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 $(x_i)\in \ell^p, p>1$ $$ \sum_{i=1}^\infty\big|\sum_{j=1}^\infty a_{ij}x_j\Big|^p \leq C \Vert x \Vert_{p}.$$ 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 ?