Using the Bargmann transform, we give a proof of that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties 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 in our recent works and those of Bhimani, Cordero, Gröchenig, Manna, Thangavelu, and others. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable on so-called Pilipovic spaces and their distribution spaces. Especially we show that fractional Fourier transforms of any complex order can be defined, and that these transforms are continuous on any Pilipovic space and corresponding distribution space, which are not Gelfand–Shilov spaces. (The talk is based on a joint work with Divyang Bhimani and Ramesh Manna.)