## 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
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