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Local operator theory, random matrices and Banach spaces. (English) Zbl 1067.46008
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 1. Amsterdam: Elsevier (ISBN 0-444-82842-7/hbk). 317-366 (2001).
This paper surveys some highlights of the connection between Banach space theory on the one hand and random matrices and local operator theory on the other. The last topic pertains to aspects of operator theory which reduce to the study of the asymptotic behaviour of parameters associated with finite matrices. The connection between Banach spaces and the other two topics is sometimes by application of one of the fields to solve problems in the other and sometimes by analogies and transfer of methods. The paper is divided into two parts. The first deals with local operator theory; among the subjects discussed are: 1. Discussion of the problem (solved affirmatively by {\it H.-X. Lin} [Fields Inst. Commun. 13, 193--233 (1997; Zbl 0881.46042)]) of whether the smallness of the commutant of two Hermitian matrices implies that they are close to a commuting pair of matrices. 2. When and in what sense is it true that if two matrices are close together then their spectra are close together as well. 3. The paving problem: Given $\varepsilon>0$, is there a $k$ such that for all matrices $T$ of norm one (as an operator from $\ell_2^n$ to itself) and with diagonal zero, so one can find a decomposition of the canonical basis into $k$ subsets such that $\sum P_iTP_i$ is at most $\varepsilon$, where $P_i$ is the orthogonal projection onto the span of the $i$-s subset? This problem is still open. {\it J. Bourgain} and {\it L. Tzafriri} [J. Reine Angew. Math. 420, 1--43 (1991; Zbl 0729.47028)] have some highly nontrivial results in this direction. The second part deals with some topics involving random matrices: Estimates of the asymptotic distribution of the norm and of the spectrum of some ensembles of random matrices, including discussions on the Wigner semicircle law, deviation inequalities for the singular values of a Gaussian matrix, the Slepian--Gordon lemma and its application for estimating norms of Gaussian matrices. There is also a section on the connection between random matrices and free probability. Some corrections and additions to this paper appear in Vol. 2 of the same handbook. For the entire collection see [Zbl 0970.46001].

46B09Probabilistic methods in Banach space theory
60G42Martingales with discrete parameter
60G46Martingales and classical analysis
46-00Reference works (functional analysis)