Let $\Gamma_1$ and $\Gamma_2$ be two infinite disjoint sets of the same cardinality. Let $i:\Gamma_1\to \Gamma_2$ be a bijection, then define $\Gamma=\Gamma_1\bigcup \Gamma_2$. One can easily show that $$ V:\ell_2(\Gamma)\to \ell_2(\Gamma):x\mapsto\left(\gamma\mapsto\begin{cases} x_{i(\gamma)}&\quad\text{ if }\quad \gamma\in\Gamma_1\\\0&\quad\text{ if }\quad \gamma\in\Gamma_2\end{cases}\right) $$ is a partial isometry. Moreover $$ n_{+}(V) =\dim(\operatorname{dom}(V)^{\perp}) =\dim(\ell_2(\Gamma_1)^\perp) =\dim(\ell_2(\Gamma_2)) =|\Gamma_2|\geq\aleph_0\\\ n_{-}(V) =\dim(\operatorname{ran}(V)^{\perp}) =\dim(\ell_2(\Gamma_2)^\perp) =\dim(\ell_2(\Gamma_1)) =|\Gamma_1|\geq\aleph_0\\\ $$