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

 

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
Published electronically: December 28, 2000
MathSciNet review: 1823910
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Abstract: Let $T$ be a compact operator on a Hilbert space such that the operators $A = \frac{1}{2} (T + T^{*})$ and $B = \frac{1}{2i}(T-T^{*})$ are positive. Let $\{ s_{j}\} $ be the singular values of $T$ and $\{ \alpha _{j}\} , \{ \beta _{j}\} $ the eigenvalues of $A,B$, all enumerated in decreasing order. We show that the sequence $\{ s^{2}_{j}\} $is majorised by $\{ \alpha ^{2}_{j} + \beta ^{2}_{j}\} $. An important consequence is that, when $p \ge 2, ~\Vert T\Vert ^{2}_{p}$ is less than or equal to $\Vert A\Vert ^{2}_{p} + \Vert B\Vert ^{2}_{p}$, and when $ 1\le p \le 2,$ 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: http://dx.doi.org/10.1090/S0002-9939-00-05832-9
PII: S 0002-9939(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