Let $e_1,e_2,\dots$ be a Schauder basis for a Hilbert space $(V , \langle \cdot , \cdot \rangle)$. Let $A:V \to V$ be an operator. Finally, let $V_n = {\rm span}( e_1, \dots, e_n)$. Let $i_n : V_n \to V$ be the injection so that $i_n^\dagger$ is the orthogonal projection. Finally, define $A_n = i_n^\dagger \circ A \circ i_n : V_n \to V_n$.

1) Are necessary and sufficient conditions known for the spectrum of $A_n$ to converge to the spectrum of $A$?

2) Same question, but for the eigen-spaces?

(p.s. I am an engineer with a fair knowledge of differential geometry. I apologize if this question is trivial. Functional analysis is a weakness for me.)