Joint paper with Paul Bourdon & Joel H. Shapiro
Submitted
dvi file pdf file
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.