a couple of hilbert spaces problems
(1) Let be an orthonormal base of the Hilbert space X. We define and :
Prove that isn't closed.
I've been trying to find a convergent sequence in , with a limit outside , but no success. (Trying to prove that its complement is open would be much harder, I think.)
(2) Let be an orthonormal base of the Hilbert space X with the inner product
We define the operator as follows:
Is bounded, normal?
Is it true that , where S is the unilateral shift?
This one is just too messy. :) I can post what I've been trying to do, but didn't get anywhere and I think it wouldn't do any good.
Thank you once again for all your help.