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Spectral properties of certain operators on the free Hilbert space ℑ [H1, . . . , HN] and the semicircular law

Cho Ilwoo

Opuscula Mathematica

2021

Vol. 41, no. 6

755-803

In this paper, we fix N -many l2-Hilbert spaces Hk whose dimensions are [formula] for k=1,…, N, for N ∈N\{1}. And then, construct a Hilbert space ℑ = ℑ [H1 , . . . , HN] induced by H1 , . . . , HN, and study certain types of operators on ℑ. In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by [formula], where Bk are the orthonormal bases of Hk , for k = 1, . . . , N.

Banach *-algebras generated by semicircular elements induced by certain orthogonal projections

Cho I., Jorgensen P. E. T.

Opuscula Mathematica

2018

Vol. 38, no. 4

501--535

The main purpose of this paper is to study structure theorems of Banach *-algebras generated by semicircular elements. In particular, we are interested in the cases where given semicircular elements are induced by orthogonal projections in a C*-probability space.

Semicircular elements induced by p-adic number fields

Cho I., Jorgensen P. E. T.

Opuscula Mathematica

2017

Vol. 37, no. 5

665--703

In this paper, we study semicircular-like elements, and semicircular elements induced by p-adic analysis, for each prime p. Starting from a p-adic number field Qp, we construct a Banach *-algebra [formula], for a fixed prime p, and show the generating elements Qpj of [formula] form weighted-semicircular elements, and the corresponding scalar-multiples Θpj of Qpj become semicircular elements, for all j ∈ Z. The main result of this paper is the very construction of suitable linear functionals [formula] on [formula], making Qpj be weighted-semicircular, for all j ∈ Z.