Clark theory in the Drury–Arveson space

We extend the basic elements of Clark's theory of rank-one perturbations of backward shifts, to row-contractive operators associated to de Branges–Rovnyak type spaces H(b) contrastively contained in the Drury–Arveson space on the unit ball in Cd. The Aleksandrov–Clark measures on the circle are...

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Bibliographic Details
Published inJournal of functional analysis Vol. 266; no. 6; pp. 3855 - 3893
Main Author Jury, Michael T.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.03.2014
Subjects
Aleksandrov–Clark measure
de Branges–Rovnyak space
Gleason problem
Row contraction
Gleason problem
Aleksandrov–Clark measure
Row contraction
de Branges–Rovnyak space
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Summary:We extend the basic elements of Clark's theory of rank-one perturbations of backward shifts, to row-contractive operators associated to de Branges–Rovnyak type spaces H(b) contrastively contained in the Drury–Arveson space on the unit ball in Cd. The Aleksandrov–Clark measures on the circle are replaced by a family of states on a certain noncommutative operator system, and the backward shift is replaced by a canonical solution to the Gleason problem in H(b). In addition we introduce the notion of a “quasi-extreme” multiplier of the Drury–Arveson space and use it to characterize those H(b) spaces that are invariant under multiplication by the coordinate functions.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2013.11.018