Basis Properties of Root Functions of Differential Operators with Spectral Parameter in the Boundary Conditions

Abstract

Let B = B ⊕ C^N be a finite-dimensional extension of a Banach space B, and let B be equipped with the norm ||u|| = (||u||² + ||a||²)^1/2, where u = {u, a}, u ∈ B, a ∈ C^N. The element u is called the projection of u onto B. We find a criterion for the simultaneous completeness and minimality (respectively, for the basis property) of the system {u_k}_k=N+1⁸ of projections under the condition that the system {u_k}_k=1⁸ is complete and minimal (respectively, is a basis) in the space B. This criterion is used to study the basis property of root functions of second- and fourth-order ordinary differential operators in the space L₂.