Closed Range and Preserving Frames of Weighted Composition Operator on Quaternionic Fock Space

In this paper, we completely describe the closed range of bounded weighted composition operator on the quaternionic Fock space \(\mathcal {F}^2(\mathbb {H})\) ${\mathcal{F}}^2(\mathbb{H})$ . This result further facilitates the characterizations for \(C_{f, \varphi }\) $C_{f,\phi }$ or \(C_{f, \varphi }^*\) $C_{f,\phi }^{\ast }$ preserving frames, Riesz bases or tight frames on \(\mathcal {F}^2(\mathbb {H})\) ${\mathcal{F}}^2(\mathbb{H})$ . Especially, we creatively obtain a useful formula of the adjoint operator \(C_{f, \varphi }^*\) $C_{f,\phi }^{\ast }$ on \(\mathcal {F}^2(\mathbb {H})\) ${\mathcal{F}}^2(\mathbb{H})$ , which is crucial for showing the equivalence of the unitary, isometry and co-isometry of the weighted composition operator on \(\mathcal {F}^2(\mathbb {H})\) ${\mathcal{F}}^2(\mathbb{H})$ . This result is new even on the ordinary complex Fock space. Moreover, a series of new equivalent characterizations for quaternionic composition operators on \(\mathcal {F}^2(\mathbb {H})\) ${\mathcal{F}}^2(\mathbb{H})$ follow as a special case.