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An operator inequality and self-adjointness

✍ Scribed by Bojan Magajna; Marko Petkovšek; Aleksej Turnšek

Publisher
Elsevier Science
Year
2004
Tongue
English
Weight
218 KB
Volume
377
Category
Article
ISSN
0024-3795

No coin nor oath required. For personal study only.

✦ Synopsis

Given bounded positive invertible operators A and B on a Hilbert space H, it is shown that the inequality AXA -1 + B -1 XB 2 X holds for all bounded operators X of rank 1 if and only if B = f (A) for some increasing function f satisfying a certain simple inequality, which in the case when the spectrum of A is connected implies that B is a scalar multiple of A. As an application some consequences of the Corach-Porta-Recht type inequality in operator ideals are studied.

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