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79
Integral Transform and SegalBargmann Representation associated to qCharlier Polynomials
, 2001
"... Let µ (q) be the qdeformed Poisson measure in the sense of SaitohYoshida [24] and νp be the measure given by Equation (3.6). In this short paper, we introduce the qdeformed analogue of the SegalBargmann transform associated with µ (q) p. We prove that our SegalBargmann transform is a unitary m ..."
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Cited by 8 (7 self)
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Let µ (q) be the qdeformed Poisson measure in the sense of SaitohYoshida [24] and νp be the measure given by Equation (3.6). In this short paper, we introduce the qdeformed analogue of the SegalBargmann transform associated with µ (q) p. We prove that our SegalBargmann transform is a unitary map of L²(µ (q) p) onto the Hardy space H²(νq). Moreover, we give the SegalBargmann representation of the multiplication opera), which is a linear combination of the qcreation, qannihilation, qnumber, and scalar operators. tor by x in L²(µ (q) p
Hypercontractivity in noncommutative holomorphic spaces
 Commun. Math. Phys
, 2005
"... ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of noncommutative “holomorphic ” algebras. Our setting is the qGaussian algebras Γq associated to the qFock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a qSeg ..."
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Cited by 8 (6 self)
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ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of noncommutative “holomorphic ” algebras. Our setting is the qGaussian algebras Γq associated to the qFock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a qSegalBargmann transform, and prove Janson’s strong hypercontractivity L 2 (Hq) → L r (Hq) for r an even integer. 1.
Rosenthal type inequalities for free chaos
, 2005
"... Let A denote the reduced amalgamated free product of a family ..."
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Cited by 8 (5 self)
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Let A denote the reduced amalgamated free product of a family
P.: Quadratic bosonic and free white noises
 Commun. Math. Phys
, 2000
"... Abstract. We discuss the meaning of renormalization used for deriving quadratic bosonic commutation relations introduced by Accardi [ALV] and find a representation of these relations on an interacting Fock space. Also, we investigate classical stochastic processes which can be constructed from nonco ..."
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Cited by 7 (0 self)
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Abstract. We discuss the meaning of renormalization used for deriving quadratic bosonic commutation relations introduced by Accardi [ALV] and find a representation of these relations on an interacting Fock space. Also, we investigate classical stochastic processes which can be constructed from noncommutative quadratic white noise. We postulate quadratic free white noise commutation relations and find their representation on an interacting Fock space. 1.
Symmetric Hilbert spaces arising from species of structures
"... Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' K are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial s ..."
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Cited by 6 (2 self)
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Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' K are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. Any such species F gives rise to an endofunctor F of the category of Hilbert spaces with contractions mapping a Hilbert space K to a symmetric Hilbert space F (K) with the same symmetry as the species F . A general framework for annihilation and creation operators on these spaces is developed, and compared to the generalised Brownian motions of R. Speicher and M. Bo _ zejko. As a corollary we nd that the commutation relation a i a j a j a i = f(N) ij with Na i a i N = a i admits a realization on a symmetric Hilbert space whenever f has a power series with infinite radius of convergence and positive coefficients.
Linearization coefficients for orthogonal polynomials using stochastic processes. Annals of Probability 33
, 2005
"... ABSTRACT. Given a basis for a polynomial ring, the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients. These coefficients have combinatorial significance for many classical families of orthogonal polynomials. Starting with ..."
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ABSTRACT. Given a basis for a polynomial ring, the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients. These coefficients have combinatorial significance for many classical families of orthogonal polynomials. Starting with a stochastic process and using the stochastic measures machinery introduced by Rota and Wallstrom, we calculate and give an interpretation of linearization coefficients for a number of polynomial families. The processes involved may have independent, freely independent, or qindependent increments. The use of noncommutative stochastic processes extends the range of applications significantly, allowing us to treat Hermite, Charlier, Chebyshev, free Charlier, and Rogers and continuous big qHermite polynomials. We also show that the qPoisson process is a Markov process. 1.
The bi  Poisson process: a quadratic harness
 Annals of Probability
"... This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a twoparameter ..."
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Cited by 6 (3 self)
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This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a twoparameter extension of the AlSalam–Chihara polynomials and a relation between these polynomials for different values of parameters. 1. Introduction. The
Quadratic harnesses, qcommutations, and orthogonal martingale polynomials
 Trans. Amer. Math. Soc
, 2007
"... Abstract. We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a qcommutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical ..."
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Abstract. We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a qcommutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest. 1.
Rdiagonal dilation semigroups
"... ABSTRACT. This paper addresses extensions of the complex OrnsteinUhlenbeck semigroup to operator algebras in free probability theory. If a1,..., ak are ∗free Rdiagonal operators in a II1 factor, then Dt(ai1 · · · ain) = e −nt ai1 · · · ain defines a dilation semigroup on the nonselfadjoint ..."
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Cited by 4 (4 self)
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ABSTRACT. This paper addresses extensions of the complex OrnsteinUhlenbeck semigroup to operator algebras in free probability theory. If a1,..., ak are ∗free Rdiagonal operators in a II1 factor, then Dt(ai1 · · · ain) = e −nt ai1 · · · ain defines a dilation semigroup on the nonselfadjoint operator algebra generated by a1,..., ak. We show that Dt extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by a1,..., ak. Moreover, we show that Dt satisfies an optimal ultracontractive property: �Dt: L 2 → L ∞ � ∼ t −1 for small t> 0. 1.
Classical versions of qGaussian processes: conditional moments and Bell’s inequality
 Comm. Math. Physics
, 2001
"... We show that classical processes corresponding to operators what satisfy a qcommutative relation have linear regressions and quadratic conditional variances. From this we deduce that Bell’s inequality for their covariances can be extended from q = −1 to the entire range −1 ≤ q < 1. 1 ..."
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We show that classical processes corresponding to operators what satisfy a qcommutative relation have linear regressions and quadratic conditional variances. From this we deduce that Bell’s inequality for their covariances can be extended from q = −1 to the entire range −1 ≤ q < 1. 1