# Two Functional Analysis Questions

1. May 27, 2012

### Combinatorics

1. The problem statement, all variables and given/known data
1. Given an operator $$H$$ , and a sequence $$\{ H_n \} _{n\geq 1 }$$ in an arbitrary Hilbert Space , such that both $$H$$ and $$H_n$$ are self-adjoint .

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

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

How can I prove that $$||(H_n+1)^{-1} - (H+1) ^ {-1} || \to 0$$ is equivalent to
$$||(H_n+i)^{-1} - (H+i) ^ {-1} || \to 0$$ ?

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