I am interested in finding literature pertaining the problem posed by [this question](https://mathoverflow.net/questions/138030/finite-dimensional-approximations-of-operators-on-hilbert-spaces), 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.



Literature Request: Finite Dimensional "Approximations" of Linear Operators
A canonical reference is : Spectral Approximation of Linear Operators by Chatelin.


<http://epubs.siam.org/doi/book/10.1137/1.9781611970678>


