# Journal of Operator Theory

Volume 84, Issue 1, Summer 2020  pp. 67-97.

Poisson type operators on the Fock space of type $B$ and in the Blitvic model

Authors:  Wiktor Ejsmont
Author institution:Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland

Summary: Biane proposed a new statistic on set partitions which he called restricted crossings. In a series of papers Anshelevich showed that this statistic is an essential tool to investigate stochastic processes on $q$-Fock space. In particular, Anshelevich constructed operators whose moments count restricted crossings and used these operators to develop a beautiful theory of noncommutative $q$-Levy processes. In the present paper following Anshelevich we define gauge operators on $(\alpha,q)$-Fock and cumulants which are governed by statistics on partitions of type $B$.

DOI: http://dx.doi.org/10.7900/jot.2018dec30.2247
Keywords: noncommutative probability, $q$-Poisson type operators, Fock spaces