  If we add a simple rotation term to both the Ornstein-Uhlenbeck semigroup and
the definition of the H-derivative, then analogue to the classical Malliavin
calculus on the real Wiener space [I. Shigekawa, Stochastic analysis, 2004], we
get a normal but nonsymmetric Ornstein-Uhlenbeck operator $L$ on the complex
Wiener space. The eigenfunctions of the operator $L$ are given. In addition,
the hypercontractivity for the nonsymmetric Ornstein-Uhlenbeck semigroup is
shown.
