Generalized study of the operator $\alpha {\mathrm{\partial }}^k{\bar{\mathrm{\partial }}}^k+\beta {\bar{\mathrm{\partial }}}^k+\gamma {\mathrm{\partial }}^k+c$ in weighted Hilbert space $L^2(\mathbb{C},{\mathrm{e}}^{-\mathrm{\mid }z{\mathrm{\mid }}^2})$

By Hörmander’s $L^2$-method, we study the operator $\alpha {\mathrm{\partial }}^k{\bar{\mathrm{\partial }}}^k+\beta {\bar{\mathrm{\partial }}}^k+\gamma {\mathrm{\partial }}^k+c$ for any order $k$ with $\alpha ,\beta ,\gamma \in \mathbb{R}$ such that $(\alpha ,\beta ,\gamma )\mathrm{\ne }(0,0,0)$ in the weighted Hilbert space $L^2(\mathbb{C},{\mathrm{e}}^{-\mathrm{\mid }z{\mathrm{\mid }}^2})$. We prove the existence of its right inverse which is also a bounded operator. Subsequently we will study two cases that arise from this operator, namely: (1) Case where $\alpha =\gamma =0$: The operator $\beta {\bar{\mathrm{\partial }}}^k+c$ with $\mathrm{\mid }\beta \mathrm{\mid }\ge 1$. (2) Case where $\beta =\gamma =0$: The operator $\alpha {\mathrm{\partial }}^k{\bar{\mathrm{\partial }}}^k+c$ with $\mathrm{\mid }\alpha \mathrm{\mid }\ge 1$.