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Perturbation of Spectrums of 2 × 2 Operator Matrices

Du Hong-Ke and Pan Jin
Proceedings of the American Mathematical Society
Vol. 121, No. 3 (Jul., 1994), pp. 761-766
DOI: 10.2307/2160273
Stable URL: http://www.jstor.org/stable/2160273
Page Count: 6
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Perturbation of Spectrums of 2 × 2 Operator Matrices
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Abstract

In this paper, we study the perturbation of spectrums of 2 × 2 operator matrices such as $M_C = \begin{bmatrix}A & C \\ 0 & B\end{bmatrix}$ on the Hilbert space $H \oplus K$. For given A and B, we prove that $\bigcap_{C \in B(K, H)} \sigma(M_C) = \sigma_\pi(A) \cup \sigma_\delta(B) \cup \{\lambda \in C: n(B - \lambda) \neq d(A - \lambda)\},$ where σ(T), σπ(T), σδ(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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