Disjoint Cesàro-hypercyclicity was introduced recently on separable Banach spaces, where it was shown that a family of unilateral weighted backward shifts on the space $ \ell^{2}(\mathbb{N}) $ which is disjoint Cesàro-hypercyclic is already disjoint hypercyclic. We determine how far that collapse reaches. Our main result is unconditional and uses no structure on the operator. If distinct powers $ T^{r_1}, T^{r_2}, \dots, T^{r_N} $ of a single continuous operator on a separable Fréchet space form a disjoint Cesàro-hypercyclic family, then each associated difference family is disjoint hypercyclic, with a dense $ G_\delta $ set of vectors, and $ T $ consequently is hypercyclic, so an operator that is Cesàro hypercyclic without being hypercyclic can never separate the two disjoint notions through its own powers. Pairs of scalar multiples are excluded for a different reason. Outside these classes, the notions do come apart, and they do so on a Hilbert space. We construct bilateral weighted backward shifts $ B_\alpha, B_\beta $ on $ \ell^{2}(\mathbb{Z}) $ with weights taking only the values $ \tfrac12 $, $ 1 $, and $ 2 $ such that $ (B_\alpha, B_\beta^{\, 2}) $ has a dense $ G_\delta $ set of disjoint Cesàro-hypercyclic vectors while failing to be disjoint hypercyclic. What divides the two behaviours is the axis on which the shifts act, not the presence of a norm. Supporting this, we transpose the transitivity characterisation and the disjoint Cesàro-hypercyclicity criterion to separable Fréchet spaces, stated through a generating sequence of seminorms, and verify the criterion for distinct powers of the differentiation operator on $ H(\mathbb{C}) $ and of the backward shift on $ \omega $.
Citation: Fadelah Almohammedali. Disjoint Cesàro-hypercyclicity: a collapse theorem and a separating pair of bilateral shifts[J]. AIMS Mathematics, 2026, 11(11): 36188-36232. doi: 10.3934/math.20261402
Disjoint Cesàro-hypercyclicity was introduced recently on separable Banach spaces, where it was shown that a family of unilateral weighted backward shifts on the space $ \ell^{2}(\mathbb{N}) $ which is disjoint Cesàro-hypercyclic is already disjoint hypercyclic. We determine how far that collapse reaches. Our main result is unconditional and uses no structure on the operator. If distinct powers $ T^{r_1}, T^{r_2}, \dots, T^{r_N} $ of a single continuous operator on a separable Fréchet space form a disjoint Cesàro-hypercyclic family, then each associated difference family is disjoint hypercyclic, with a dense $ G_\delta $ set of vectors, and $ T $ consequently is hypercyclic, so an operator that is Cesàro hypercyclic without being hypercyclic can never separate the two disjoint notions through its own powers. Pairs of scalar multiples are excluded for a different reason. Outside these classes, the notions do come apart, and they do so on a Hilbert space. We construct bilateral weighted backward shifts $ B_\alpha, B_\beta $ on $ \ell^{2}(\mathbb{Z}) $ with weights taking only the values $ \tfrac12 $, $ 1 $, and $ 2 $ such that $ (B_\alpha, B_\beta^{\, 2}) $ has a dense $ G_\delta $ set of disjoint Cesàro-hypercyclic vectors while failing to be disjoint hypercyclic. What divides the two behaviours is the axis on which the shifts act, not the presence of a norm. Supporting this, we transpose the transitivity characterisation and the disjoint Cesàro-hypercyclicity criterion to separable Fréchet spaces, stated through a generating sequence of seminorms, and verify the criterion for distinct powers of the differentiation operator on $ H(\mathbb{C}) $ and of the backward shift on $ \omega $.
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Fadelah Almohammedali. Disjoint Cesàro-hypercyclicity: a collapse theorem and a separating pair of bilateral shifts[J]. AIMS Mathematics, 2026, 11(11): 36188-36232. doi: 10.3934/math.20261402
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