Antitonicity of the inverse for selfadjoint matrices, operators, and relations

Abstract

Let $H_1$ and $H_2$ be selfadjoint operators or relations (multivalued operators) acting on a separable Hilbert space and assume that the inequality $H_1\le H_2$ holds. Then the validity of the inequalities $-H_1^{-1}\le -H_2^{-1}$ and $H_2^{-1}\le H_1^{-1}$ is characterized in terms of the inertia of $H_1$ and $H_2$. 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.

To appear in Proc. Amer. Math. Soc.; 13 pages