Higher powers of analytical operators and associated ∗-Lie algebras

Abstract

We introduce a new product of two test functions denoted by [Formula: see text] (where [Formula: see text] and [Formula: see text] in the Schwartz space [Formula: see text]). Based on the space of entire functions with [Formula: see text]-exponential growth of minimal type, we define a new family of infinite dimensional analytical operators using the holomorphic derivative and its adjoint. Using this new product [Formula: see text], such operators give us a new representation of the centerless Virasoro–Zamolodchikov-[Formula: see text]∗-Lie algebras (in particular the Witt algebra) by using analytical renormalization conditions and by taking the test function [Formula: see text] as any Hermite function. Replacing the classical pointwise product [Formula: see text] of two test functions [Formula: see text] and [Formula: see text] by [Formula: see text], we prove the existence of new ∗-Lie algebras as counterpart of the classical powers of white noise ∗-Lie algebra, the renormalized higher powers of white noise (RHPWN) ∗-Lie algebra and the second quantized centerless Virasoro–Zamolodchikov-[Formula: see text]∗-Lie algebra.