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Remarks on Catalan and super-Catalan numbers

Anna Dorota Krystek, Łukasz Jan Wojakowski (2011)

Banach Center Publications

In this article we discuss the Catalan and super-Catalan (or Schröder) numbers. We start with some combinatorial interpretations of those numbers. We study two probability measures in the context of free probability, one whose moments are super-Catalan, and another, whose even moments are super-Catalan and odd moments are zero. With the use of the latter we also show some new formulae for evaluation of the Catalans in terms of super-Catalans and vice-versa.

Remarks on conditional moments of the free deformed Poisson random variables

Wiktor Ejsmont (2011)

Banach Center Publications

We will show that the conditional first moment of the free deformed Poisson random variables (q = 0) corresponding to operators fulfilling the free relation is a linear function of the regression and the conditional variance also is a linear function of the regression. For this purpose we will first demonstrate some properties of the Wick product and then we will concentrate on the free deformed Poisson random variables.

Remarks on q-CCR relations for |q| > 1

Marek Bożejko (2007)

Banach Center Publications

In this paper we give a construction of operators satisfying q-CCR relations for q > 1: A ( f ) A * ( g ) - A * ( g ) A ( f ) = q N f , g I and also q-CAR relations for q < -1: B ( f ) B * ( g ) + B * ( g ) B ( f ) = | q | N f , g I , where N is the number operator on a suitable Fock space q ( ) acting as Nx₁ ⊗ ⋯ ⊗ xₙ = nx₁ ⊗ ⋯ ⊗xₙ. Some applications to combinatorial problems are also given.

Remarks on the boolean convolution and Kerov's α-transformation

Anna Dorota Krystek (2006)

Banach Center Publications

This paper consists of two parts. The first part is devoted to the study of continuous diagrams and their connections with the boolean convolution. In the second part we investigate the rectangular Young diagrams and respective discrete measures. We recall the definition of Kerov's α-transformation of diagrams, define the α-transformation of finitely supported discrete measures and generalize the notion of the α-transformation.

Restrictions of CP-semigroups to maximal commutative subalgebras

Franco Fagnola, Michael Skeide (2007)

Banach Center Publications

We give a necessary and sufficient criterion for a normal CP-map on a von Neumann algebra to admit a restriction to a maximal commutative subalgebra. We apply this result to give a far reaching generalization of Rebolledo's sufficient criterion for the Lindblad generator of a Markov semigroup on ℬ(G).

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