Compact operators whose real and imaginary parts are positive

Authors:
Rajendra Bhatia and Xingzhi Zhan

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2277-2281

MSC (2000):
Primary 47A30, 47B10; Secondary 15A18, 15A60

DOI:
https://doi.org/10.1090/S0002-9939-00-05832-9

Published electronically:
December 28, 2000

MathSciNet review:
1823910

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact operator on a Hilbert space such that the operators and are positive. Let be the singular values of and the eigenvalues of , all enumerated in decreasing order. We show that the sequence is majorised by . An important consequence is that, when is less than or equal to , and when this inequality is reversed.

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Additional Information

**Rajendra Bhatia**

Affiliation:
Indian Statistical Institute, New Delhi 110 016, India

Email:
rbh@isid.ac.in

**Xingzhi Zhan**

Affiliation:
Institute of Mathematics, Peking University, Beijing 100871, China

Email:
zhan@sxx0.math.pku.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-00-05832-9

Keywords:
Compact operator,
positive operator,
singular values,
eigenvalues,
majorisation,
Schatten $p$-norms

Received by editor(s):
January 5, 1999

Received by editor(s) in revised form:
November 20, 1999

Published electronically:
December 28, 2000

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society