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Abstract

We show that there are linear operators on Hilbert space that have $n$-dimensional subspaces with dense orbit, but no $(n-1)$-dimensional subspaces with dense orbit. This leads to a new class of operators, called the $n$-supercyclic operators. We show that many cohyponormal operators are $n$-supercyclic. Furthermore, we prove that for an $n$-supercyclic operator, there are $n$ circles centered at the origin such that every component of the spectrum must intersect one of these circles.