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Maps preserving absolute continuity and singularity of positive operators

Gehér, G., Tarcsay, Z. and Titkos, T. (2020) Maps preserving absolute continuity and singularity of positive operators. New York Journal of Mathematics, 26. pp. 129-137. ISSN 1076-9803

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Official URL: http://nyjm.albany.edu/j/2020/26-6.html

Abstract/Summary

In this paper we consider the cone of all positive, bounded operators acting on an infinite dimensional, complex Hilbert space, and examine bijective maps that preserve absolute continuity in both directions. It turns out that these maps are exactly those that preserve singularity in both directions. Moreover, in some weak sense, such maps are always induced by bounded, invertible, linear- or conjugate linear operators of the underlying Hilbert space. Our result gives a possible generalization of a recent theorem of Molnar which characterizes maps on the positive cone that preserve the Lebesgue decomposition of operators.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:89515
Publisher:State University of New York

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