Convolutions related to $q$-deformed commutativity

Tom 89 / 2010

Anna Kula Banach Center Publications 89 (2010), 189-200 MSC: Primary 28D05, 46L53; Secondary 05A30, 42A85. DOI: 10.4064/bc89-0-11


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.


  • Anna KulaInstitute of Mathematics
    Jagiellonian University
    Łojasiewicza 4
    30-348 Kraków, Poland

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