Invertibility of operator $-\Delta -\lambda_{1} I: W_{*}^{2,p}(\Omega)\rightarrow L_{*}^{p}(\Omega)$

Question

The operator $-\Delta -\lambda_{1} I: W_{*}^{2,p}(\Omega)\rightarrow L_{*}^{p}(\Omega)$ is invertible? $W_{*}^{2,p}(\Omega)=\{u\in W^{2,p}(\Omega);\int_{\Omega}u\varphi_{1}dx=0\}, \,L_{*}^{p}(\Omega)=\{u\in L^{p}(\Omega);\int _{\Omega}u\varphi_{1}dx=0\}$, where $\varphi_{1}$ is the first eigenfunction associated with the first eigenvalue of the Laplacian.