[Paper Review] Weighted shifts on directed trees. Their multiplier algebras, reflexivity and decompositions

Piotr Budzyński, Piotr Dymek+2 more|arXiv (Cornell University)|Feb 2, 2017

Advanced Topics in Algebra15 references4 citations

TL;DR

This paper investigates bounded weighted shifts on directed trees using an analytic function theory approach, establishing that the multiplier algebra of an injective weighted shift coincides with the weak/strong operator topology closure of its polynomials. Key results include reflexivity for injective weighted shifts with positive path-induced spectral radii and a Wold-type decomposition, where orthogonality of components is equivalent to the shift being balanced.

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.

Motivation & Objective

• To characterize the multiplier algebra of injective weighted shifts on rooted directed trees.
• To establish reflexivity conditions for such weighted shifts based on path-induced spectral radii.
• To investigate Wold-type decompositions of weighted shifts on directed trees.
• To resolve open problems on multiplier algebras posed in prior work.
• To clarify the role of balancing in the orthogonality of decomposition components.

Proposed method

• Utilizes an analytic function theory approach, modeling the weighted shift via coefficients of analytic functions.
• Defines the multiplier algebra as the set of multiplication operators corresponding to symbols in the algebra of coefficients.
• Proves that the WOT/SOT closure of polynomials in the shift equals the set of multiplication operators from the multiplier algebra.
• Applies spectral radius analysis along paths in the tree to derive reflexivity conditions.
• Establishes a Wold-type decomposition by decomposing the space into orthogonal ranges of powers of the shift.
• Uses boundedness from below to prove uniqueness of decomposition representations.

Experimental results

Research questions

• RQ1 Under what conditions does the multiplier algebra of an injective weighted shift on a rooted directed tree coincide with the WOT/SOT closure of its polynomial algebra?
• RQ2 When is a weighted shift on a rooted directed tree reflexive, particularly in relation to path-induced spectral radii?
• RQ3 When does a weighted shift on a directed tree admit a Wold-type decomposition into orthogonal components?
• RQ4 What is the relationship between the balancing condition and the orthogonality of the components in the Wold decomposition?
• RQ5 How does boundedness from below affect the uniqueness of the decomposition representation?

Key findings

• The set of multiplication operators associated with an injective weighted shift on a rooted directed tree equals the WOT/SOT closure of the polynomials in the shift.
• Injective weighted shifts on rooted directed trees are reflexive if all path-induced spectral-like radii are positive.
• Weighted shifts with positive weights on rooted directed trees admit a Wold-type decomposition.
• The pairwise orthogonality of the components in the Wold decomposition is equivalent to the shift being balanced.
• If the shift is bounded from below, the Wold-type decomposition representation is unique.
• Counterexamples show that the decomposition may fail to exist without injectivity or balancing, even for locally balanced shifts.