Starting from an extension of the classical moment problem to linear operators, sufficient conditions for the existence of solutions of some moment problems are obtained. As a corollary, for positive selfadjoint operators on a Hilbert space having the spectrum on an interval, some solutions from the moment point of view are found. Via the Jensen inequality, some interesting inequalities are obtained in the case when the moments are real numbers. Also, a moment problem is considered in a space of analytic functions on an open disk, which are continuous on the closed disk.