Lebesgue type decompositions for nonnegative forms

Seppo Hassi*, Zoltan Sebestyen, Henk de Snoo

*Corresponding author for this work

    Research output: Contribution to journalArticleAcademicpeer-review

    52 Citations (Scopus)
    9 Downloads (Pure)

    Abstract

    A nonnegative form t on a complex linear space is decomposed with respect to another nonnegative form tu: it has a Lebesgue decomposition into an almost dominated form and a singular form. The part which is almost dominated is the largest form majorized by t which is almost dominated by w. The construction of the Lebesgue decomposition only involves notions from the complex linear space. An important ingredient in the construction is the new concept of the parallel sum of forms. By means of Hilbert space techniques the almost dominated and the singular parts are identified with the regular and a singular parts of the form. This decomposition addresses a problem posed by B. Simon. The Lebesgue decomposition of a pair of finite measures corresponds to the present decomposition of the forms which are induced by the measures. T. Ando's decomposition of a nonnegative bounded linear operator in a Hilbert space with respect to another nonnegative bounded linear operator is a consequence. It is shown that the decomposition of positive definite kernels involving families of forms also belongs to the present context. The Lebesgue decomposition is an example of a Lebesgue type decomposition, i.e., any decomposition into an almost dominated and a singular part. There is a necessary and sufficient condition for a Lebesgue type decomposition to be unique. This condition is inspired by the work of Ando concerning uniqueness questions. (C) 2009 Elsevier Inc. All rights reserved.

    Original languageEnglish
    Pages (from-to)3858-3894
    Number of pages37
    JournalJournal of functional analysis
    Volume257
    Issue number12
    DOIs
    Publication statusPublished - 15-Dec-2009

    Keywords

    • Nonnegative sesquilinear forms
    • Parallel sum
    • Almost dominated part
    • Singular part
    • Lebesgue type decomposition
    • Lebesgue decomposition
    • Pairs of nonnegative finite measures
    • Pairs of nonnegative bounded operators
    • CANONICAL DECOMPOSITION
    • UNBOUNDED OPERATORS
    • POSITIVE OPERATORS
    • SHORTED OPERATORS
    • PARALLEL ADDITION
    • LINEAR-OPERATORS

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