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Spectral Synthesis on the Reduced Heisenberg Group

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Abstract

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 \(U(n)\).

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991, Russia

    V. V. Volchkov & Vit. V. Volchkov

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  1. V. V. Volchkov
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  2. Vit. V. Volchkov
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Correspondence to V. V. Volchkov.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 46–57 https://doi.org/10.4213/mzm13617.

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Volchkov, V.V., Volchkov, V.V. Spectral Synthesis on the Reduced Heisenberg Group. Math Notes 113, 49–58 (2023). https://doi.org/10.1134/S0001434623010066

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