  We study the characteristics of thermalizing and non-thermalizing operators
in integrable theories as we turn on a non-integrable deformation.
Specifically, we show that $\sigma^z$, an operator that thermalizes in the
integrable transverse field Ising model, has mean matrix elements that resemble
ETH, but with fluctuations around the mean that are sharply suppressed. This
suppression rapidly dwindles as the Ising model becomes non-integrable by the
turning on of a longitudinal field. We also construct a non-thermalizing
operator in the integrable regime, which slowly approaches the ETH form as the
theory becomes non-integrable. At intermediate values of the non-integrable
deformation, one distinguishes a perturbatively long relaxation time for this
operator.
