Uncertainty inequality for weighted Fock spaces

In this paper we introduce a weighted Fock space \(\mathscr {F}_{\beta }\) ${\mathcal{F}}_{\beta }$ . This space which gives a generalization of some Hilbert spaces of analytic functions on the complex plane \(\mathbb {C}\) $\mathbb{C}$ like, the classical Fock space \(\mathscr {F}\) $\mathcal{F}$ , the Dunkl type Fock space \(\mathscr {F}_{\nu }\) ${\mathcal{F}}_{\nu }$ and the Bessel-Struve type Fock space \(\mathbb {F}_{\nu }\) ${\mathbb{F}}_{\nu }$ , it plays a background to our contribution. Especially, we study the multiplication operator M by \(z^2\) $z^2$ and its adjoint operator \(L_{\mathscr {F}_{\beta }}\) $L_{{\mathcal{F}}_{\beta }}$ on \(\mathscr {F}_{\beta }\) ${\mathcal{F}}_{\beta }$ , and we deduce a general uncertainty inequality of Heisenberg type for this space.