Fractional symplectic Fourier transforms and their actions on function and distribution spaces

We consider symplectic analogues of the fractional Fourier transforms and obtain convenient representations in terms of Bargmann transforms and Wigner distributions. We show that they act as homeomorphisms on Schwartz spaces, the space of square integrable functions, the space of tempered distributions, and on Fourier-invariant Gelfand-Shilov spaces and their corresponding distribution spaces. More generally, we establish their homeomorphism properties on a broad class of spaces defined through (Wigner-)Hermite expansions. In addition, we prove that these transforms preserve norms on Schatten-von Neumann symbol classes associated with the Weyl calculus.