Originally Posted by

**Zeke** Hi! I am kinda going mad trying to find the following operator. Here goes the problem: Let $\displaystyle (z_n)$ be a complex seq. such that $\displaystyle \lim z_n=0$ and let $\displaystyle (x_n)$ be a seq. of pos. numbers such that $\displaystyle \lim x_n=\infty$. Show that there exists (i.e. find) a compact operator $\displaystyle P$ on $\displaystyle l^2$ so that the following two are satisfied:

(a) $\displaystyle \lim_n \frac{\|(z_n I-P)^{-1}\|}{ x_n}=\infty$

and

(b) All the $\displaystyle z_n$ lie in the resolvent of $\displaystyle P$.

Thank You!