Spectral Synthesis on the Reduced Heisenberg Group

V. V. Volchkov ; Vit. V. Volchkov

Matematičeskie zametki, Tome 113 (2023) no. 1, pp. 46-57

Résumé

The spectral synthesis problem for the phase space $\mathbb{C}^n$ associated with the reduced Heisenberg group $H^n_{\rm{red}}$ is studied. The paper deals with the case of subspaces in $\mathcal{E}(\mathbb{C}^n)$ invariant under the twisted shifts $$ f(z)\to f(z-w)e^{(i/2)\operatorname{Im}\langle z,{w}\rangle},\qquad w\in\mathbb{C}^n, $$ and the action of the unitary group $U(n)$. It is shown that any such subspace is generated by the root vectors of a special Hermite operator contained in this subspace. As a corollary, we obtain the spectral synthesis theorem for subspaces in $\mathcal{E}(H^n_{\rm{red}})$ invariant under the unilateral shifts and the action of the unitary group $U(n)$.

Détail

Keywords: spherical harmonics, Heisenberg group, transmutation operators.