Radial Bargmann representation for the Fock space of type B (Q2795525)

The desire to construct new Fock type spaces by modifying the inner product of the usual full Fock space culminated in the general definitions of so-called interacting Fock spaces by \textit{L. Accardi}, \textit{Y. G. Lu} and \textit{I. V. Volovich} [Interacting Fock spaces and Hilbert module extensions of the Heisenberg commutation relations. Kyoto: IIAS Publications (1997)] (building directly on the idea of modifying the inner product) and by \textit{L. Accardi} and \textit{M. Skeide} [Commun. Stoch. Anal. 2, No. 3, 423--444 (2008; Zbl 1331.46051)] (concentrating more on the structure of the arising Fock-type space). So-called \(q\)-deformed Fock spaces and their ramifications are among the most prominent example classes. Their history goes back at least as far as to \textit{M. Arik} and \textit{D. D. Coon} [J. Math. Phys. 17, 524--526 (1976; Zbl 0941.81549)] who discussed the \(q\)-Fock space over a one-dimensional Hilbert space and determined the corresponding Bargmann measure. While the one-dimensional case is quite easy to compute, the general case is quite a bit harder; it has been dealt with in [\textit{M. Bożejko} and \textit{R. Speicher}, Commun. Math. Phys. 137, No. 3, 519--531 (1991; Zbl 0722.60033)]. (While it is easy to guess how the inner product has to be defined, it is hard to prove that it is actually positive. The Bargmann measure, on the other hand, follows from that of the one-dimensional case.)NEWLINENEWLINEIn the present paper, the authors determine the Bargmann measure for \((\alpha,q)\)-Fock spaces, a generalization of \(q\)-Fock spaces.