Q-functions of Quasi-selfadjoint Contractions

Yury M. Arlinskiĭ, Seppo Hassi, Henk de Snoo

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    A bounded everywhere defined operator T in a Hilbert space H is said to be a quasi-selfadjoint contraction or (for short) a qsc-operator, if T is a contraction and ker (T − T*) ≠ {0}. For a closed linear subspace N of H containing ran (T − T*) the operator-valued function QT(z) = PN(T − zI)^−1↾N, |z| > 1, where PN is the orthogonal projector from H onto N, is said to be a Q-function of T acting on the subspace N. The main properties of such Q-functions are studied, in particular the underlying operator-theoretical aspects are considered by using some block representations of the contraction T and analytical characterizations for such functions QT(z) are established. Also a reproducing kernel space model for QT(z) is constructed. In the special case where T is selfadjoint QT(z) coincides with the Q-function of the symmetric operator A := T↾(H ⊖ N) and its selfadjoint extension T = T* in the usual sense.
    Original languageEnglish
    Title of host publicationOperator Theory and Indefinite Inner Product Spaces
    PublisherUniversity of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science
    Number of pages32
    ISBN (Electronic)978-3-7643-7516-4
    ISBN (Print)978-3-7643-7515-7
    Publication statusPublished - 2005

    Publication series

    NameOperator Theory: Advances and Applications


    • resolvent
    • operator model
    • Q-function
    • quasi-selfadjoint operator
    • contractive extension
    • symmetric contraction

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