A Note on $C^*$-Algebra of Toeplitz Operators with $\mathcal {L}$-Invariant Symbols

Abstract

Let $\mathcal {L}\subset {\mathbb {C}}^n$ be a Lagrangian plane. In this article, we give structure for $\mathcal {L}$-invariant operators on the Fock space $F^2({\mathbb {C}}^n)$. With the help of this structure, we study Toeplitz operators $T_{\textbf{a}}$ on $F^2(\mathbb {C}^n)$ with $\mathcal {L}$-invariant symbols $\textbf{a}\in L^\infty (\mathbb {C}^n)$. We show that every operator in the $C^*$-algebra generated by Toeplitz operators with $\mathcal {L}$-invariant symbols, denoted by $\mathcal {T}_\mathcal {L}(L^\infty )$, can be represented as an integral operator of the form

$$\begin{aligned} (H_\varphi ^Xf)(z) = \int _{\mathbb {C}^n} f(w)\varphi (z+X^*\overline{Xw}) e^{z\overline{w}}d\lambda (w) \end{aligned}$$

for some $\varphi \in F^2(\mathbb {C}^n)$ and $X\in \mathcal {U}(n, \mathbb {C})$ such that $X\mathcal {L} = i\mathbb {R}^n$. In fact, we prove that $H_\varphi ^X\in \mathcal {T}_\mathcal {L}(L^\infty )$ if and only if there exists $m\in {\mathcal {C}_{b,u}(\mathbb {R}^n)}$ such that

$$\begin{aligned} \varphi (z) = \bigg (\frac{2}{\pi }\bigg )^{n/2} \int _{\mathbb {R}^n}m(x) e^{-2(x-\frac{Xz}{2})^2} dx,\quad z\in \mathbb {C}^n. \end{aligned}$$

Here ${\mathcal {C}_{b,u}(\mathbb {R}^n)}$ denotes all functions on $\mathbb {R}^n$ which are bounded uniformly continuous with respect to the standard metric on $\mathbb {R}^n$.