The numerical range of linear relations and stability theorems

Abstract

The nullity, deficiency and index of a linear relation A from a linear space X to a linear space Y are defined as the dimension of the null space of A, the codimension of the range of A and the difference between the nullity and deficiency respectively. A is said to have an index if at least one of the nullity or deficiency is finite. Otherwise, A is said to have no index. Let X, Y be Banach spaces, A, B closed linear relations from X to Y and z a complex number. Stability of the nullity and deficiency of A when perturbed by -zB is established. We show that there exists a constant k > 0 for which both the nullity and deficiency of A (and therefore the index) remain constant under perturbation by -zB for any z inside the Disk |z| < k. We also establish a disk inside which the nullity and deficiency of a closed linear relation A may not be stable under perturbation by -zB for any z inside the disk, yet, on the same disk, the index of the same linear relation is stable under a similar perturbation. The stability of the defect number dA of a closed linear relation A which is related to the solvability of the inclusion y ∈ A(x) is also considered. It is shown that if B is relatively bounded with respect to A then dA = dA+B. Convexity for the numerical range of a linear relation A relative to B on a Hilbert space H where B is injective is proved together with the Spectral Inclusion Theorem. The convexity is used here to study the stability of the index of A when perturbed by a relatively bounded linear relation B. It is shown that for the constant k determined above and z outside the numerical range, the nullity and deficiency do not increase but the index remains constant.