Ascent, descent, nullity, defect, and related notions for linear relations in linear spaces

Abstract

For a linear relation in a linear space the concepts of ascent, descent, nullity, and defect are introduced and studied. It is shown that the results of A.E. Taylor and M.A. Kaashoek concerning the relationship between ascent, descent, nullity, and defect for the case of linear operators remain valid in the context of linear relations, sometimes under the additional condition that the linear relation does not have any nontrivial singular chains. In particular, it is shown for a linear relation A with a trivial singular chain manifold whose ascent p is finite and whose nullity and defect are equal and finite that the linear space H is a direct sum of ker A p and ran A p . Furthermore it is shown that the various results which require the absence of singular chains are not valid when such chains are present.