Weighted composition operator on quaternionic Fock space (Q2218278)

The (slice regular) quaternionic Fock space \(\mathcal{F}^2(\mathbb{H})\) was introduced in [\textit{D. Alpay} et al., in: Hypercomplex analysis: new perspectives and applications. Selected papers presented at the session on Clifford and quaternionic analysis at the 9th congress of ISAAC, Krakow, August 2013. New York, NY: Birkhäuser/Springer. 43--59 (2014; Zbl 1314.30092)]. The authors investigate the properties of the right linear weighted composition operators denoted by \(W_{f,\varphi}\) and acting on \(\mathcal{F}^2(\mathbb{H})\). These operators are induced by suitable entire (left) slice regular functions \(f\) and \(\varphi\) and are defined by means of the \(\Large\star\)-product (for this one, see, for instance, [\textit{F. Colombo} et al., Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions. Basel: Birkhäuser (2011; Zbl 1228.47001)]). In particular, the authors determine sufficient and necessary conditions on the functions \(f\) and \(\varphi\) for boundedness and compactness of \(W_{f,\varphi}\) (Theorems 3.1 and 3.2). These conditions are significantly simplified in the particular cases of composition and \(\Large\star\)-multiplication operators. In the appendix, the authors give a nice closed expression for the quaternionic exponential function \(\sum_{n=0}^{+\infty} \dfrac{p^n q^n}{n!}\) in terms of the real coordinates of the quaternions \(p\) and \(q\).