Publishing house / Banach Center Publications / All volumes

Abstract

Two important examples of $q$-deformed commutativity relations are: $aa^*-qa^*a=1$, studied in particular by M. Bożejko and R. Speicher, and $ab=qba$, studied by T. H. Koornwinder and S. Majid. The second case includes the $q$-normality of operators, defined by S. Ôta ($aa^*=qa^*a$). These two frameworks give rise to different convolutions. In particular, in the second scheme, G. Carnovale and T. H. Koornwinder studied their $q$-convolution. In the present paper we consider another convolution of measures based on the so-called $(p,q)$-commutativity, a generalization of $ab=qba$. We investigate and compare properties of both convolutions (associativity, commutativity and positivity) and corresponding Fourier transforms.