Consider the Hermite operator $H=-/Delta+|x|^2$  on the Euclidean space $/mathbb{R}^n$. In this talk, we develop a theory of homogeneous and inhomogeneous Besov and Triebel-Lizorkin spaces associated to the Hermite operator.
As applications, we show the boundedness of negative powers and spectral multipliers of the Hermite operators on some appropriate Besov and Triebel-Lizorkin spaces. This is joint work with The Anh Bui, to appear in Journal of Fourier Analysis and Applications.