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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.

Citations:

Zbl 0274.47004
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