I am interested in finding literature pertaining the problem posed by this question, which is the degree to which an operator $A$ on an infinite dimensional (separable) Hilbert $X$ space can be "approximated" by matrices $A_n$ corresponding to the restriction of $A$ to the $n$th member of an ascending chain of subspaces $U_1\subset U_2 \subset \cdots \subset U_n \subset \cdots \subset X$ for which

(1) $\dim(U_n) = n$, and

(2) $\lim_{n\rightarrow \infty} U_n = X.$

It is mentioned that such an "approximation" theory was developed around the 1900s, but I have no functional analysis text which mentions any such development. Specific questions of interest would be

(i) Let $\sigma(P)$ be the spectrum of $P$. When must we have $\lim_{n\rightarrow \infty}\sigma(A_n) = \sigma(A)$? (a statement is made without proof that this is true for compact operators).

(ii) Is there ever a case for which $\lim_{n\rightarrow \infty}\|A_n\|_{op} = A$? (another statement is made that there are no such cases).

(iii) Let $x \in X$ and suppose $x_n$ is its restriction to $U_n$. When is it true (if ever) that $\lim_{n\rightarrow\infty} A_nx_n = Ax$? (this may just be a restatement of (ii).)

Does anyone know of a good standard reference here? A text or detailed review article would be the best case.