Nullity and deficiency of linear relations under certain perturbations

Let A and B be two closed linear relations acting between two Banach spaces X and Y and let λ be a complex number. We study the behavior of the nullity and deficiency of A when perturbed by λB. In particular, we show the existence of a constant > 0 for which both the nullity and deficiency of A do not remain constant when A is perturbed by λB for all λ inside the disk jλj <. It turns out, however, that these quantities do not depend on λ in the specified disk, that is, both the nullity and deficiency of A - λB are uniform on the specified disk.