Local spectral theory for 2×2 operator matrices
We discuss the spectral properties of the operator MC∈ℒ(X⊕Y) defined by MC:=(AC0B), where A∈ℒ(X), B∈ℒ(Y), C∈ℒ(Y,X), and X, Y are complex Banach spaces. We prove that (SA∗∩SB)∪σ(MC)=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 gen...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203012043 |
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| Summary: | We discuss the spectral properties of the operator MC∈ℒ(X⊕Y) defined by MC:=(AC0B), where A∈ℒ(X), B∈ℒ(Y), C∈ℒ(Y,X), and X, Y are complex Banach spaces. We prove that (SA∗∩SB)∪σ(MC)=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. |
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| ISSN: | 0161-1712 1687-0425 |