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

Behrndt, Jussi; Hassi, Seppo; Wietsma, Hendrik; de Snoo, Henk

doi:10.1090/S0002-9939-2014-12115-0

Antitonicity of the inverse for selfadjoint matrices, operators, and relations
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by Jussi Behrndt, Seppo Hassi, Hendrik Wietsma and Henk de Snoo PDF

Proc. Amer. Math. Soc. 142 (2014), 2783-2796 Request permission

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 $H_1 \leq H_2$ 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.

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