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

الأصل · EN

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 -H₁⁻¹ ≤ -H₂⁻¹ and H₂⁻¹ ≤ H₁⁻¹ 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.