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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