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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)= n=0 a n z n , g(z)= n=0 c n z n with inner product

(f,g)= n=0 a n c ¯ n n!

The present paper studies a generalized Fock space F α for α>-1/2 in which the inner product is

(f,g) α = n=0 a n c ¯ n b n (α),

where b n is a quotient of gamma functions. The authors show that the Dunkl operator

(Λ α f)(z)=d dzf(z)+2α+1 zf ( z ) - f ( - z )/2

defines a reproducing kernel K α (w,z) in F α such that

f (z) , K α (w,z) α =f(w)·

Among other results, they show that (f,f)(f,f) α so that F α is a subspace of F. They study the multiplication operator Q on F α defined by (Qf)(z)=zf(z) and show that (Λ α f,g) α =(f,Qg) α , so that Λ α and Q are adjoints of one another. This leads to various results on commutator relations and generalized Weyl relations between Λ α and Q.

MSC:
46E22Hilbert spaces with reproducing kernels
47B32Operators in reproducing-kernel Hilbert spaces
30H05Bounded analytic functions