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
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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.References
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Additional Information
- Jussi Behrndt
- Affiliation: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
- MR Author ID: 760074
- 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
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3209333