Perturbation of ordinary differential operators

AbstractA perturbation theorem is proved which enables us to extend the results of J. Schwartz and others, to the effect that a wide class of boundary value problems for ordinary differential operators with operator valued coefficients generate unbounded spectral operators in the L2 space over a finite interval. Thus under mild restrictions on the boundary conditions allowed, an operator (1iddx)n + Cn−1 (ddy)n−1 + Bn−2 (ddx)n−2 + ··· + B0 in L2[0, 1], where B0, …, Bn−2 are arbitrary bounded operators and Cn−1 is any compact operator plus a bounded one with small norm, is shown to have a complete set of generalized eigenfunctions (φn). Moreover, the series expansions in the φn are unconditionally convergent