Finite dimensional approximations of operators on Hilbert spaces

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.)

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$A$ is in the norm closure of finite dimensional operators iff $A$ is a compact operator. Then the spectrum and the eigenspaces of $A_n$ converge to that of $A$.

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This is a very natural question, of course! In 1900, this would have been a hot topic!

But, as we find, and as @PeterMichor's answer indicates, the finite-dimensional "approximations" to a given operator do not approximate it in operator norm. I think it is reasonable to be surprised at this.

But the compelling significance of the question does not allow us to stop with a technical objection, in my sense of this. That is, we "should" ask what inferences can be made about the spectrum of perturbations of an operator. This turns out to be a highly non-trivial issue, but the answer is not "nothing".

In particular, for example, as in Kato's book on Perturbation Theory, compact perturbations are understandable.

In physics, "singular" perturbations of reasonable operators are understandable.

So, not knowing the larger context of the question, nevertheless one can reasonably say that while the spectrum of an operator is not usually related to that of its "finite-dimensional (bad-) approximations", useful things can be said.

Perhaps the questioner can clarify the context...

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Thankyou very much Paul. What I say might be a bit vague, but I've been advised against releasing too many details. A colleague and I are making some numerical methods for a linear evolution PDE. –  hoj201 Jul 29 '13 at 7:39
If you are approximating (linear) differential operators by finite-dimensional operators, you'd find that taking larger and larger finite-dimensional approximations tend to blow up in various ways in the limit, since if there are infinitely-many eigenvalues (=point spectrum) they are unbounded, and there may be continuous spectrum, which sometimes is approximated by denser-and-denser seeming eigenvalues, which disappear in the limit. Depending on your context, there are different names for "approximate eigenfunctions/eigenvalues". So interpretation of finite approximations is required... –  paul garrett Jul 29 '13 at 12:42
... but/and if you are approximating the resolvent for an unbounded operator, and are in a nice situation so that the resolvent is compact (as for diffops on a bounded domain...), then you can successfully approximate the resolvent by finite-dimensional operators (on a Hilbert space). –  paul garrett Jul 29 '13 at 12:44