\(n\)-supercyclic operators. (English) Zbl 1006.47008

Let \(T: H\to H\) be a continuous linear operator on a separable Hilbert space. The orbit of a subset \(C\) of \(H\) under \(T\) is the union of the sets \(C,T(C),T^2(C),\dots\) . For a natural number \(n\), an operator is called \(n\)-supercyclic if there is an \(n\)-dimensional subspace of \(H\) whose orbit under \(T\) is dense in \(H\). The 1-supercyclic operators coincide with the supercyclic operators introduced in 1974 by Hilden and Wallen. In this article the author shows that there are natural operators (in fact adjoints of multiplication operators) which are \(n\)-supercyclic but not \((n-1)\)-supercyclic. The concept of an infinitely supercyclic operator is also introduced.
Section 2 of the article states known results that are needed in the rest of the paper.
In Section 3 sufficient conditions for operators to be supercyclic are presented. One of the main results asserts that the direct sum of \(n\) supercyclic operators is \(n\)-supercyclic if each of the operators satisfies the so-called supercyclicity criterion with respect to the same sequence of natural numbers. We refer the reader to the paper for details.
Section 4 is devoted to necessary conditions for an operator to be \(n\)-supercyclic. The main result of the section is an analogue of a “circle theorem” due to the author, T. L. Miller and V. G. Miller. It is proved that if \(T\) is \(n\)-supercyclic, then there are \(n\) circles centered at the origin such that every “part of the spectrum” of \(T^*\) must intersect one of the circles. This necessary condition is essential in the construction of the examples mentioned above.
Local spectral theory techniques are used to take a first step towards the characterization of cohyponormal \(n\)-supercyclic operators in Section 5. The last Section mentions several open problems concerning \(n\)-supercyclic operators.


47A16 Cyclic vectors, hypercyclic and chaotic operators
47B20 Subnormal operators, hyponormal operators, etc.
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
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