The range of operators on von Neumann algebras

Abstract

Abstract. We prove that for every bounded linear operator T : X → X, where X is a non-reflexive quotient of a von Neumann algebra, the point spectrum of T * is non-empty (i.e., for some λ ∈ C the operator λI − T fails to have dense range). In particular, and as an application, we obtain that such a space cannot support a topologically transitive operator.

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

“…Theorem 4.1 is valid for the l p (N) spaces over the reals as well. Concerning the non-separable Banach space l ∞ (N) we stress that this space does not support topologically transitive operators, see [2]. On the other hand there exist operators acting on l ∞ (N) which are locally topologically transitive, see [5].…”

Topological Transitivity of Shift Similar Operators on Nonseparable Hilbert Spaces

“…A lot of work on hypercyclicity has been based on the well-known so-called hypercyclicity criterion (the Kitai-Gethner-Shapiro theorem). In [5], Bermúdez and Kalton observed that similar criterion holds for topologically transitive operators on Banach spaces.…”

“…Problem 2 (Bermúdez and Kalton [5]). Is there any characterization of nonseparable Banach spaces which support a topologically transitive operator?…”

“…This fact remains to be true if we replace ℓ 2 into ℓ p for 1 ≤ p < ∞: Bermúdez and Kalton in [5] showed that spaces like ℓ ∞ and Lðℓ 2 Þ) do not support topologically transitive operators, where Lðℓ 2 Þ is the space of all bounded operators on ℓ 2 :…”

A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to linear dynamics

“…Salas in [14], of topological transitivity of a backward unilateral weighted shift on l 2 (H), where H is a (not necessarily separable) Hilbert space, in terms of its weight sequence. The motivation of this problem comes from a work of T. Bermúdez and N.J. Kalton [5]. In this paper they showed that spaces like l ∞ (N) and l ∞ (Z) do not support topologically transitive operators.…”