Thursday, October 16, 2008

Unbounded operators are not defined on all of H

I am looking for an elementary proof of the fact that an unbounded operator cannot have the whole Hilbert space as its domain of definition. In the textbooks I had a look at this follows from the closed graph theorem which then is proved using somewhat heavy functional analysis machinery. What I am looking for is something that is accessible to physicists that have just learned about unbounded operators and that could be turned into a homework problem. If you know such a reference or could give me a hint (in the comments or to I would greatly appreciate it!

1 comment:

Anonymous said...

Don't you need that the operator is both unbounded and closed? Otherwise there are such operators but they cannot be given explicitly as the construction requires the axiom of choice. So probably you can't escape the closed graph theorem.