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Hypercyclicity properties of commutator maps. (English) Zbl 1522.47021

Summary: We investigate the hypercyclic properties of commutator operators acting on separable Banach ideals of operators. As the main result we prove the commutator map induced by scalar multiples of the backward shift operator fails to be hypercyclic on the space of compact operators on \(\ell^2\). We also establish several necessary conditions which identify large classes of operators that do not induce hypercyclic commutator maps.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
47B47 Commutators, derivations, elementary operators, etc.
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