  During the recent developments of quantum theory it has been clarified that
the observable quantities (like energy or position) may be represented by
operators (with real spectra) which are manifestly non-Hermitian. The
mathematical consistency of the resulting models of stable quantum systems
requires a reconstruction of an alternative, amended, physical inner product of
states. We point out the less known fact that for more than one observable the
task is not always feasible. The difficulty is re-analyzed and its elementary
linear-algebraic interpretation and treatment are outlined.
