The Sz.-Nagy dilation theorem says that for a Hilbert space $H$ with nonexpansive operator $T$, there is a larger space $H'$ containing $H$ and a unitary operator $U$ on $H'$ such that for all $x \in H$ and all $n$, $T^n x = P U^n x$ where $P$ is the projection from $H'$ to $H$.
Is there anything analogous to the Sz.-Nagy dilation theorem except for uniformly convex Banach spaces?
A few more details:
I have a result for linear isometries on uniformly convex Banach spaces with "power type" modulus of convexity, $\eta(\varepsilon) = C \varepsilon^p$. I want to extend the result to linear nonexpansive operators on the same class of spaces. I need the larger space to be uniformly convex, ideally with the same power type.
It would be a bonus if this also holds for power-bounded operators $T$ (i.e. $\|T^n x\| \leq C\|x\|$). I don't actually need the other map to be an isometry, just something that is power bounded from above and below (i.e. $c\|x\| \leq \|T^n x\| \leq C\|x\|$).