Abstract. Let T be a continuous linear operator on a locally convex topological vector space X. We show that if x ∈ X has orbit under T that is somewhere dense in X, then the orbit of x under T must be everywhere dense in X, answering a question raised by Alfredo Peris.

In this paper we prove that there are hypercyclic (n + 1)-tuples of diagonal matrices on C n and that there are no hypercyclic n-tuples of diagonalizable matrices on C n . We use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions. We then show that on real Hilbert spaces there are tuples with somewhere dense orbits that are not dense, but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit, on a real or complex space, must be dense.

Abstract. Let T be a continuous linear operator on a Hausdorff topological vector space X over the field C. We show that if T is N -supercyclic, i.e., if X has an N -dimensional subspace whose orbit under T is dense in X , then T * has at most N eigenvalues (counting geometric multiplicity). We then show that N -supercyclicity cannot occur nontrivially in the finite-dimensional setting: the orbit of an N -dimensional subspace cannot be dense in an (N + 1)-dimensional space. Finally, we show that a subnormal operator on an infinite-dimensional Hilbert space can never be N -supercyclic.

Abstract. We show that there are linear operators on Hilbert space that have ndimensional 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. Introduction. If T : H → H is a bounded linear operator on a separable Hilbert space and C ⊆ H, then the orbit ofAn operator T is said to be hypercyclic if there is a vector with dense orbit. The first example of a hypercyclic operator on a Hilbert space was given by Rolewicz [11] in 1969. He showed that if B is the backward shift, then λB is hypercyclic for any scalar λ ∈ C with |λ| > 1. In 1974 Hilden and Wallen [8] introduced the class of supercyclic operators as those operators that have a vector whose scaled orbit is dense. That is, T is supercyclic if there is a vector x such that {αT n x : n ≥ 0, α ∈ C} is dense. Hilden and Wallen showed, among other things, that any unilateral backward weighted shift is supercyclic. Hypercyclic and supercyclic operators have received considerable attention recently, especially since they arise in familiar classes of operators, such as weighted shifts [12], [13], composition operators [3], adjoints of multiplication operators on spaces of analytic functions [6] and adjoints of subnormal and hyponormal operators [5]. For a general survey of hypercyclicity, see [7].Notice that an operator is supercyclic if and only if it has a one-dimensional subspace with dense orbit. We shall say that an operator T is nsupercyclic (1 ≤ n < ∞) if there is an n-dimensional subspace whose orbit under T is dense.In this paper we shall prove that for every n ≥ 2, there are very natural operators (adjoints of multiplication operators) that are n-supercyclic but

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