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

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

1st Edition - February 1, 2023

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  • Authors: Ilwoo Cho, Hemen Dutta
  • Paperback ISBN: 9780443151750

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In Spectral Properties of Certain Operators on a Free Hilbert Space and the Semicircular Law, the authors consider 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 previous theory showing how the Semicircular Law is deformed by actions "from outside"
  • Explores free Hilbert Spaces and their modeling applications
  • Authored by two leading researchers in Operator Theory and Operator Algebra


Researchers in computational modelling, mathematicians, Computer Scientists, as well as researchers in Biomedical Engineering

Table of Contents

  • Chapter-1: Introduction
    1.1. Motivation
    1.2. Overview

    Chapter-2:  Free Hilbert Space

    Chapter-3: Jump Operators on Free Hilbert Spaces
    3.1. Operators J in B(Hj,  Hi)
    3.2. Jumps in B(Hj,  Hi) for a fixed permutation
    3.3. Jump Operators on a Free Hilbert Space

    Chapter-4: Semicircular Elements Induced by a Free Hilbert Space
    4.1. Free Probability and Semicircular Elements
    4.2. The Free Hilbert Space induced by l2(Z)
    4.3. Semicircular Elements Induced by l2(Z)
    4.4. Semicircular Elements Induced by {H1,…,HN}

    Chapter-5: Jump Operators and Semicircularity
    5.1. The Case where ni = nj
    5.2. The Case where ni < nj

    Chapter-6: The Semicircular Law on LQ [H1,…,HN]

    Chapter-7: Shift Operators on Free Hilbert Spaces
    7.1. Shiftings on ONBs
    7.2. The n-th Shifts on Hk
    7.3. The Shift Operator on a Free Hilbert Space

    Chapter-8: Shift Operators and Semicircularity
    8.1. The Case where nk < infinity
    8.2. The Case where nk = infinity

    Chapter-9: Shift Operators Acting on LQ [H1,…,HN]

    Chapter-10: Jump-Shift Operators on Free Hilbert Spaces

    Chapter-11: Jump-Shift Operators and Semicircularity
    11.1. Banach-Space Operators Induced by Jump-Shift Operators
    11.2. The Case where ni = nj < infinity
    11.3. The Case where ni = nj = infinity
    11.4. The Case where ni < nj

Product details

  • Language: English
  • Copyright: © Academic Press 2023
  • Published: February 1, 2023
  • Imprint: Academic Press
  • Paperback ISBN: 9780443151750

About the Authors

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

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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