Spectral Properties of Certain Operators on a Free Hilbert Space and the Semicircular Law

Presents the construction of free Hilbert-Space Operators and how they affect the Semicircular Law

Ilwoo Cho (Author), Hemen Dutta (Author)

9780443151750, Elsevier Science

Paperback / softback, published 26 April 2023

164 pages
23.5 x 19 x 1.2 cm, 0.45 kg

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.

1. Fundamentals 2. Semicircular Elements Induced by Orthogonal Projections 3. Semicircular Elements Induced by Projections On l2-Spaces 4. Jump Operators on Free Hilbert Spaces and Deformed Semicircular Laws 5. Shift Operators on Free Hilbert Spaces and Deformed Semicircular Laws 6. Jump-Shift Operators on Free Hilbert Spaces and Deformed Semicircular Laws

Subject Areas: Applied mathematics [PBW]