On the Solutions of a Class of Linear Selfadjoint Differential Equations

Let be a linear selfadjoint ordinary differential operator with coefficients which are real and sufficiently regular on <!-- MATH $( - \infty ,\infty )$ --> . Let <!-- MATH ${A^ + }({A^ - })$ --> denote the subspace of the solution space of such that <!-- MATH $y \in {A^ + }(y \in {A^ - })$ --> iff <!-- MATH ${D^k}y \in {L^2}[0,\infty )({D^k}y \in {L^2}( - \infty ,0])$ --> for <!-- MATH $k = 0,1, \ldots ,m$ --> where is the order of . A sufficient condition is given for the solution space of to be the direct sum of and . This condition which concerns the coefficients of reduces to a necessary and sufficient condition when these coefficients are constant. In the case of periodic coefficients this condition implies the existence of an exponential dichotomy of the solution space of .