Adjoint characterisations of quasi-weakly compact linear relations

In this paper, the class of all quasi-weakly compact linear relations is introduced and described in terms of their first and second adjoints. Complete characterisations are obtained in the case when the adjoint is continuous. We investigate the connection between a quasi-weakly compact linear relation and its adjoint. We also characterise the quasi-reflexive spaces in terms of quasi-weak compactness of operators. Examples of linear relations belonging to this class are exhibited.