New studies on a family of $q$-weighted Bergman spaces on the unit disk and applications

Abstract

In this paper, we give a family of $q$-weighted
Bergman spaces
$\left\{\mathcal{A}_{{\alpha,n,q}}\right\}_{n\in\N}$ which
satisfies the continuous inclusion
$\mathcal{A}_{{\alpha,n,q}}\subset...\subset\mathcal{A}_{{\alpha,1,q}}\subset
\mathcal{A}_{{\alpha,0,q}}=\mathcal{A}_{{\alpha,q}}$, where
$\mathcal{A}_{{\alpha,q}}$ the $q$-weighted Bergman space.
Moreover, a more general uncertainty inequality of the
Heisenberg-type for the space $\mathcal{A}_{{\alpha,n,q}}$ is
given by considering the operators
$\nabla_{\alpha,n,q}:=\nabla^n_{\alpha,q}$ and
$L_{\alpha,n,q}:=L^n_{\alpha,q}$. Also, we study on
$\mathcal{A}_{{\alpha,q}}$ the $q$-Toeplitz operators, the
$q$-Hankel operators and the $q$-Berezin operators. Finally, an
application of the theory of extremal function and reproducing
kernel of Hilbert space is given and we use it to establish the
extremal function associated to an bounded linear operator
$T:\mathcal{A}_{{\alpha,q}}\rightarrow H$, for any Hilbert space
$H$. As application, we come up with some results regarding the
extremal functions associated to the difference operator
$Tf(z):=\frac{1}{z}(f(z)-f(0))$ and
$Tf(z):=\frac{1}{1+q}(f(z)-f(-z)).$