Common hypercyclic subspaces
The author proves a general criterion that guarantees that a family of operators \(\{T_\lambda: \lambda \in \Lambda\}\) have a common hypercyclic subspace, that is, an infinite-dimensional closed subspace that, except for the zero vector, is hypercyclic for all the operators in the family. The first examples of an operator on Banach space having a hypercyclic subspace were given in the reviewer's paper [Mich.\ Math.\ J., 43, 419--436 (1996; Zbl 0907.47023)], in which it was proved that if \(T\) satisfies Kitai's criterion for hypercyclicity and \(T^nx \to 0\) for each \(x\) on an infinite subspace, then \(T\) has a hypercyclic subspace. The author of the present paper shows that if the same is true for all the elements in the family \(\Lambda\), then the family has a common hypercyclic subspace. The methods of proof are those used by \textit{K.\,C.\thinspace Chan} and \textit{R.\,D.\thinspace Taylor jun.} [Integral Equations Oper.\ Theory 41, No.\,4, 381--388 (2001; Zbl 0995.46014)] and the reviewer and \textit{M.\,C.\thinspace Romero--Moreno} [Stud.\ Math.\ 150, No.\,3, 201--213 (2002; Zbl 1006.47009)] to give a simpler proof of the result by the reviewer. These operator algebra methods were first used in this context by \textit{K.\,C.\thinspace Chan} [J.~Oper.\ Theory 42, No.\,2, 231--244 (1999; Zbl 0997.47058)].
scientific article; zbMATH DE number 5005103