Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line. (English) Zbl 1012.46033

The ordinary Fock space is the Hilbert space \(F\) of entire functions \(f(z)= \sum^\infty_{n=0} a_nz^n\), \(g(z)= \sum^\infty_{n=0} c_nz^n\) with inner product \[ (f,g)= \sum^\infty_{n=0} a_n\overline c_nn! \] The present paper studies a generalized Fock space \(F_\alpha\) for \(\alpha> -1/2\) in which the inner product is \[ (f,g)_\alpha= \sum^\infty_{n=0} a_n\overline c_nb_n (\alpha), \] where \(b_n\) is a quotient of gamma functions. The authors show that the Dunkl operator \[ (\Lambda_\alpha f)(z)={d\over dz}f(z) +{2\alpha+ 1\over z} \bigl(f(z)- f(-z)\bigr)/2 \] defines a reproducing kernel \(K_\alpha(w,z)\) in \(F_\alpha\) such that \[ \bigl(f(z), K_\alpha(w,z) \bigr)_\alpha= f(w). \] Among other results, they show that \((f,f)\leq (f,f)_\alpha\) so that \(F_\alpha\) is a subspace of \(F\). They study the multiplication operator \(Q\) on \(F_\alpha\) defined by \((Qf)(z)= zf(z)\) and show that \((\Lambda_\alpha f,g)_\alpha =(f, Qg)_\alpha\), so that \(\Lambda_\alpha\) and \(Q\) are adjoints of one another. This leads to various results on commutator relations and generalized Weyl relations between \(\Lambda_\alpha\) and \(Q\).


46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
30H05 Spaces of bounded analytic functions of one complex variable
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