The Heat Kernel and Green Function of the Generalized Hermite Operator, and the Abstract Cauchy Problem for the Abstract Hermite Operator

Following Wong's point of view (see [14], by Wong) we give a formula for the heat kernel of the generalized Hermite operator L-lambda on R-2, lambda is an element of R \ {0}. This formula is derived by means of pseudo-differential operators of the Weyl type, i.e., Weyl transforms, Fourier-Wigner transforms and Wigner transforms of generalized Hermite functions, which are the eigenfunctions of the generalized Hermite operators and form an orthonormal basis of L-2(R-2) (see [2], by Catana). By means of the heat kernel, we give a formula for the Green function of L-lambda, lambda is an element of R \ {0}. Using the Green function and the heat kernel we give some applications concerning the global hypoellipticity of L-lambda in the sense of Schwartz distributions, the ultracontractivity and the hy-L-lambda percontractivity of the strongly continuous one-parameter semigroup e-(tL lambda), t > 0, lambda is an element of R \ {0}. We also give a formula for the one-parameter strongly continuous semigroup e(-tA) generated by the abstract Hermite operator A. The formula is derived by means of the abstract Weyl operators, the abstract Fourier-Wigner operator and the abstract Wigner operators.