  We consider a rather general version of ladder operator $Z$ used by some
authors in few recent papers, $[H_0,Z]=\lambda Z$ for some
$\lambda\in\mathbb{R}$, $H_0=H_0^\dagger$, and we show that several interesting
results can be deduced from this formula. Then we extend it in two ways: first
we replace the original equality with formula $[H_0,Z]=\lambda Z[Z^\dagger,
Z]$, and secondly we consider $[H,Z]=\lambda Z$ for some
$\lambda\in\mathbb{C}$, $H\neq H^\dagger$. In both cases many applications are
discussed. In particular we consider factorizable Hamiltonians and Hamiltonians
written in terms of operators satisfying the generalized Heisenberg algebra or
the $\D$ pseudo-bosonic commutation relations.
