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Linear structure of hypercyclic vectors. (English) Zbl 0999.47009

Summary: A vector \(x\) in a Banach space \({\mathcal B}\) is called hypercyclic for a bounded linear operator \(T:{\mathcal B}\rightharpoonup{\mathcal B}\) if the orbit \(\{T^n x:n\geq 1\}\) is dense in \({\mathcal B}\). Our main result states that if \(T\) is a compact perturbation of an operator of norm \(\leq 1\) and satisfies an appropriate extra hypothesis, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for \(T\). In particular the result applies to compact perturbations of the identity. We also include applications to some weighted backward shifts and compact perturbations of the identity by weighted backward shifts. This last result in combination with a recent one that states that every Banach space admits an operator with a hypercyclic vector proves that in all Banach spaces there is an operator \(T\) with an infinite-dimensional closed subspace consisting, except for zero, of hypercyclic vectors. The main result also applies to the differentiation operator and the translation operator \(T:f(z)\rightharpoonup f(z+1)\) on certain Hilbert spaces consisting of entire functions.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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