Local spectral theory for 2 × 2 operator matrices

H. Elbjaoui and E. H. Zerouali

International Journal of Mathematics and Mathematical Sciences, 2003, vol. 2003, 1-6

Abstract:

We discuss the spectral properties of the operator M C ∈ ℒ ( X ⊕ Y ) defined by M C : = ( A C 0 B ) , where A ∈ ℒ ( X ) , B ∈ ℒ ( Y ) , C ∈ ℒ ( Y , X ) , and X , Y are complex Banach spaces. We prove that ( S A ∗ ∩ S B ) ∪ σ ( M C ) = σ ( A ) ∪ σ ( B ) for all C ∈ ℒ ( Y , X ) . This allows us to give a partial positive answer to Question 3 of Du and Jin (1994) and generalizations of some results of Houimdi and Zguitti (2000). Some applications to the similarity problem are also given.

Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:893151

DOI: 10.1155/S0161171203012043

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