Antitonicity of the inverse for selfadjoint matrices, operators, and relations
Jussi Behrndt, Seppo Hassi, Hendrik Luit Wietsma, Henk S. V. de Snoo · Proceedings of the American Mathematical Society · 2014
Let H 1 H_1 and H 2 H_2 be selfadjoint operators or relations (multivalued operators) acting on a separable Hilbert space and assume that the inequality H 1 ≤ H 2 H_1 \leq H_2 holds. Then the validity of the inequalities − H 1 − 1 ≤ − H 2 − 1 -H_1^{-1} \le -H_2^{-1} and H 2 − 1 ≤ H 1 − 1 H_2^{-1} \le H_1^{-1} is characterized in terms of the inertia of H 1 H_1 and H 2 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.