$$\begin{array}{c} \lambda\in\text{Sp}(u)\\ \Updownarrow\\ \exists x\ne0_{E}\;/\;u(x)=\lambda x\\ \Updownarrow\\ \exists x\ne0_{E}\;/\;(u-\lambda\text{Id}_{E})(x)=0_{E}\\ \Updownarrow\\ u-\lambda\text{Id}_{E}\notin\mathcal{GL}(E)\\ \Updownarrow\\ \text{rg}(u-\lambda\text{Id}_{E})<\dim(E)\\ \Updownarrow\\ \dim(E_{\lambda}(u))=\dim(\text{Ker}(u-\lambda\text{Id}_{E}))=\dim(E)-\text{rg}(u-\lambda\text{Id}_{E})>0 \end{array}$$