  We show that harmonic oscillator propagators and fractional Fourier
transforms are essentially the same. We deduce continuity properties and fix
time estimates for such operators on modulation spaces, and apply the results
to prove Strichartz estimates for the harmonic oscillator propagator when
acting on modulation spaces. Especially we extend some results by Balhara,
Cordero, Nicola, Rodino and Thangavelu. We also show that general forms of
fractional harmonic oscillator propagators are continuous on suitable Pilipovic
spaces.
