Basic estimates for the generalized $\partial$-complex

We study certain densely defined unbounded operators on the Segal–Bargmann space, related to the annihilation and creation operators of quantum mechanics. We consider the corresponding D-complex and study properties of the complex Laplacian $\tilde{\Box}_D = DD^\ast + D^\ast D$, where $D$ is a differential operator of polynomial type, in particular we discuss the corresponding basic estimates, where we express a commutator term as a sum of squared norms.

Keywords

$\partial$-complex, Segal–Bargmann space, sum of squared norms

2010 Mathematics Subject Classification

Primary 30H20, 32A36, 32W50. Secondary 47B38.

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The author is supported by the Austrian Science Fund (FWF) project P28154.

Received 12 March 2021

Received revised 12 July 2021

Accepted 22 July 2021

Published 13 May 2022