Extremal extensions for the sum of nonnegative selfadjoint relations

Hassi, Seppo; Sandovici, Adrian; de Snoo, Henk; Winkler, Henrik

doi:10.1090/S0002-9939-07-08827-2

Extremal extensions for the sum of nonnegative selfadjoint relations
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by Seppo Hassi, Adrian Sandovici, Henk de Snoo and Henrik Winkler

Proc. Amer. Math. Soc. 135 (2007), 3193-3204

DOI: https://doi.org/10.1090/S0002-9939-07-08827-2

Published electronically: May 14, 2007

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Abstract:

The sum $A+B$ of two nonnegative selfadjoint relations (multi-valued operators) $A$ and $B$ is a nonnegative relation. The class of all extremal extensions of the sum $A+B$ is characterized as products of relations via an auxiliary Hilbert space associated with $A$ and $B$. The so-called form sum extension of $A+B$ is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with $A$ and $B$. Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on $A$ and $B$.

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