We give a formula for the one-parameter strongly continuous semigroups $${e^{-tL^{\lambda}}}$$ and $${e^{-t \tilde{A}}}$$ , t > 0 generated by the generalized Hermite operator $${L^{\lambda}, \lambda \in {\bf R}\backslash \{0\}}$$ respectively by the generalized Landau operator Ã. These formula are derived by means of pseudo-differential operators of the Weyl type, i.e. Weyl transforms, Fourier-Wigner transforms and Wigner transforms of some orthonormal basis for L 2(R 2n ) which consist of the eigenfunctions of the generalized Hermite operator and of the generalized Landau operator. Applications to an L 2 estimate for the solutions of initial value problems for the heat equations governed by L λ respectively Ã, in terms of L p norm, 1 ≤ p ≤ ∞ of the initial data are given.