Hello!

I have a problem that comes in two parts.

(1) is a Hilbert space with orthonormal basis . We are given a positivie strictly increasing sequence with the following property . Our task is to show that there exists a unique operator such that .

(2) Determine and the spectrum of M (hint: use the eigenvalues of the adjoint ).

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My attempt

(1) For the first one I was thinking that I could just simply find the operator by using where . Here it goes,

gives me .

So,

(the last equality by Parseval's formula and the second by continuity (boundedness) of ).

Well I now know what a bounded linear operator with the above stated property looks like. But is it unique? Futhermore, its "look" complicates things in the second part.

(2) I know the definition . But this doesn't seem all that easy to compute using what I know about my operator.

And what will the eigenvalues of tell me? I know that (where I by denote the spectrum of ) and this does not help me unless I can use the eigenvalues of to find . This would be easy if was compact, then I would know that .

Thanks!