On a Family of q-Weighted Bergman Spaces and Applications

Abstract

In this paper, we introduce a q-weighted Bergman spaces {A_α,n,q}_n∈N . For n = 0 an uncertainty inequality of the Heisenberg-type for the space A_α,q is given by considering the operators ∇_α,q := ∇_α,q and L_α,q := L_α,q. Also, we study on this space the q-Toeplitz operators, the q-Hankel operators. At the end, we study the theory of extremal function and reproducing kernel of Hilbert space and we use it to establish the extremal function associated to an bounded linear operator T : A_α,q → H, for any Hilbert space H.