**151**(2002), 141-159.

An n-supercyclic operator is one that has an n-dimensional subspace with dense orbit. Hence, a 1-supercyclic operator is simply a supercylic operator. In this paper we show that, for every n > 1, there are (very natural) linear operators on Hilbert space that are n-supercyclic but not (n-1)-supercyclic. We also prove that for an n-supercyclic operator T there are n circles centered at the origin such that every component of the spectrum of T intersects at least one of these circles.

It is shown that if T_1,...,T_n are each supercyclic operators and satisfy the supercyclicity criterion with respect to the same sequence {n_k}, then their direct sum (T_1 \oplus ... \oplus T_n) is n-supercyclic. Other conditions for n-supercyclicity are also given. For cohyponormal operators some sufficient spectral conditions are given, for example:

If T is a pure cohyponormal operator and there exists an r > 0 such that for every \epsilon > 0, span{ ker(T-z) : r - \epsilon < |z| < r + \epsilon, |z| \neq r } is dense, then T is 2-supercyclic.