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80
Free martingale polynomials
 Journal of Functional Analysis
"... ABSTRACT. In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. ..."
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Cited by 35 (3 self)
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ABSTRACT. In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials. Finally, we indicate how Rota’s finite operator calculus can be modified for the free context.
H∞ FUNCTIONAL CALCULUS AND SQUARE FUNCTIONS ON Noncommutative L^Pspaces
, 2006
"... In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semig ..."
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Cited by 30 (11 self)
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In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, qOrnsteinUhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
Appell polynomials and their relatives
 Int. Math. Res. Not
, 2004
"... ABSTRACT. This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials among polynomials in noncommuting va ..."
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Cited by 20 (1 self)
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ABSTRACT. This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials among polynomials in noncommuting variables. They also fit well into the framework of free probability. For the free Appell polynomials, a number of combinatorial and “diagram ” formulas are proven, such as the formulas for their linearization coefficients. An explicit formula for their generating function is obtained. These polynomials are also martingales for free Lévy processes. For more general free Sheffer families, a necessary condition for pseudoorthogonality is given. Another family investigated are the KailathSegall polynomials. These are multivariate polynomials, which share with the Appell polynomials nice combinatorial properties, but are always orthogonal. Their origins lie in the Fock space representations, or in the theory of multiple stochastic integrals. Diagram formulas are proven for these polynomials as well, even in the qdeformed case. 1.
qLévy processes
 J. Reine Angew. Math
, 2004
"... ABSTRACT. We continue the investigation of the Lévy processes on a qdeformed full Fock space started in [1]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals ..."
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Cited by 17 (1 self)
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ABSTRACT. We continue the investigation of the Lévy processes on a qdeformed full Fock space started in [1]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras. 1.
Factoriality of qgaussian von neumann algebras
 Comm. Math. Phys
"... We prove that the von Neumann algebras generated by n qGaussian elements, are factors for n � 2. 1 ..."
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Cited by 15 (1 self)
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We prove that the von Neumann algebras generated by n qGaussian elements, are factors for n � 2. 1
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 15 (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.
Strong Haagerup inequalities for free Rdiagonal elements
 J. FUNCT. ANAL
, 2007
"... In this paper, we generalize Haagerup’s inequality [H] (on convolution norm in the free group) to a very general context of Rdiagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its originial form. We give ..."
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Cited by 15 (6 self)
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In this paper, we generalize Haagerup’s inequality [H] (on convolution norm in the free group) to a very general context of Rdiagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its originial form. We give combinatorial proofs of two important special cases of our main result, and then generalize these techniques. En route, we prove a number of moment and cumulant estimates for Rdiagonal elements that are of independent interest. Finally, we use our strong Haagerup inequality to prove a strong ultracontractivity theorem, generalizing and improving the one in [Bi2].
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 13 (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