Page "Spectral theory" Paragraph 31

Some natural questions are: under what circumstances does this formalism work, and for what operators are expansions in series of other operators like this possible?

Can any function be expressed in terms of the eigenfunctions ( are they a complete set ) and under what circumstances does a point spectrum or a continuous spectrum arise?

How do the formalisms for infinite dimensional spaces and finite dimensional spaces differ, or do they differ?

Can these ideas be extended to a broader class of spaces?

Answering such questions is the realm of spectral theory and requires considerable background in functional analysis and matrix algebra.