(Untitled)

[dhilbert83]

This is the dumbest little problem I have. It's really just second year Undergrad linear algebra, but seemed to trick me up last night. So my dog woke me up at 6:45am, took him out, fiddled online, so now I'm pissed and must solve this now :p.

T is self adjoint iff is real for all x in H (H some hilbert space, but the proof is probably exactly

Comments

[wit_power_force]

This well-known fact puzzled me for about an hour (because I didn't have the comprehensive book by MS Birman, MZ Solomyak, "Spectral theory of self-adjoint operators in Hilbert space" at hand), but at last I've figured out the following proof.
Suppose is real for all x. We are to show that = for all x, y. Our assumption gives us that
= + + ( + ) and
= + + i( - ) are also real, and hence
Im ( + ) = 0 and Re ( - ) = 0. Therefore
Re ( - ) = Re ( - + ( - ) ) = 0, on the other hand Im ( - )= Im ( + - ( + ) ) = 0,
so = and we are done. The converse is trivial, since
Conj () = .

[dhilbert83]

Hey yea it's an easy little annoying arguement. Basic operator theory is filled with stuff like this. I think it's the kind of proof where you do it, understand it, and almost completely forget it afterwards, but can recover it if you sit down for a bit of time and think about it.