Operator Machines on Directed Graphs

Abstract

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator ${R\,:\,X\longrightarrow\, X}$ such that the set

$$A = \{x \in X\,:\,{\left|\left|{R^n x}\right|\right|}\rightarrow \infty\}$$

is non-empty and nowhere norm-dense in X. Moreover, if ${x \in X\setminus A}$ then some subsequence of ${(R^n x)_{n=1}^\infty}$ converges weakly to x. This answers in the negative a recent conjecture of Prǎjiturǎ. The result can be extended to any Banach space containing an infinite-dimensional, complemented subspace with a symmetric basis; in particular, all ‘classical’ Banach spaces admit such an operator.