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Some exponential inequalities for semisimple Lie groups. (English) Zbl 1192.15008

Ball, Joseph A. (ed.) et al., Topics in operator theory. Volume 1: Operators, matrices and analytic functions. Proceedings of the 19th international workshop on operator theory and applications (IWOTA), College of William and Mary, Williamsburg, VA, USA, July 22–26, 2008. A tribute to Israel Gohberg on the occasion of his 80th birthday. Basel: Birkhäuser (ISBN 978-3-0346-0157-3/hbk; 978-3-0346-0163-4/set; 978-3-0346-0158-0/ebook). Operator Theory: Advances and Applications 202, 539-552 (2010).
Summary: Let \(|||\cdot|||\) be any given unitarily invariant norm. We obtain some exponential relations in the context of semisimple Lie group. On one hand they extend the inequalities (1) \(|||e^A|||\leq |||e^{\operatorname{Re}A}|||\) for all \(A\in\mathbb C_{n\times n}\), where \(\operatorname{Re}A\) denotes the Hermitian part of \(A\), and (2) \(|||e^{A+B}|||\leq llle^Ae^B|||\)A, where \(A\) and \(B\) are \(n\times n\) Hermitian matrices. On the other hand, the inequalities of Weyl, Ky Fan, Golden-Thompson, Lenard-Thompson, Cohen, and So-Thompson are recovered. Araki’s relation on \((e^{A/2}e^Be^{A/2})^r\)A and \(e^{rA/2}e^{rB}e^{rA/2}\), where \(A\), \(B\) are Hermitian and \(r\in\mathbb R\), is extended.
For the entire collection see [Zbl 1181.47002].

MSC:

15A45 Miscellaneous inequalities involving matrices
22E46 Semisimple Lie groups and their representations
15A42 Inequalities involving eigenvalues and eigenvectors
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