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Weighted shifts on directed trees: their multiplier algebras, reflexivity and decompositions

Volume 244 / 2019

P. Budzyński, P. Dymek, A. Płaneta, M. Ptak Studia Mathematica 244 (2019), 285-308 MSC: Primary 47B37; Secondary 47L75. DOI: 10.4064/sm170220-20-9 Published online: 29 June 2018

Abstract

We study bounded weighted shifts on directed trees. We show that the set of multiplication operators associated with an injective weighted shift on a rooted directed tree coincides with the WOT/SOT closure of the set of polynomials of the weighted shift. From this fact we deduce reflexivity of those weighted shifts on rooted directed trees whose all path-induced spectral-like radii are positive. We show that weighted shifts with positive weights on rooted directed trees admit a Wold-type decomposition. We prove that the pairwise orthogonality of the factors in the decomposition is equivalent to the weighted shift being balanced.

Authors

  • P. BudzyńskiKatedra Zastosowań Matematyki
    Uniwersytet Rolniczy w Krakowie
    Balicka 253c
    30-198 Kraków, Poland
    e-mail
  • P. DymekKatedra Zastosowań Matematyki
    Uniwersytet Rolniczy w Krakowie
    Balicka 253c
    30-198 Kraków, Poland
    e-mail
  • A. PłanetaKatedra Zastosowań Matematyki
    Uniwersytet Rolniczy w Krakowie
    Balicka 253c
    30-198 Kraków, Poland
    e-mail
  • M. PtakKatedra Zastosowań Matematyki
    Uniwersytet Rolniczy w Krakowie
    Balicka 253c
    30-198 Kraków, Poland
    e-mail

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