DOI: https://doi.org/10.31972/ticma22.01

Abstract

In this paper, we study new classes of operators on separable Banach spaces which are called extended-cyclic operators and extended-transitive operators. We study some properties of their vectors which are called extended-cyclic vectors. We show that if  is an extended-cyclic vector for , then  is also an extended-cyclic vector for  for all . Then, we show the extended-cyclicity is preserved under qsuasi-similarity. Moreover, we prove that an operator is extended-cyclic if and only if it is extended-transitive. As a consequence, the set of all extended-cyclic vectors is a dense and  set. Finally, we find some spectral properties of these operators. Particularly, the point spectrum of the adjoint of an extended-cyclic operator has at most one element of modules greater than one. Moreover, if the spectrum of an operator has a connected component subset of , then  is not extended-cyclic.

Key Words: Extended-Cyclic, Banach spaces, G__δ set.

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