Interesting approaches to study statical and dynamic properties of quantum systems, e.g., the quantum transition state theory and the centroid molecular dynamics, have been previously derived using fixed centroid path integrals. We show that these constrained propagators can be alternatively defined using an operator formalism. An interesting result is the finding of the differential equations that determine the temperature dependence of these propagators. One equation applies to path integrals with fixed-centroid position (i.e., those used in quantum transition state theory), and the other one to path integrals with fixed-centroid position and momentum (i.e., those used in centroid molecular dynamics). Both equations are solved for a harmonic oscillator, so that the spectral decomposition of the operators represented by fixed-centroid path integrals is derived for this particular case. Their eigenvalues build an alternating geometric series, showing explicitly the impossibility of considering such operators as true density operators, i.e., some eigenfunctions are associated to "negative probabilities." The eigenfunctions are shown to be a generalization of the coherent and squeezed states of the harmonic oscillator. The physical meaning of centroid molecular dynamics, an approximation to study the time evolution of these mixed states, is clarified by considering the time evolution of the corresponding eigenfunctions. The mixed states constructed with "negative probabilities" display vanishing small position and momentum dispersion in the high temperature limit.