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

Agmon's type estimates of exponential behavior of solutions of systems  of elliptic partial differential equations.Applications to Schr\"{o}dinger,  Moisil-Theodorescu and Dirac operators