(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

1. Given an operator [tex]H[/tex] , and a sequence [tex]\{ H_n \} _{n\geq 1 } [/tex] in an arbitrary Hilbert Space , such that both [tex]H[/tex] and [tex] H_n [/tex] are self-adjoint .

How can I prove that if [tex]||(H_n+i)^{-1} - (H+i) ^ {-1} || \to 0 [/tex] and if [tex]H[/tex] has an isolated eigenvalue [tex]\lambda[/tex] of multiplicity one, then for large enough [tex]n[/tex], [tex]H_n[/tex] also have isolated eigenvalues [tex]\lambda _ n[/tex] of multiplicty one that converge to [tex]\lambda[/tex].

2. Given an operator [tex]H[/tex] , and a sequence [tex]\{ H_n \} _{n\geq 1 } [/tex] in an arbitrary Hilbert Space , such that both [tex]H[/tex] and [tex] H_n [/tex] are self-adjoint and non-negative.

How can I prove that [tex]||(H_n+1)^{-1} - (H+1) ^ {-1} || \to 0 [/tex] is equivalent to

[tex]||(H_n+i)^{-1} - (H+i) ^ {-1} || \to 0 [/tex] ?

2. Relevant equations

3. The attempt at a solution

I really have no idea about it... I assume it has something to do with the self-adjointness...maybe some estimations on the inner-product...but can't figure out what

Thanks in advance

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Two Functional Analysis Questions

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**