On isomorphism of ordinary differential operators

The article deals with an ordinary linear differential operator L of even order 2m with constant coefficients which defines a natural mapping of the space (Formula presented.) to itself. The operator L is considered under an extra condition that its characteristic polynomial has no real roots and exactly m roots with strictly positive imaginary part. This work prepares the treatment of properly elliptic partial differential operators in bounded domains of (Formula presented.) It turns out that there exists an extension operator to L which is an isomorphism between the standard Sobolev spaces H^m (a, b) and (Formula presented.) and such that the inverse operator can be represented in an extremely simple form using the Fourier transform.