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.
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