  In the present paper we show properties of a little-known Laplacian operator
acting on symmetric tensors. This operator is an analogue of the well known
Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover,
this operator admits the Weitzenb\"ock decomposition and we study it using the
analytical method, due to Bochner, of proving vanishing theorems for the null
space of a Laplace operator admitting a Weitzenb\"ock decomposition and further
of estimating its lowest eigenvalue.
