This paper is devoted to studying some fundamental properties of the left-evaluation operator L(0) and the compression operator \(S_{z}\) on \(N_{\varphi }^{(2)}\) -type quotient modules, generated by \((z-\varphi (w))^{2}\) , of the Hardy module over the bidisk for bounded analytic function \(\varphi (w)\) . In the present paper, we give some characterizations of the spectral of \(S_{z}\) , the compactness of \(L(0)|_{N_{\varphi }^{(2)}}\) , the kernels of \(S_{z}\) and \(L(0)|_{N_{\varphi }^{(2)}}\) . Furthermore, we provide a comprehensive analysis of the reducibility of \(S_{z}\) on \(N_{\varphi }^{(2)}\) in the case where \(\varphi (w)\) is a nonconstant inner function.
