I am presently doing research concerned with operator algebras and operator theory and I thought to write here in the hopes of seeking expert advice on an idea I had here. The classic Sz.-Nagy dilation theorem says

Given a contraction T on a Hilbert space, we are guaranteed the existence of a lager Hilbert space K containing the original H as a subspace, we have are guaranteed the existence of an isometric dilation V on K such the for all $ n \in N $ we have the equality: $ T^n = P_H V^n P_H $ where $ P_H $ is the orthogonal projection from K onto H.

This is a well known result but I wanted to look at the possibility for expanding it to many operators, meaning that if we are given contractions on H, $ T_1 ,..., T_k $ then can we guarantee the existence of a larger Hilbert space K containing he original H as a Hilbert subspace with isometric dilations defined on K, $ V_1,...,V_k $ such that this equality holds: $ \prod_{i=1} ^ {k} T_i = P_H \prod_{i=1} ^ {k} V_i P_H $

This seems too good to be true all on its own, but I thought perhaps via adding additional constraints on the Hilbert space H or perhaps on the contractions T we are given here, perhaps we can achieve this result. Perhaps I can receive advice here on the minimal conditions you think are necessary to generalize the theorem to many operators. I thank all helpers.