Dynamics of the Commutator Operator

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http://urn.fi/URN:NBN:fi-fe2016092324278
Title: Dynamics of the Commutator Operator
Author: Gilmore, Clifford
Contributor: University of Helsinki, Faculty of Science, Department of Mathematics and Statistics
Thesis level: Licentiate thesis
Abstract: Linear dynamics is a rapidly evolving area of operator theory, however the only results related to the dynamics of the commutator operators have hitherto been on the characterisation of the hypercyclicity of the left and right multiplication operators. This text introduces the requisite background theory of hypercyclicity before surveying the hypercyclicity of the left and right multiplication operators. It expands on this to prove sufficient conditions for the hypercyclicity of the two-sided multiplication operator. Conditions are established under which the general class of elementary operators are never hypercyclic on Banach algebras and notably it is shown that elementary operators are never hypercyclic on the space of bounded linear operators of the Argyros-Haydon Banach space. In the first main result of this text, large classes of operators for which the induced commutator operators are never hypercyclic on separable Banach ideals are identified. In particular it is proven that commutator operators induced by compact and Riesz operators are never hypercyclic on the ideal of compact operators and that commutator operators are also never hypercyclic on the ideal of compact operators of the Argyros-Haydon Banach space. In the Hilbert space setting it is demonstrated that commutator operators induced by hyponormal operators are never hypercyclic on the ideal of Hilbert-Schmidt operators. In the second main result, nonzero scalar multiples of the backward shift operator on the Hilbert space H is identified as a strong candidate to induce a hypercyclic commutator operator on the separable ideal of compact operators on the separable Hilbert space H. However it is proven that it cannot have a dense orbit and hence is never hypercyclic. This study indicates that the commutator operator typically behaves in a non-hypercyclic fashion and that if they exist, instances of hypercyclic commutator operators are rare.
URI: URN:NBN:fi-fe2016092324278
http://hdl.handle.net/10138/167039
Date: 2014-06-10
Rights: This publication is copyrighted. You may download, display and print it for Your own personal use. Commercial use is prohibited.


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