Weighted shifts on directed trees: their multiplier algebras, reflexivity and decompositions
Volume 244 / 2019
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.
