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

By Ilwoo Cho and Hemen Dutta

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.

• 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

• Language: English
• Publisher: Academic Press
• Release date: Apr 21, 2023
• ISBN: 9780443151767

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.

Book preview

Preface

The main purpose of this monograph is to consider the semicircular law induced by the canonical rank-1 mutually orthogonal projections on multi separable Hilbert spaces. Starting from a fixed separable Hilbert space, containing countably many orthonormal-basis elements, we construct the corresponding mutually orthogonal rank-1 projections. From the family of such projections, mutually free, semicircular elements are constructed and studied, as elements of C*-algebra elements. From these semicircular elements whose free distributions are the semicircular law, we study how the semicircularity (or, semicircular property) of them is preserved, or distorted, by certain actions. Our main results show how such semicircularity is deformed by certain natural actions, classified by the jump operators, the shift operators, and the jump-shift operators, on so-called the free Hilbert space induced by fixed multi Hilbert spaces. And the deformations are characterized.

The book is presented in 6 chapters and each chapter contains several sections. Chapter 1 consists of Sections 1.1–1.3, Chapter 2 consists of Sections 2.1–2.3, Chapter 3 consists of Sections 3.1–3.4, Chapter 4 consists of Sections 4.1–4.5, Chapter 5 consists of Sections 5.1–5.3, and Chapter 6 consists of Sections 6.1–6.2. Some sections are further divided into subsections to make the presentation more understandable to readers. We highlight some major aspects of the book section wise below.

In Section 1.2, we briefly introduce combinatorial free probability of Speicher. And semicircular elements whose free distributions are the semicircular law are considered in Section 1.3. In Sections 2.1, 2.2, and 2.3, we consider how to construct semicircular elements, as elements of certain Banach *-probability spaces, from a fixed family of mutually orthogonal, integer-many projections in a C*-probability space.

In the following Sections 3.1 and 3.2, based on the main results of the previous sections, we study how to construct semicircular elements from the mutually orthogonal rank-1 projections induced by the orthonormal basis elements of a separable infinite-dimensional Hilbert space, which is isomorphic to the usual -Hilbert space. In Section 3.3, we enlarge the semicircularity of Sections 3.1 and 3.2 under tensor-product with an arbitrarily chosen C*-probability space. Special interesting cases are considered in Section 3.4.

In Section 4.1, our main object of this monograph, free Hilbert spaces induced by multi separable Hilbert spaces are introduced and studied. In Sections 4.2, 4.3, and 4.4, jump operators on a free Hilbert space are introduced-and-studied. In particular, the operator-theoretic spectral properties of them are characterized. Motivated by the main results of Sections 3.1 and 3.2, the semicircularity induced by mutually orthogonal projections on the free Hilbert space is considered in Section 4.5. In particular, we show how our jump operators of Section 4.2 deform the semicircular law induced by the projections.

In Section 5.1, a new type of operators on our free Hilbert spaces, called shift operators, is introduced. And the operator-theoretic spectral properties of them are characterized. In Sections 5.2 and 5.3, we show how the shift operators of Section 5.1 deform the semicircular law induced by the projections on our free Hilbert spaces. The deformations are characterized case-by-case.

By using our jump operators and shift operators, a new type of operators, called jump-shift operators, is defined in Section 6.1. We characterize how these operators affect the semicircular law in Section 6.2.

The action of operators introduced here affects the semicircularity induced by the mutually orthogonal projections on the free Hilbert space. It illustrates how certain inside actions deform the semicircular law.

It is expected that readers will find this book useful and interesting, and any constructive criticism will be taken seriously. The authors are grateful to their family members, colleagues, friends, and well-wishers for their encouragement and support in developing this book. The authors are also grateful to Elsevier's editors and support staff for providing timely support.

1: Fundamentals

Abstract

In this chapter, we introduce motivations and preliminaries of our works. We here briefly review combinatorial free probability theory. In particular, the importance and usages of semicircular elements, whose free distributions are the semicircular law, are reviewed.

Keywords

Hilbert spaces; Operators; Free probability; Free distributions; Free moments; Free cumulants; Semicircular elements; The semicircular law

1.1 Introduction

In this monograph, we study spectral properties of three types of (bounded linear) operators on a Hilbert space,

generated by N-many multi -Hilbert spaces,

equipped with their orthonormal bases (in short, ONBs, from below),

for k = 1, ..., N, for N ∈ ∖ {1}. In other words, the Hilbert space is induced by the multi -dimensional separable Hilbert spaces , for k = 1, ..., N.

In particular, we are interested in operators on assigning ONB vectors to other ONB vectors, preserving patterns of them. We here not only characterize the operator-theoretic spectral properties of such operators in the operator algebra , which is a -algebra with its operator-norm topology, but also show how these operators affect the semicircular law induced by some ONB vectors of .

Recall that the spectral properties on the operator algebra on an arbitrary Hilbert space H are characterized by the types of them, i.e.,

(i)  an operator T is said to be self-adjoint, if ;

(ii)  an operator T is normal, if ;

(iii)  an operator T is an isometry, if ;

(iv)  an operator T is a unitary, if ;

(v)  an operator T is a partial isometry, if is a projection on H, where an operator P is a projection on H, if it is both self-adjoint and idempotent in the sense that on H;

etc., where is the adjoint of T in . The spectra of such operators are well-characterized generally. So, by characterizing the spectral types (or, the spectral properties) of operators, one can verify the spectra of operators in a universal manner (e.g., see [14]).

Meanwhile, free probability allows us to study statistical properties of operators under fixed linear functionals. In particular, the semicircular law is the noncommutative analogue of the Gaussian distribution (or, the normal distribution) in classical statistical analysis. Whenever a linear functional on a topological ⁎-algebra (e.g., a -algebra, or a von Neumann algebra, or a Banach ⁎-algebra), studying semicircular elements, whose free distributions are the semicircular law, is one of the most interesting topics in free probability theory.

1.1.1 Motivation

For more about fundamental operator theory and its applications, e.g., see [9], [14], and [21], and see [12], [11], [2], [10], [17], [18], [22], [23], [24], and [13] for connections between classical measure theory and free probability theory. Also, for more about semicircularity of operators, or the semicircular law, see [1], [3], [15], [16], [19], [20], and [25].

In [5] and [4], the first-named author considered how to construct semicircular elements, whose free distributions are the semicircular law, from mutually orthogonal -many projections in a -probability space, and studied the corresponding Banach ⁎-algebras generated by such semicircular elements. As an application, the first named author and Jorgensen constructed semicircular elements, as Banach-space operators (in the sense of [9] acting on an operator algebra , from mutually orthogonal -many rank-1 projections in [8], where H is the -dimensional Hilbert space. Independently, the joint free distributions of mutually free, multi semicircular elements are not only re-characterized, but also estimated and asymptotically estimated in [7], by computing the joint free moments of them combinatorially. By using the computational techniques of [7], the main results of [6] demonstrate that if a topological ⁎-algebra is generated by multi, mutually free semicircular elements, and if the free distributions of the generators (which are the semicircular elements) are preserved by certain multiplicative actions, then the original free-distributional data are preserved by such