Extended-Cyclic Operators

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 x is an extended-cyclic vector for T, then T^n x is also an extended-cyclic vector for T for all n∈N. 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 G_δ 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 B_0 (1), then T is not extended-cyclic.