Resolvent growth and Birkhoff-regularity

We prove a long standing conjecture in the theory of two-point boundary value problems that unconditional basisness implies Birkhoff-regularity. It is a corollary of our two main results: minimal resolvent growth along a sequence of points implies nonvanishing of a regularity determinant, and sparseness of nthorder roots of eigenvalues in small sectors provided that eigen and associated functions of the boundary value problem form an unconditional basis. Considerations are based on a new direct method, exploiting almost orthogonality of Birkhoff’s solutions of the equation l(y) = λy. This property was discovered earlier by the author. © 2005 Elsevier Inc. All rights reserved.