  We prove a generalization of the quantum Markovian equation for observables.
In this generalized equation, we use superoperators that are fractional powers
of completely dissipative superoperators. We prove that the suggested
superoperators are infinitesimal generators of completely positive semigroups
and describe the properties of this semigroup. We solve the proposed fractional
quantum Markovian equation for the harmonic oscillator with linear friction. A
fractional power of the Markovian superoperator can be considered a parameter
describing a measure of "screening" of the environment of the quantum system:
the environmental influence on the system is absent for $\alpha=0$, the
environment completely influences the system for $\alpha=1$, and we have a
powerlike environmental influence for $0<\alpha<1$.
