First order differential operators in white noise analysis

Let $(E)$ be the space of test white noise functionals. We first introduce a family $\{ \diamond _{\gamma}\,\,;\,\,\gamma\in {\Bbb C} \}$ of products on $(E)$ including Wiener and Wick products, and then show that with each product $\diamond _{\gamma}$, we can associate a first order differential operator, called a first order $\gamma$-differential operator. We next show that a first order $\gamma$-differential operator is indeed a continuous derivation under the product $\diamond _{\gamma}$. We finally characterize $\gamma\Delta _G+N$ by means of rotation-invariance and continuous derivation under the product $\diamond _{\gamma}$. Here $\Delta _G$ and $N$ are the Gross Laplacian and the number operator on $(E)$, respectively.