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

- 1st Edition - April 21, 2023
- Authors: Ilwoo Cho, Hemen Dutta
- Language: English
- Paperback ISBN:9 7 8 - 0 - 4 4 3 - 1 5 1 7 5 - 0
- eBook ISBN:9 7 8 - 0 - 4 4 3 - 1 5 1 7 6 - 7

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

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Request a sales quote*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

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 l^{2}-Spaces

3.1. The Cloned C*-Algebra of B (l^{2} (N_{0}))

3.2. Semicircular Elements Induced by *Q*3.3. Semicircular Elements Induced by

*A*x

*B*(l

^{2}(N

_{0}))

3.4. Applications

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

4.1. Hilbert Spaces *F* [H_{1}, ..., H_{N}]

4.2. Jump Operators on *F* [H_{1}, ..., H_{N}]

4.3. Semicircular Elements Induced by *F* [H_{1}, ..., H_{N}]

4.4. Jump Operators of B (*F*) and Semicircularity on L_{Q} [H_{1}, ..., H_{N}]

4.5. The Semicircular Law on L_{Q} [H_{1}, ..., H_{N}]

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

5.1. Shift Operators on *F* [H_{1}, ..., H_{N}]

5.2. Shift Operators on *F* and Semicircularity on L_{Q} [H_{1}, ..., H_{N}]

5.3. Shift Operators of B (*F*) Acting on L_{Q} [H_{1}, ..., H_{N}]

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

6.1. Jump-Shift Operators on *F* [H_{1}, ..., H_{N}]

6.2. Jump-Shift Operators on *F* and Semicircularity on L_{Q} [H_{1}, ..., H_{N}]

- No. of pages: 164
- Language: English
- Edition: 1
- Published: April 21, 2023
- Imprint: Academic Press
- Paperback ISBN: 9780443151750
- eBook ISBN: 9780443151767

IC

### Ilwoo Cho

HD

### Hemen Dutta

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