n-supercyclic operators

Abstract

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

Hyponormal Operators, Weighted Shifts and Weak Forms of Supercyclicity

“…It was proved by Feldman [6,Theorem 4.9] that a normal operator is never N -supercyclic. We extend this result to the case of hyponormal operators.…”

“…Questions 1.1 and 1.2 are reminiscent of two results of Kitai [11] (in the hypercyclic case) and Bourdon [2]: no hyponormal operator can be supercyclic. Question 1.3 seems interesting because N -supercyclic operators which appear in [6] or in [3] are constructed as direct sums of supercyclic operators, and it would be nice to obtain some other, less 'ad hoc' examples. Moreover, hypercyclic and supercyclic weighted shifts have already been characterized by Salas [15,16], so it is natural to consider N -supercyclic case.…”

“…A more successful generalization of supercyclicity is the notion of N -supercyclicity, which was introduced in [6] and also studied in [3]. Observe that an operator T ∈ L(X) is supercyclic if and only if there exists a one-dimensional subspace of X whose orbit under T is dense in X.…”

“…The operator T is said to be N -supercyclic (1 N < ∞) if X has an N -dimensional subspace with dense orbit. It is shown in [6] that, for every N 2, there exist natural examples of N -supercyclic Hilbert-space operators that are not (N − 1)-supercyclic.…”

n-supercyclic and strongly n-supercyclic operators in finite dimensions

“…Later, in 1974, Hilden and Wallen [9] worked on a different set E = Kx which is a one-dimensional subspace of X, and if O(E, T ) is dense in X, then T is said to be supercyclic. Several generalisations of supercyclicity were proposed since like the one introduced by Feldman [5] in 2002. Rather than considering orbits of lines, Feldman defines an n-supercyclic operator as being an operator for which there exists an n-dimensional subspace E such that O(E, T ) is dense in X.…”