Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization

Abstract

Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it (e.g. $T={\mathbb{R}}^d$ and $\sigma$ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of $T$, with freely independent values. Such a process (field), $\omega =\omega (t)$, $t\in T$, is given a rigorous meaning through smearing out with test functions on $T$, with ${\int }_T\sigma (dt)f(t)\omega (t)$ being a (bounded) linear operator in a full Fock space. We define a set $\mathbf{CP}$ of all continuous polynomials of $\omega$, and then define a con-commutative $L^2$-space $L^2(\tau )$ by taking the closure of $\mathbf{CP}$ in the norm $\Vert P{\Vert }_{L^2(\tau )}:=\Vert P\Omega \Vert$, where $\Omega$ is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between $L^2(\tau )$ and a (Fock-space-type) Hilbert space $\mathbb{F}=\mathbb{R}\oplus {⨁}_{n=1}^{\infty }{L}^2(T^n,{\gamma }_n)$, with explicitly given measures ${\gamma }_n$. We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set $\mathbf{CP}$ invariant. (Note that, in the general case, the projection of a continuous monomial of oder $n$ onto the $n$-th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions $\lambda$ and $\eta \ge 0$ on $T$, such that, in the $\mathbb{F}$ space, $\omega$ has representation $ω(t)=\di_t^†+λ(t)\di_t^†\di_t+\di_t+η(t)\di_t^†\di^2_t$, where $\di_t^†$ and $\di_t$ are the usual creation and annihilation operators at point $t$.

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• Orthogonal polynomials with a resolvent-type generating function
• Free Meixner states
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Cited by in corpus (7)

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• Meixner class of non-commutative generalized stochastic processes with freely independent values II. The generating function
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• Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations
• Characterizations of some free random variables by properties of conditional moments of third degree polynomials
• Semigroups of distributions with linear Jacobi parameters
• Some Fock spaces with depth two action