Stević-Sharma type operators on Fock spaces in several variables

Let ϕ be an entire self-map of ℂ^N, u₀ be an entire function on ℂ^N and u = (u₁, …, u_N) be a vector-valued entire function on ℂ^N. We extend the Stević-Sharma type operator to the classcial Fock spaces, by defining an operator \(T_{u_0,{\bf u},\varphi}\) $$T_{u_0,\mathbf{u},\phi }$$ as follows

\(T_{u_0,{\bf u},\varphi}f=u_0\cdot f\circ\varphi+\sum_{i=1}^Nu_i\cdot\frac{\partial f}{\partial z_i}\circ\varphi.\) $$T_{u_0,\mathbf{u},\phi }f=u_0\cdot f\circ \phi +\sum _{i=1}^N{u}_i\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }{z}_i}\circ \phi .$$

We investigate the boundedness and compactness of \(T_{u_0,{\bf u},\varphi}\) $$T_{u_0,\mathbf{u},\phi }$$ on Fock spaces. The complex symmetry and self-adjointness of \(T_{u_0,{\bf u},\varphi}\) $$T_{u_0,\mathbf{u},\phi }$$ are also characterized.