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OPERATORS WITH COMMON HYPERCYCLIC R. ARON, J. B`ES, F. LEON AND A. PERIS
 

Summary: OPERATORS WITH COMMON HYPERCYCLIC
SUBSPACES
R. ARON, J. B`ES, F. LEŽON AND A. PERIS
Abstract. We provide a reasonable sufficient condition for a fam-
ily of operators to have a common hypercyclic subspace. We also
extend a result of the third author and A. Montes [22], thereby
obtaining a common hypercyclic subspace for certain countable
families of compact perturbations of operators of norm no larger
than one.
1. Introduction
It is known that for any separable infinite dimensional Banach space
X, there is a continuous linear operator T : X X which is hyper-
cyclic; that is, there is a vector x such that the set {x, Tx, . . . , Tn
x, . . .}
is norm dense in X ([2], [5]). Moreover, a simple Baire category argu-
ment shows that the set HC(T) of such so-called hypercyclic vectors
x is a dense G in X [21], and its linear structure is well understood:
While HC(T) must always contain a dense subspace ([9], [20]), it not
always contains a closed infinite dimensional one; see [16] for a complete
characterization of when this occurs. (Throughout, when we say that

  

Source: Aron, Richard M. - Department of Mathematics, Kent State University

 

Collections: Mathematics