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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Perturbation of spectrums of $ 2\times 2$ operator matrices


Authors: Hong Ke Du and Pan Jin
Journal: Proc. Amer. Math. Soc. 121 (1994), 761-766
MSC: Primary 47A10; Secondary 47A62
MathSciNet review: 1185266
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Abstract: In this paper, we study the perturbation of spectrums of $ 2 \times 2$ operator matrices such as $ {M_C} = \left[ {\begin{array}{*{20}{c}} A & C \\ 0 & B \\ \end{array} } \right]$ on the Hilbert space $ H \oplus K$. For given A and B, we prove that

$\displaystyle \bigcap\limits_{C \in B(K,H)} {\sigma ({M_C}) = {\sigma _\pi }(A)... ...ma _\delta }(B) \cup \{ \lambda \in C:n(B - \lambda ) \ne d(A - } \lambda )\} ,$

where $ \sigma (T),{\sigma _\pi }(T),{\sigma _\delta }(T),n(T)$, and $ d(T)$ denote the spectrum of T, approximation point spectrum, defect spectrum, nullity, and deficiency, respectively. Some related results are obtained.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1185266-2
PII: S 0002-9939(1994)1185266-2
Keywords: Perturbation of spectrums, $ 2 \times 2$ operator matrices
Article copyright: © Copyright 1994 American Mathematical Society