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


Authors: Jussi Behrndt, Seppo Hassi, Hendrik Wietsma and Henk de Snoo
Journal: Proc. Amer. Math. Soc. 142 (2014), 2783-2796
MSC (2010): Primary 47A06, 47A63, 47B25; Secondary 15A09, 15A45, 15B57
DOI: https://doi.org/10.1090/S0002-9939-2014-12115-0
Published electronically: May 8, 2014
MathSciNet review: 3209333
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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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Additional Information

Jussi Behrndt
Affiliation: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
Email: behrndt@tugraz.at

Seppo Hassi
Affiliation: Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, FI-65101 Vaasa, Finland
Email: sha@uwasa.fi

Hendrik Wietsma
Affiliation: Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, FI-65101 Vaasa, Finland
Email: rwietsma@uwasa.fi

Henk de Snoo
Affiliation: Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen, Netherlands
Email: desnoo@math.rug.nl

DOI: https://doi.org/10.1090/S0002-9939-2014-12115-0
Keywords: Selfadjoint operator, selfadjoint relation, inertia, matrix inequality, operator inequality, ordering
Received by editor(s): May 13, 2011
Received by editor(s) in revised form: September 7, 2012
Published electronically: May 8, 2014
Additional Notes: This research was supported by grants from the Academy of Finland (project 139102) and the German Academic Exchange Service (DAAD), PPP Finland project 50740090.
The third author would like to thank the Deutsche Forschungsgemeinschaft (DFG) for the Mercator visiting professorship at the Technische Universität Berlin.
Communicated by: Marius Junge
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.