Shifts on trees versus classical shifts in chain recurrence

Abstract

We construct continuous (and even invertible) linear operators acting on Banach (even Hilbert) spaces whose restrictions to their respective closed linear subspaces of chain recurrent vectors are not chain recurrent operators. This construction completely solves in the negative a problem posed by Nilson C. Bernardes Jr. and Alfred Peris on chain recurrence in Linear Dynamics. In particular: we show that the non-invertible case can be directly solved via relatively simple weighted backward shifts acting on certain unrooted directed trees; then we modify the non-invertible counterexample to address the invertible case, but falling outside the class of weighted shift operators; and we finally show that this behaviour cannot be achieved via classical (unilateral neither bilateral) weighted backward sifts (acting on $N$ and $Z$ respectively) by noticing that a classical shift is a chain recurrent operator as soon as it admits a non-zero chain recurrent vector.