In this paper we study new L p-boundedness properties and Parseval-type relations concerning the operators with complex Gaussian kernels over the spaces L p (R, w(x)dx), 1 ≤ p ≤ ∞, where w represents any function greater than or equal to one almost everywhere on R. Here, the Gauss-Weierstrass semigroup is considered as a particular case of this analysis,