  The aim of this paper is to derive Agmon's type exponential estimates for
solutions of elliptic systems of partial differential equations on $\sR^n$. We
show that these estimates are related with the essential spectra of a family of
associated differential operators which depend on the original operator, and
with exponential weights which describe the decrease of solutions at infinity.
The essential spectra of the involved operators are described by means of their
limit operators. The obtained results are applied to study the problem of
exponential decay of eigenfunctions of matrix Schr\"{o}dinger,
Moisil-Theodorescu, and Dirac operators.
