A vector x in a Hilbert space H is called irregular for an operator T : H → H provided that sup n T n x = ∞ and inf n T n x = 0. We establish some basic properties of operators having irregular vectors and present examples that highlight the relationship, or lack thereof, between irregularity and hypercyclicity.

No abstract

Abstract. We give a spectral characterization of the norm closure of the class of all weakly hypercyclic operators on a Hilbert space. Analogous results are obtained for weakly supercyclic operators.2000 Mathematics Subject Classification. Primary 47A16. Secondary 47B37.

Abstract. For a scalar λ, two operators T and S are said to λ-commute if T S = λST . In this note we explore the pervasiveness of the operators that λ-commute with a compact operator by characterizing the closure and the interior of the set of operators with this property.For a scalar λ, two operators T and S are said to λ-commute if T S = λST . This notion has received attention in the past. In particular, both [B] and [CC] examined the concept. [B] shows that if T λ-commutes with a compact operator, then T has a non-trivial hyperinvariant subspace. In [CC] it is shown that for an integer n, the commutants of T and T n are different if and only if there is a non-zero operator Y and an nth root of unity, λ = 1, such that T Y = λY T . In [L] some additional properties and examples of λ-commuting operators are explored. In this note we explore the pervasiveness of the operators that λ-commute with a compact operator by characterizing the closure and the interior of the set of operators with this property.Throughout we will denote by B(H) the set of all operators on the Hilbert space H, and by B 0 (H) and B 00 (H), respectively, the set of all compact and finite rank operators on H. For a complex number λ define the following classes of operators:Note that when λ = 1, these classes and their topological properties were examined in [CP]. If the scalar λ is understood, write C 1 , C 2 , and C 3 .It is clear that C 1 ⊂ C 2 ⊂ C 3 and that all three sets are invariant under similarities. Every operator in C 1 has a finite-dimensional invariant subspace

No abstract