Spectral Properties of Certain Operators on a Free Hilbert Space and the Semicircular Law - by Ilwoo Cho & Hemen Dutta (Paperback)

Book Synopsis

In Spectral Properties of Certain Operators on a Free Hilbert Space and the Semicircular Law, the authors consider the so-called free Hilbert spaces, which are the Hilbert spaces induced by the usual l² Hilbert spaces and operators acting on them. The construction of these operators itself is interesting and provides new types of Hilbert-space operators. Also, by considering spectral-theoretic properties of these operators, the authors illustrate how "free-Hilbert-space" Operator Theory is different from the classical Operator Theory. More interestingly, the authors demonstrate how such operators affect the semicircular law induced by the ONB-vectors of a fixed free Hilbert space. Different from the usual approaches, this book shows how "inside" actions of operator algebra deform the free-probabilistic information--in particular, the semicircular law.

Review Quotes

"The present monograph considers semicircular elements whose free distributions are the semicircular law induced by some mutually orthogonal projections on the free Hilbert space induced by fixed multi Hilbert spaces.... This motivates the authors of the present monograph to introduce certain operators which preserve or distort the semicircular law.... The book will be interesting and useful for mathematicians working in this area." --Ali Talebi, zbMATHOpen