Hiai F., Kosaki H. - Means of Hilbert Space Operators.pdf - Analysis - Kuya

Lecture Notes in Mathematics

1820

Editors:

J.--M. Morel, Cachan

F. Takens, Groningen

B. Teissier, Paris

Berlin

Heidelberg

New York

Hong Kong

London

Milan

Paris

Tokyo

Fumio Hiai

Hideki Kosaki

Means of

Hilbert Space Operators

Authors

Fumio Hiai

Graduate School of Information Sciences

Tohoku University

Aoba-ku, Sendai

980-8579

Hideki Kosaki

Graduate School of Mathematics

Kyushu University

Higashi-ku, Fukuoka

812-8581

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Mathematics Subject Classification (2000):

47A30, 47A64, 15A60

ISSN

0075-8434

ISBN

3-540-40680-8

Springer-Verlag Berlin Heidelberg New York

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Springer-Verlag Berlin Heidelberg

2003

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e-mail: hiai@math.is.tohoku.ac.jp

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e-mail: kosaki@math.kyushu-u.ac.jp

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Preface

Roughly speaking two kinds of operator and/or matrix inequalities are known,

of course with many important exceptions. Operators admit several natural

notions of orders (such as positive semideﬁniteness order, some majorization

orders and so on) due to their non-commutativity, and some operator in-

equalities clarify these order relations. There is also another kind of operator

inequalities comparing or estimating various quantities (such as norms, traces,

determinants and so on) naturally attached to operators.

Both kinds are of fundamental importance in many branches of math-

ematical analysis, but are also sometimes highly non-trivial because of the

non-commutativity of the operators involved. This monograph is mainly de-

voted to means of Hilbert space operators and their general properties with

the main emphasis on their norm comparison results. Therefore, our operator

inequalities here are basically of the second kind. However, they are not free

from the ﬁrst in the sense that our general theory on means relies heavily on

a certain order for operators (i.e., a majorization technique which is relevant

for dealing with unitarily invariant norms).

In recent years many norm inequalities on operator means have been in-

vestigated. We develop here a general theory which enables us to treat them in

a uniﬁed and axiomatic fashion. More precisely, we associate operator means

to given scalar means by making use of the theory of Stieltjes double integral

transformations. Here, Peller’s characterization of Schur multipliers plays an

important role, and indeed guarantees that our operator means are bounded

operators. Basic properties on these operator means (such as the convergence

property and norm bounds) are studied. We also obtain a handy criterion (in

terms of the Fourier transformation) to check the validity of norm comparison

among operator means.

Sendai, June 2003

Fumio Hiai

Fukuoka, June 2003

Hideki Kosaki

Hiai F., Kosaki H. - Means of Hilbert Space Operators.pdf

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