The weighted Weiss conjecture and reproducing kernel theses for generalized Hankel operators

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Abstract

The weighted Weiss conjecture states that the system theoretic property of weighted admissibility can be characterized by a resolvent growth condition. For positive weights, it is known that the conjecture is true if the system is governed by a normal operator; however, the conjecture fails if the system operator is the unilateral shift on the Hardy space (discrete time) or the right-shift semigroup on (continuous time). To contrast and complement these counterexamples, in this paper, positive results are presented characterizing weighted admissibility of linear systems governed by shift operators and shift semigroups. These results are shown to be equivalent to the question of whether certain generalized Hankel operators satisfy a reproducing kernel thesis.

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Subject classification (UKÄ) – MANDATORY

  • Mathematics

Keywords

  • One parameter semigroups, admissibility, Hardy space, weighted Bergman, space, Hankel operators, reproducing kernel thesis
Original languageEnglish
Pages (from-to)85-120
JournalJournal of Evolution Equations
Volume14
Issue number1
Publication statusPublished - 2014
Publication categoryResearch
Peer-reviewedYes