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

  • 1st Edition - April 21, 2023
  • Latest edition
  • Authors: Ilwoo Cho, Hemen Dutta
  • Language: English

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… Read more

Description

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.

Key features

  • Presents the spectral properties of three types of operators on a Hilbert space, in particular how these operators affect the semicircular law
  • Demonstrates how the semicircular law is deformed by actions "from inside", as opposed to actions "from outside" considered by previous theory
  • Explores free Hilbert spaces and their modeling applications
  • Authored by two leading researchers in Operator Theory and Operator Algebra

Readership

The primary audience includes researchers in computational modelling, mathematicians, Computer Scientists, as well as researchers in Biomedical Engineering.
Other interested audiences will be comprised of researchers in scientific computing.

Table of contents

1. Fundamentals

1.1. Introduction

1.2. Free Probability

1.3. Semicircular Elements



2. Semicircular Elements Induced by Orthogonal Projections

2.1. The Banach *-Algebra Induced by Orthogonal Projections

2.2. Weighted-Semicircular Elements Induced by Q

2.3. Semicircular Elements Induced by Q

3. Semicircular Elements Induced by Projections On l2-Spaces

3.1. The Cloned C*-Algebra of B (l2 (N0))

3.2. Semicircular Elements Induced by Q

3.3. Semicircular Elements Induced by A x B (l2 (N0))

3.4. Applications

4. Jump Operators on Free Hilbert Spaces and Deformed Semicircular Laws

4.1. Hilbert Spaces F [H1, ..., HN]

4.2. Jump Operators on F [H1, ..., HN]

4.3. Semicircular Elements Induced by F [H1, ..., HN]

4.4. Jump Operators of B (F) and Semicircularity on LQ [H1, ..., HN]

4.5. The Semicircular Law on LQ [H1, ..., HN]

5. Shift Operators on Free Hilbert Spaces and Deformed Semicircular Laws

5.1. Shift Operators on F [H1, ..., HN]

5.2. Shift Operators on F and Semicircularity on LQ [H1, ..., HN]

5.3. Shift Operators of B (F) Acting on LQ [H1, ..., HN]

6. Jump-Shift Operators on Free Hilbert Spaces and Deformed Semicircular Laws

6.1. Jump-Shift Operators on F [H1, ..., HN]

6.2. Jump-Shift Operators on F and Semicircularity on LQ [H1, ..., HN]

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

Product details

  • Edition: 1
  • Latest edition
  • Published: April 21, 2023
  • Language: English

About the authors

IC

Ilwoo Cho

Dr. Ilwoo Cho is a Professor in the Department of Mathematics and Statistics at St. Ambrose University, Davenport, Iowa, USA. He holds a PhD in Mathematics from the University of Iowa. His research is focused in the areas of Free Probability, Operator Theory, Operator Algebra, Noncommutative Dynamical Systems, and Combinatorics. He has contributed chapters to several books, including Methods of Mathematical Modelling and Computation for Complex Systems, Springer; New Directions in Function Theory: From Complex to Hypercomplex to Noncommutative, Birkhäuser; Nonlinear Analysis: Problems, Applications and Computational Methods, Springer; Complex Function Theory, Operator Theory, Schur Analysis and Systems Theory, Birkhäuser; and Mathematical Methods and Modelling in Applied Sciences, Springer.
Affiliations and expertise
Professor Department of Mathematics and Statistics, St. Ambrose University, USA

HD

Hemen Dutta

Dr. Hemen Dutta PhD is a Professor at Gauhati University, India. He also served three other higher learning academic institutions in different capacities prior to joining the Gauhati University. His current research interests are in the areas of nonlinear analysis and mathematical modeling. He is a regular and guest editor of several international indexed journals. He has published 25 books, including Mathematical Modelling and Analysis of Infectious Diseases, New Trends in Applied Analysis and Computational Mathematics, Current Trends in Mathematical Analysis and Its Interdisciplinary Applications from Springer, Concise Introduction to Basic Real Analysis, Topics in Contemporary Mathematical Analysis and Applications, and Mathematical Methods in Engineering and Applied Sciences from CRC Press, and Fractional Order Analysis: Theory, Methods and Applications from Wiley, among others. Dr. Dutta is also an honorary research affiliate and speaker for several international and national events.
Affiliations and expertise
Professor, Department of Mathematics, Gauhati University, India

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