finding a specific operator

Hi! I am kinda going mad trying to find the following operator. Here goes the problem: Let be a complex seq. such that and let be a seq. of pos. numbers such that . Show that there exists (i.e. find) a compact operator on so that the following two are satisfied:

(a)

and

(b) All the lie in the resolvent of .

Thank You!

Re: finding a specific operator

I've not done the computations, but did you try with diagonal?

Re: finding a specific operator

yes I did try... I'm not sure how that helps.

Re: finding a specific operator

It would reduce the problem to find a sequence instead of an operator.

Re: finding a specific operator

I'm sorry but I just can't get it to work. How am I supposed to find it?

Re: finding a specific operator

There is something I don't understand: if for all , then has to be both compact and invertible in order to make the expression have a sense. It's impossible since is infinite dimensional.

Re: finding a specific operator

The assumptio is not , it is that

Re: finding a specific operator

Yes, but for all n is a particular case of . So we have to assume that for all otherwise would be invertible.

Re: finding a specific operator

I see what you mean. Ok, how would I proceed after removing that special case.

Re: finding a specific operator