  We establish two conditions equivalent to coamenability for type I locally
compact quantum groups. The first condition is concerned with the spectra of
certain convolution operators on the space
$\operatorname{L}^2(\operatorname{Irr}(\mathbb{G}))$ of functions which are
square integrable with respect to the Plancherel measure. The second condition
involves spectra of character-like operators associated with direct integrals
of irreducible representations. As examples we study special classes of quantum
groups: classical, dual to classical, compact or given by a certain bicrossed
product construction.
