Typical operators admit common cyclic vectors

TL;DR

This paper proves that in the space of bounded operators on a separable Hilbert space, most operators have the property that all non-zero vectors in a dense subset are hypercyclic or weakly supercyclic, depending on the topology.

Contribution

It shows that the set of operators with this universal hypercyclic property is residual in the strong and weak operator topologies.

Findings

Residual set of operators with hypercyclic vectors in dense subset

Residual set of operators with weakly supercyclic vectors in dense subset

Results hold in the unit ball of L(H) multiplied by R>1 or R>0

Abstract

Given a countable dense subset D of an infinite-dimensional separable Hilbert space H the set of operators for which every vector in D except zero is hypercyclic (weakly supercyclic) is residual for the strong (resp. weak) operator topology in the unit ball of L(H) multiplied by R>1 (resp. R>0)