TITLE: Is the spectrum of a "self adjoint" operator real on $\ell^p$?
QUESTION [10 upvotes]: There might be an obvious answer to the question, but it doesn't come to mind.
Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all sequences $(x_i)\in \ell^p, p>1$
$$ \sum_{i=1}^\infty\big|\sum_{j=1}^\infty a_{ij}x_j\Big|^p \leq C \Vert x \Vert_{p}^p.$$
For some positive constant $C$.
Furthermore, assume that $a_{ij}=\overline{a_{ji}}$. Is it true that the spectrum of $A$ on $\ell^p $ is real ?

REPLY [4 votes]: It seems that I have found a counter example myself. 
For the Hilbert matrix 
$$ H_\lambda:= \big( \frac{1}{1-\lambda+k+n} \big)_{k,n\geq 0}, \lambda < 1 $$
Rosenblum in "On the Hilbert Matrix I, Proceedings of the AMS" proves that the pointspectrum  considered as an operator on $\ell^p, p>2$ contains the set 
$$ \{ \pi \sec(\pi u ) : | \Re ( u )| < 1/2-1/p \}. $$
If one could provide a more elementary counterexample I would be interested in looking at.