 2 × 2 upper triangular operator matrix acting on the Banach space \(\mathcal {X} \oplus \mathcal {Y}\), then the passage from σLD(A) ∪ σLD(B) to σLD(MC) is accomplished by removing certain open subsets of σd(A) ∩ σLD(B) from the former, that is, there is the equality σLD(A) ∪ σLD(B) = σLD(MC) ∪ℵ, where ℵ is the union of certain of the holes in σLD(MC) which happen to be subsets of σd(A) ∩ σLD(B). Generalized Weyl’s theorem and generalized Browder’s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how generalized Weyl’ theorem, generalized Browder’s theorem, generalized a-Weyl’s theorem and generalized a-Browder’s theorem survive for 2 × 2 upper triangular operator matrices on the Banach space.

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  1. Department of Mathematics & Statistics, Mu’tah University Al-Karak-Jordan, Faculty of Science P. O. Box (7), Mu’tah, Jordan

    Mohammad H. M. Rashid

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Rashid, M.H.M. Variations of Weyl Type Theorems for Upper Triangular Operator Matrices. Acta Math Vietnam 46, 719–735 (2021). https://doi.org/10.1007/s40306-021-00431-4

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