Weighted shifts on directed trees. Their multiplier algebras, reflexivity and decompositions

2017/02/02 by Budzynski, Piotr, Dymek, Piotr, Planeta, Artur +1 · 1 citation
#47B37 #47L75 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1702.00765

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.

Cited by

• Unitary equivalence of balanced weighted shifts on rooted directed trees

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