ON DIRECT SUM OF TOPOLOGICALLY TRANSITIVE OPERATORS
By
Peter Slaa1, Santosh Kumar2 and Marco Mpimbo3
1,2Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, Tanzania-35091
3Department of Mathemtics, School of Physical Sciences, North Eastern Hill University, Shillong, Meghalaya, India-793022
Email: masongslaa@gmail.com, drsengar2002@gmail.com, kmpimbo33@gmail.com
(Received: July 29, 2023; In format: August 21, 2023; Revised: September 19, 2023; Accepted: October 05, 2023)
DOI: https://doi.org/10.58250/jnanabha.2023.53216
Abstract
The purpose of this paper is to answer the question posed by Feldman [9] on topological transitivity which states that "If E is transitive, does it follows that direct sum E ⊕ E is topologically transitive?" We will show that this question has a positive answer under certain conditions. In particular, we dene topologically transitive operators and use them to show that the direct sum E ⊕ E of two operators is topologically transitive whenever E is topologically transitive. Then, we give some examples of a topologically transitive operator which does not satisfy topologically transitive criterion and so not topologically transitive.
2020 Mathematical Sciences Classification: 47A16, 47B02.
Keywords and Phrases: Hypercyclic operator, topologically transitive, direct sum, transitivity criterion.