## Orbits of hyponormal operators.(English)Zbl 0896.47020

A bounded linear operator $$A$$ on the Hilbert space $$H$$ is hypercyclic if there is a vector $$x\in H$$ such that $\text{Orb}(A,x)=\{A^nx: n=0,1,2,\cdots\}$ is dense in $$H$$. In this case we say that $$x$$ is a hypercyclic vector for $$A$$. We say that $$A$$ is supercyclic if there is a vector $$x\in H$$ such that $\{\lambda A^nx:\lambda\in\mathbb{C},n=0,1,2,\cdots\}$ is dense in $$H$$. Finally, if $\bigvee\{A^nx:n=0,1,2,\cdots\}=H$ for some $$x\in H$$ then $$A$$ is cyclic with cyclic vector $$x$$.
H. M. Hilden and L. J. Wallen [Indiana Univ. Math. J. 23, 557-565 (1974; Zbl 0274.47004)] have shown that normal operators are never supercyclic. Kitai [Invariant closed sets for linear operators, dissertation, University of Toronto (1982)] asks whether a hyponormal operator can be supercyclic. The present author shows that a hyponormal operator cannot be supercyclic thus generalizing a result of Hilden and Wallen and answering the question of Kitai. It is also shown that every hyponormal operator $$A$$ is power regular in the sense that the sequence $$\{\| A^nx\|^{1/n}\}$$ converges for every $$x\in H$$.

### MSC:

 47B20 Subnormal operators, hyponormal operators, etc.

### Keywords:

hyponormal operator; cyclic; supercyclic; hypercyclic; orbit

Zbl 0274.47004
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