For a compact self-adjoint operator $T$, the spectral operator can be defined as follows: $$E(\lambda) = \frac{1}{2\pi i}\int_{\Gamma}(z-T)^{-1}dz$$.
For a finite dimensional approximate operator $T_h$, similarly, we have $$E_h(\lambda) = \frac{1}{2\pi i}\int_{\Gamma}(z-T_h)^{-1}dz$$.
My question is then what do you mean by $E_h(\lambda)u$ for any $u\in V$, since $T_h:L^2 \rightarrow V_h$?