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



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> 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. 
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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.



On the possibility of extending the Sz.-Nagy dilation theorem for multiple contraction operators on Hilbert spaces
This is a much-studied problem. If you do not require the $V$'s to commute, then a dilation (even a unitary dilation) always exists, this is a theorem of Bozejko. For commuting operators $T$ (and seeking a commuting dilation) the problem is more subtle. A theorem of Ando says that two commuting contractions $T\_1, T\_2$ have a commuting unitary dilation $U\_1,U\_2$. However this result is known to fail for three or more operators; the first example was due to Parrot. As of now a characterization of which commuting tuples have a commuting dilation is still unknown. A good place to read about these results is the book "Completely Bounded Maps and Operator Algebras" by Vern Paulsen, Chapters 4 and 5. 


