I have two questions.
Let , a bounded linear operator of a Hilbert space , and let be the adjoint of .
Firstly, is it true that ?
Secondly, I want to show that for all , for some . Is it enough to show that ? As then surely this means that the limit exists...
Thanks in advance!
so then we just take the limit? (It is the limit bit I'm confused by. Sorry for what could be a really silly question - analysis really isn't my forte!)
But if you only know that then that is not enough to ensure convergence of . For example, let be an orthonormal basis for H, and define by . Then , which has norm . But , which does not converge.
The key thing here is that the sequence should be increasing (or decreasing and bounded below). For the series that is of course equivalent to each operator being positive. I'm not sure if that is what you mean by saying that your sum is monotone.
Strong convergence means exactly that in norm.
Okay, I have, after many dead ends, proven (in a very elementary way) that , so I believe this gives us absolute convergence, and so convergence? I am, however, suspicious, as it is too elementary...