Note on epsilon-cyclic operator

In this paper, we investigated the concept of ε-diskcyclic operators on a separable infinite-dimensional Hilbert space . A bounded linear operator is called -diskcyclic if there exists a vector in such that its disk orbit visits every cone of aperture . That is, for every non-zero vector in , there exist in where and in such that . Such a vector is then called an ε-diskcyclic vector for . We established several properties of ε-diskcyclic operators. In particular, we showed that every -diskcyclic operator is cyclic. Moreover, we examined the relationship between ε-diskcyclic vectors of and eigenvectors of the adjoint operator that cannot be orthogonal to each other. We also proved that if is a bounded linear operator on ; , then the direct sum ​ is -diskcyclic provided each is -diskcyclic. Finally, we presented a criterion for determining -diskcyclicity