Higher powers of analytical operators and associated ∗-Lie algebras

We introduce a new product of two test functions denoted by f□g (where f and g in the Schwartz space 𝒮(ℝ)). Based on the space of entire functions with θ-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 f□g, such operators give us a new representation of the centerless Virasoro–Zamolodchikov-ω∞∗-Lie algebras (in particular the Witt algebra) by using analytical renormalization conditions and by taking the test function f as any Hermite function. Replacing the classical pointwise product f ⋅ g of two test functions f and g by f□g, 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-ω∞∗-Lie algebra.