Operators of the $q$-oscillator

Volume 78 / 2007

Franciszek Hugon Szafraniec Banach Center Publications 78 (2007), 293-307 MSC: Primary 47B20, 81S05. DOI: 10.4064/bc78-0-22


We scrutinize the possibility of extending the result of [19] to the case of $q$-deformed oscillator for $q$ real; for this we exploit the whole range of the deformation parameter as much as possible. We split the case into two depending on whether a solution of the commutation relation is bounded or not. Our leitmotif is subnormality. The deformation parameter $q$ is reshaped and this is what makes our approach effective. The newly arrived parameter, the operator $C$, has two remarkable properties: it separates in the commutation relation the annihilation and creation operators from the deformation as well as it $q$-commutes with those two. This is why introducing the operator $C$ may have far-reaching consequences.


  • Franciszek Hugon SzafraniecInstytut Matematyki
    Uniwersytet Jagielloński
    Reymonta 4
    30-059 Kraków, Poland

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