Form Sums of Nonnegative Selfadjoint Operators

S. Hassi, A. Sandovici, H.S.V. de Snoo, Henrik Winkler, 27740 Sandovici

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    The sum of two unbounded nonnegative selfadjoint operators is a nonnegative operator which is not necessarily densely defined. In general its selfadjoint extensions exist in the sense of linear relations (multivalued operators). One of its nonnegative selfadjoint extensions is constructed via the form sum associated with A and B. Its relations to the Friedrichs and Krein--von Neumann extensions of A + B are investigated. For this purpose, the one-to-one correspondence between densely defined closed semibounded forms and semibounded selfadjoint operators is extended to the case of nondensely defined semibounded forms by replacing semibounded selfadjoint operators by semibounded selfadjoint relations. In particular, the inequality between two closed nonnegative forms is shown to be equivalent to a similar inequality between the corresponding nonnegative selfadjoint relations.

    Original languageEnglish
    Pages (from-to)81-105
    Number of pages25
    JournalActa mathematica hungarica
    Issue number1-2
    Publication statusPublished - Apr-2006


    • nonnegative selfadjoint relation
    • representation theorem
    • Friedrichs extension
    • Krein-von Neumann extension
    • sum of nonnegative selfadjoint operators
    • form sum extension


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