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Some Properties of n-Supercyclic Operators

Joint paper with Paul Bourdon & Joel H. Shapiro
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An n-supercyclic operator is one that has an n-dimensional subspace with dense orbit.   That is, T is n-supercyclic if there is an n-dimensional subspace M such that ${T^n(x) : n \geq 0, x \in M}$ is dense.   Hence, a 1-supercyclic operator is simply a supercylic operator.  In a previous paper several examples and some basic properties of n-supercyclic operators were established.

In this paper it is shown that an nxn matrix on C^n cannot be (n-1)-supercyclic.  It follows that if T is an n-supercyclic operator, then its adjoint can have at most n eigenvalues.  We also proved that a subnormal operator on an infinite dimensional Hilbert space cannot be n-supercyclic.

It was left as a question as to whether or not a hyponormal operator can be n-supercyclic or not and whether there are any (n-2)-supercyclic operators on R^n.