ANTITONICITY OF THE INVERSE FOR SELFADJOINT MATRICES, OPERATORS, AND RELATIONS.

Let H1 and H2 be selfadjoint operators or relations (multivalued operators) acting on a separable Hilbert space and assume that the inequality H1 = H2 holds. Then the validity of the inequalities -H-1 1 = -H-12 and H-12 = H-11 is characterized in terms of the inertia of H1 and H2. Such results are known for matrices and boundedly invertible operators. In the present paper those results are extended to selfadjoint, in general unbounded, not necessarily boundedly invertible, operators and, more generally, for selfadjoint relations in separable Hilbert spaces. [ABSTRACT FROM AUTHOR]