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**Bi-strictly cyclic operators.**
*(English)*
Zbl 0887.47004

Summary: The genesis of this paper is the construction of a new operator that, when combined with a theorem of Herrero, settles a question of Herrero. Herrero proved that a strictly cyclic operator on an infinite dimensional Hilbert space is never triangular. He later asks whether the adjoint of a strictly cyclic operator is necessarily triangular. We settle the question by constructing an operator \(T\) for which both \(T\) and \(T^*\) are strictly cyclic. We make a detailed study of this bi-strictly cyclic operator which leads to theorems about general bi-strictly cyclic operators. We conclude the paper with a comparison of the operator space structures of the singly generated algebras \(A(S)\) and \(A(T)\), when \(S\) is strictly cyclic and \(T\) is bi-strictly cyclic.

### MSC:

47A15 | Invariant subspaces of linear operators |

46B28 | Spaces of operators; tensor products; approximation properties |

47A66 | Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal linear operators |