Representation of essentially semi regular linear relations and perturbations

In the case of linear operator the property P(B, k) was introduced by M.A. Kaashoek. In this paper, we characterize the essentially semi regular linear relation in terms of the property P(B, k). After that and as an application of this result we give some connection between essentially semi regular and semi regular linear relations. Further, we will give some supplementary conditions on essentially semi regular linear relation to be semi Fredholm. Then, we analyze the stability of the class of essentially semi regular linear relations under small perturbations and Riesz operators. Finally, we study some properties of the essentially semi regular spectrum of a linear relation and we establish a spectral mapping theorem.