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We consider the spreading of a local operator $A$ in one-dimensional systemswith Hamiltonian $H$ by calculating the $k$-fold commutator$[H,[H,[...,[H,A]]]]$. We derive bounds for the operator norm of thiscommutator in free and interacting systems with and without disorder thusdirectly connecting the operator growth hypothesis with questions oflocalization. We analytically show that an almost factorial growth of theoperator norm - as recently proven for the random Ising model - is inconsistentwith an exponential localization of $A$. Assuming that a quasi-local unitary$U$ exists which maps $H$ onto an effective Hamiltonian $\tildeH=UHU^\dagger=\sum_n E_n \tau^z_n +\sum_{i,j} J_{ij} \tau^z_i\tau^z_j+\dots$,we show that $\tilde A=UAU^\dagger$ is a quasi-local operator which in themany-body case does not remain exponentially localized in general leading to analmost factorial norm growth. Therefore the unitary $U$ in many-body systemswith maximal norm growth either does not exist and such systems are alwaysergodic or unusual non-ergodic phases described by $\tilde H$ do exist whichviolate the operator growth hypothesis and in which operators spread, implyingthat transport will eventually set in. We analytically and symbolically verifyour results for the Anderson and Aubry-Andr\'e models. For the XXX case, thesymbolic calculations are consistent with a maximal norm growth. Furthermore,we find no indication of a weakened exponential localization of $A$, expectedfor strong disorder and low commutator orders if the unitary $U$ does exist.Finally, we try to perturbatively construct $U$ by consecutive Schrieffer-Wolfftransformations. While it is straightforward to show that this constructionconverges in the Anderson case, we find no indications for a convergence in theinteracting case, suggesting that $U$ does not exist and that many-bodylocalization is absent.

Operator Growth in Disordered Spin Chains: Indications for the Absence of Many-Body Localization