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