Weighted composition operator on quaternionic Fock space

Abstract

This paper is concerned with several important properties of weighted composition operator acting on the quaternionic Fock space ${\mathcal {F}}^2({\mathbb {H}})$. Complete equivalent characterizations for its boundedness and compactness are established. As corollaries, the descriptions for composition operator and multiplication operator on ${\mathcal {F}}^2({\mathbb {H}})$ are presented, which can indicate some well-known existing theories in complex Fock space. Finally, as an appendix the closed expression for the kernel function of ${\mathcal {F}}^2({\mathbb {H}})$ is exhibited, which can deepen the understanding of ${\mathcal {F}}^2({\mathbb {H}})$.

Appendix

It is well-known that the kernel function always plays significant role in the investigations on properties of Hilbert spaces. In this appendix, we show that the quaternionic exponential function $e_{\star }^{pq}=\sum _{n=0}^\infty \frac{p^n q^n}{n!}$ admits a closed expression.

Theorem 5.1

For any $p=a+\omega b$, $q=c+\eta d$, where $\omega , \eta$ belong to the 2-sphere ${\mathbb {S}}$. Then function $e_{\star }^{pq}$ can be expressed as

$$\begin{aligned} e_{\star }^{pq}= & {} \frac{1}{2}(\cos (bd-ac)+\omega \sin (bd-ac))e^{ac+bd}(1+\omega \eta )\nonumber \\&+\frac{1}{2}(\cos (ac+bd)-\omega \sin (bd+ac))e^{ac-bd}(1-\omega \eta ). \end{aligned}$$

(17)

Proof

Denoting

$$\begin{aligned} p=a+\omega b=r_1(\cos x+ \omega \sin x) \end{aligned}$$

and

$$\begin{aligned}q=c+\eta b=r_2(\cos y+\eta \sin y), \end{aligned}$$

by direct computation, we have that

$$\begin{aligned} p^nq^n & = {} r_1^nr_2^n(\cos nx+ \omega \sin nx)(\cos ny+\eta \sin ny)\\& =r_1^nr_2^n \cos nx \cos ny + r_1^nr_2^n \omega \sin nx \cos ny \\& \quad +r_1^nr_2^n \eta \cos nx \sin ny + r_1^nr_2^n \omega \eta \sin nx \sin ny. \end{aligned}$$

Thus, the quaternionic exponential can be split into four series,

$$\begin{aligned} e_{\star }^{pq}=K_1+K_2+K_3+K_4 \end{aligned}$$

(18)

where

$$\begin{aligned} K_1:= & {} \sum _{n=0}^\infty \frac{r_1^n r_2^n \cos nx \cos ny}{n!},\\ K_2:= & {} \omega \sum _{n=0}^\infty \frac{r_1^n r_2^n \sin nx \cos ny}{n!},\\ K_3:= & {} \eta \sum _{n=0}^\infty \frac{r_1^n r_2^n \cos nx \sin ny}{n!},\\ K_4:= & {} \omega \eta \sum _{n=0}^\infty \frac{r_1^n r_2^n \sin nx \sin ny}{n!}. \end{aligned}$$

In the sequel, we compute the closed expression of each $K_i$, $i=1, 2, 3, 4$.

$$\begin{aligned} K_1= & {} \frac{1}{2} \sum _{n=0}^\infty \frac{(r_1r_2)^n(\cos n(x-y)+\cos n(x+y))}{n!}\\= & {} \frac{1}{4} \sum _{n=0}^\infty \frac{(r_1r_2)^n(e^{i n(x-y)}+e^{-i n(x-y)}+e^{i n(x+y)}+e^{-i n(x+y)}}{n!}\\= & {} \frac{1}{4}\left[ e^{r_1r_2e^{i(x-y)}}+ e^{r_1r_2e^{-i(x-y)}}+e^{r_1r_2e^{i(x+y)}}+e^{r_1r_2e^{-i(x+y)}}\right] \\= & {} \frac{1}{2} [e^{r_1r_2\cos (x-y)}\cos (r_1r_2 \sin (x-y))\\&+e^{r_1r_2\cos (x+y)}\cos (r_1r_2 \sin (x+y))].\end{aligned}$$

Similarly, we obtain

$$\begin{aligned} K_2= & {} \frac{\omega }{2} [e^{r_1r_2\cos (x-y)}\sin (r_1r_2 \sin (x-y))\\&+e^{r_1r_2\cos (x+y)}\sin (r_1r_2 \sin (x+y))]\\ K_3= & {} \frac{\eta }{2} [e^{r_1r_2\cos (x+y)}\sin (r_1r_2 \sin (x+y))\\&-e^{r_1r_2\cos (x-y)}\sin (r_1r_2 \sin (x-y))]\\ K_4= & {} \frac{\omega \eta }{2} [e^{r_1r_2\cos (x-y)}\cos (r_1r_2 \sin (x-y))\\&-e^{r_1r_2\cos (x+y)}\cos (r_1r_2 \sin (x+y))] .\end{aligned}$$

Putting the terms $K_i$, $i=1,2,3,4$ into (18), eliminating the angular variables and grouping corresponding terms, we obtain the desired closed expression (17). $\square$