Defect basis property of the eigenvectors of a selfadjoint operator-valued function and formula for defect

Abstract

A continuous selfadjoint operator-valued functionA(x) is considered. Possible changes of dimension of the positive partA+(t) are investigated ast transverses through some Fredholm eigenvalue λ0 ∈ ℝ. Some results on defect basis property are obtained, and the formula for the defect is proved. Both holomorphic and non-holomorphic cases are considered.