Disjoint Supercyclic Weighted Shifts

Document Type

Article

Language

eng

Format of Original

30 p.

Publication Date

6-2016

Publisher

Springer

Source Publication

Integral Equations and Operator Theory

Source ISSN

0378-620X

Abstract

Complementing the existing literature in d-hypercyclicity, we characterize disjoint supercyclicity for a finite family of weighted shift operators. Using this characterization, we answer Question 2 in a recent paper by Bès, Martin and Peris in the negative by constructing examples of disjoint supercyclic weighted shifts whose direct sum operator is hypercyclic, but the same shifts operators fail to be disjoint hypercyclic. We also show the Disjoint Blow-Up/Collapse Property and the Strong Disjoint Blow-Up/Collapse Property for disjoint supercyclicity are equivalent when dealing with a finite family with two or more weighted shifts. However, those weighted shifts operators will never satisfy the Disjoint Supercyclicity Criterion. This provides a sharp distinction between disjoint supercyclicity and supercyclicity for a single operator. We provide a partial answer to disjoint supercyclic version of Question 3 in a recent paper by Salas by showing that we can always select an additional operator to add to an family of d-supercyclic weighted shift operators while maintaining the d-supercyclicity. We also show that, in general, this additional operator cannot be another weighted shift.

Comments

Integral Equations and Operator Theory, Vol. 85, No. 2 (June 2016): 191-220. DOI.

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