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Disjoint Supercyclic Weighted Shifts

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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.

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Authors and Affiliations

  1. Mathematics Department, Mimar Sinan Fine Arts University, Silahşör Cad. 71, Bomonti Şişli, 34380, Istanbul, Turkey

    Özgür Martin

  2. Department of Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee, WI, 53201, USA

    Rebecca Sanders

Authors
  1. Özgür Martin
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  2. Rebecca Sanders
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Correspondence to Özgür Martin.

Additional information

Özgür Martin was supported in part by TÜBİTAK Bideb 2232, Project 114C045.

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Martin, Ö., Sanders, R. Disjoint Supercyclic Weighted Shifts. Integr. Equ. Oper. Theory 85, 191–220 (2016). https://doi.org/10.1007/s00020-016-2293-2

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  • DOI: https://doi.org/10.1007/s00020-016-2293-2

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