  In this paper, we study eigenvalues of the closed eigenvalue problem of the
differential operator $ L$, which is introduced by Colding and Minicozzi in
[4], on an $n$-dimensional compact self-shrinker in ${R}^{n+p}$. Estimates for
eigenvalues of the differential operator $ L$ are obtained. Our estimates for
eigenvalues of the differential operator $ L$ are sharp. Furthermore, we also
study the Dirichlet eigenvalue problem of the differential operator $ L$ on a
bounded domain with a piecewise smooth boundary in an $n$-dimensional complete
self-shrinker in $ {R}^{n+p}$. For Euclidean space $ {R}^{n}$, the differential
operator $ L$ becomes the Ornstein-Uhlenbeck operator in stochastic analysis.
Hence, we also give estimates for eigenvalues of the Ornstein-Uhlenbeck
operator.
