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Positive definiteness of functions with applications to operator norm inequalities
About this Title
Hideki Kosaki, Faculty of Mathematics Kyushu University 744 Motooka Nishi-ku Fukuoka 819-039, Japan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 212, Number 997
ISBNs: 978-0-8218-5307-8 (print); 978-1-4704-0614-1 (online)
DOI: https://doi.org/10.1090/S0065-9266-2010-00616-1
Published electronically: November 29, 2010
Keywords: Fourier transform,
Heinz inequality,
Hilbert space operator,
operator arithmetic-geometric mean inequality,
operator mean,
norm inequality,
positive definite function,
positive operator,
unitarily invariant norm
MSC: Primary 47A63, 47A64; Secondary 15A42, 15A60, 47A30
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Fourier transforms and positive definiteness
- 4. A certain Heinz-type inequality and related commutator estimates
- 5. Norm comparison for various operator means
- 6. Norm inequalities for $H^{\frac {1}{2}+\beta }XK^{\frac {1}{2}-\beta }+ H^{\frac {1}{2}-\beta }XK^{\frac {1}{2}+\beta }\pm H^{1/2}XK^{1/2}$
- 7. Norm comparison of Heron-type means and related topics
- 8. Operator Lehmer means and their properties
- A. A direct proof for Proposition 7.3
- B. Proof for Theorem 7.10
Abstract
Positive definiteness is determined for a wide class of functions relevant in the study of operator means and their norm comparisons. Then, this information is used to obtain an abundance of new sharp (unitarily) norm inequalities comparing various operator means and sometimes other related operators.- T. Ando, Matrix Young inequalities, Operator theory in function spaces and Banach lattices, Oper. Theory Adv. Appl., vol. 75, Birkhäuser, Basel, 1995, pp. 33–38. MR 1322498
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