Operator Machines on Directed Graphs
نویسندگان
چکیده
We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X −→ X such that the set A = {x ∈ X : ||Rx|| → ∞} is non-empty and nowhere dense in X. Moreover, if x ∈ X \ A then some subsequence of (Rx)n=1 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.
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