The only thing I know about non-closable operators can be summarised as "they exist, but they're nasty, so let's not talk about them!" This seems to be the case with everyone else I've talked to. I'd appreciate some references or ideas about the following:

What classes of non-closable operators have been studied (if any). For instance have the operators with non-empty, real, positive spectrum been studied? Are there weaker, but similar conditions to closable/symmetric which gaurentee nice properties (e.g. non-empty spectrum, adjoint with non-trivial domain, some form of polar decompostion)?

EDIT: As the comments suggest, the usual definition of spectrum doesn't make sense if the operator's not closable. Is there another way to define it - or is there a similar object which is useful?