Relationships between inessential, strictly singular, strictly cosingular and improjective linear relations

This paper is devoted to study the interrelations between inessential, improjective, strictly singular and strictly cosingular linear relations. First, we show that the classes of strictly singular and strictly cosingular linear relations with finite dimensional multivalued part are contained in the class of inessential linear relations and that for many Banach spaces these inclusions are equalities. Moreover, we prove that the class of improjective linear relations contains the class of inessential linear relations and we also see that for the most classical Banach spaces the improjective linear relations with finite dimensional multivalued part coincide with the inessential linear relations with finite dimensional multivalued part. Finally, to give value to our results we construct an example of a closed everywhere defined linear relation with finite dimensional multivalued part of each of the following types: inessential not strictly singular. inessential not strictly cosingular. improjective not inessential.