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Abstract
In this work, we introduce a class of Hilbert spaces Fq of entire functions on the disk , , with reproducing kernel given by the q-exponential function eq(z); and we prove some properties concerning Toeplitz operators on this space. The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Dq by and the q-Derivative operator on the Fock space Fq ; and we prove that these operators are adjoint-operators and continuous from this space into itself. Lastly, we study a generalized translation operators and a Weyl commutation relations on Fq .
Highlights
In 1961, Bargmann [1] introduced a Hilbert space of entire functions f z = n=0an z n on such that f 2 := an 2 n! < . n=0On this space the author study the differential operator D = d dz and the multiplication operator by z, and proves that these operators are densely defined, closed and adjoint-operators on
The Hilbert space is called Segal-Bargmann space or Fock space and it was the aim of many works [2,3]
We prove that the space q is a Hilbert space and we give an Hilbert basis
Summary
The Hilbert space is called Segal-Bargmann space or Fock space and it was the aim of many works [2,3]. Associated to the q-exponential function and we give some applications. In the first part of this work, building on the ideas of Bargmann [1], space of entire we define functions the q-Fock space a. The q-Fock space q has a reproducing kernel q given by q w, z = eq wz ; w, z
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