Weighted orbits of hypercyclic vectors on F-spaces
We study $\mathcal{F}$-hypercyclicity of weighted orbits $λ_nT^nx$ for continuous linear operators $T$ on complex F-spaces, where $\mathcal{F}$ is a Furstenberg family with the cut--shift--and--paste (CuSP) property. We characterize the scalar sequences for which weighted hypercyclicity always implies ordinary hypercyclicity: their nonzero values must be bounded and bounded away from zero. The same condition removes weights for every such family. For the converse inclusion, we prove that a bounded scalar sequence preserves all $\mathcal{F}$-hypercyclic vectors if its backward-shift $ω$-limit system is distal and does not contain the zero sequence. For chaotic operators, minimality of this system suffices. In particular, every complex nilsequence which is not identically zero preserves $\mathcal{F}$-hypercyclicity, and equality holds when its nonzero values are bounded away from zero. We also obtain simultaneous weighted return results and stability under unimodular weights with asymptotically trivial multiplicative differences.