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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A note on the operator $ X\rightarrow AX-XB$

Author: L. Fialkow
Journal: Trans. Amer. Math. Soc. 243 (1978), 147-168
MSC: Primary 47A50; Secondary 47A10
MathSciNet review: 502900
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Abstract: If A and B are bounded linear operators on an infinite dimensional complex Hilbert space $ \mathcal{H}$, let $ \tau (X)\, = \,AX\, - \,XB$ (X in $ \mathcal{L}(\mathcal{H})$). It is proved that $ \sigma (\tau )\, = \,\sigma (\tau \vert{C_p})\,(1\, \leqslant \,p\, \leqslant \infty )$, where, for $ 1\, \leqslant p\, < \,\infty $, $ {C_p}$ is the Schatten p-ideal, and $ {C_\infty }$ is the ideal of all compact operators in $ \mathcal{L}(\mathcal{H})$. Analogues of this result for the parts of the spectrum are obtained and sufficient conditions are given for $ \tau $ to be injective. It is also proved that if A and B are quasisimilar, then the right essential spectrum of A intersects the left essential spectrum of B.

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Keywords: Operator equations, Rosenblum's Theorem, quasisimilarity, essential spectrum
Article copyright: © Copyright 1978 American Mathematical Society