Based on academic work by Bjorn Dahlgren, “On the aqueous reactions of the aminyl radical with molecular oxygen and the superoxide anion" where, per his summary in Table 2.1, details applicable radical reactions, including those associated with ozone, all importantly, with the original source reference and associated reaction rates, from which I selectively cite the following reactions:
$\ce{•O2- + O3 → •O3- + O2 (1.5 × 10^9)}$
$\ce{O3 + •OH → •HO2 + O2 (1.1 × 10^8)}$
$\ce{•HO2 → H+ + •O2- (1.35 × 10^6)}$ (pKa 4.88)
$\ce{•O3- + •OH → OH- + O3 (2.55 × 10^9)}$
which, in my opinion, constitute the likely reactions of interest in the decay of ozone as could occur in natural waters (containing dissolved oxygen and transition metals) implicitly capable of inducing formation of the superoxide radical anion ($\ce{•O2-}$) and the hydroxyl radical ($\ce{•OH}$).
In strong sunlight, photo generation of electrons combing with $\ce{H+}$ creating the hydrogen atom radical from natural photocatalyst (like dissolved organic matter, DOM) may also occur leading to further pathways:
$\ce{H+ + e- → •H (2.3 × 10^10) }$
$\ce{•H + O3 -> •HO3 (3.8 × 10^10)}$
$\ce{•HO3 -> O2 + •OH (1.1 × 10^5) Slow}$
Note, in alkaline conditions, only a very limited reaction:
$\ce{OH- + O3 → •O2- + •HO2 (70) Extremely Slow}$
which I would cite also for the distilled water decay of ozone (absence transition metals, DOM and dissolved oxygen).
So, to answer the question does ozone decay in pure water, yes, but apparently, per the reaction above, slowly. Clearly, per the cited reaction rates possible in natural water, the decay can be greatly accelerated with the creation of transient reactive oxygen species including superoxide that can participate in Fenton (with iron, and Fenton-like with other transition metals) associated radical attacks on organic matter in the presence of dissolved oxygen and minerals.
I hope this helps.