Dynamics of the Commutator Operator

Visa fullständig post

Permalink

http://urn.fi/URN:NBN:fi-fe2016092324278
Titel: Dynamics of the Commutator Operator
Author: Gilmore, Clifford
Medarbetare: Helsingfors universitet, matematisk-naturvetenskapliga fakulteten, institutionen för matematik och statistik
Nivå: Licentiatsavhandling
Abstrakt: 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.
Permanenta länken (URI): URN:NBN:fi-fe2016092324278
http://hdl.handle.net/10138/167039
Datum: 2014-06-10
Rights: Publikationen är skyddad av upphovsrätten. Den får läsas och skrivas ut för personligt bruk. Användning i kommersiellt syfte är förbjuden.


Filer under denna titel

Total number of downloads: Loading...

Filer Storlek Format Granska
dynamics.pdf 1.258Mb PDF Granska/Öppna

Detta dokument registreras i samling:

Visa fullständig post