Linear operators with wild dynamics (Q2884431)

In some sense, this nice paper combines two rich subjects of Functional Analysis: hypercyclic operators and the outstanding theory, due to Gowers and Maurey, of spaces with few operators.NEWLINENEWLINELet \(X\) be a (real or complex) separable infinite-dimensional Banach space; for a bounded linear operator \(T:X\rightarrow X\), consider the set NEWLINE\[NEWLINE A_{T}=\left\{ x\in X:\left\| T^{t}x\right\| \rightarrow\infty\right\} . NEWLINE\]NEWLINE Among various other results, the author constructs a bounded linear operator \(R:X\rightarrow X\) such thatNEWLINENEWLINE(i) \(A_{R}\) is not dense,NEWLINENEWLINE(ii) \(A_{R}\) has nonempty interior,NEWLINENEWLINE(iii) \(R=I+K,\) where \(I\) is the identity and \(K\) is a compact operator.NEWLINENEWLINEThis result answers two questions of \textit{P. Hájek} and \textit{R. J. Smith} [Integral Equations Oper. Theory 67, No. 1, 15--31 (2010; Zbl 1215.47005)].