First off, I'd like to ask:
If $H$ is a Hilbert space, and we have $A$ a bounded operator from $H$ to itself, $A$ being self adjoint (or normal), then if $A$ is compact there is a eigenspace decomposition. What if A is not compact? I have noticed a number of sources I've read avoid this case, as if it's not true.
The other question is:
A document I am reading says that a possibly noncompact positive operator is the norm limit of certain linear combinations with positive coefficients of "spectral projections." I'd like to see a proof of this. First I need clarification on what "spectral projection" is likely to mean given that there isn't a spectral decomposition theorem that I know of. I am aware that in the separable $H$ case, there's something one can say about using Banach-Space valued integrals as "spectral projections" in order to extract information about generalized eigenspaces in the sense of Jordan Blocks. (you integrate the resolvent.) But I'd be a little surprised if this sort of thing was being treated as trivial in the text I'm reading, and also I know no proof of this for the nonseparable case, as the proof in the separable case clearly uses Analytic Fredholm theory.
In case context helps, this appears in the neighborhood of a discussion of the equivalence of complete additivity and normality of positive continuous linear functionals on Von Neumann Algebras. Thanks!