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 ? 



Is the spectrum of a "self adjoint" operator real on $\ell^p$?
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.


