On the possibility of extending the Sz.-Nagy dilation theorem for multiple contraction operators on Hilbert spaces

Question

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.

Answer 1

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.

Answer 2

To add to what Mike Jury wrote:

Paulsen's book also contains what he calls the Sz.-Nagy-Foias Theorem, stating that $n$ doubly commuting contractions dilate to $n$ doubly commuting unitaries. We say $T_1$ and $T_2$ doubly commute if $T_1T_2 = T_2T_1$ and $T_1T_2^* = T_2^*T_1$, that is $C^*(1,T_1)$ and $C^*(1,T_2)$ commute. This is stronger than just commuting and so avoids Parrott's example that Mike mentioned.

A second avenue for dilation is what Yemon Choi mentioned, row contractions. A tuple $(T_1,\cdots, T_n)$ is a row contraction if the row operator $T = [T_1 \cdots T_n] \in B(H^n, H)$ is a contraction. In the non-commutative case Frazho, Bunce and Popescu showed that this tuple of operators dilates to a row isometry $V = [V_1 \cdots V_n]$ where V^*V = I, that is $(V_1,\cdots, V_n)$ dilates $(T_1,\cdots,T_n)$ and $V_1,\cdots, V_n$ are isometries with pairwise orthogonal ranges.

If $T = (T_1,\cdots, T_n)$ is a commuting row contraction (i.e. the $T_i$'s pairwise commute) Muller-Vasilescu and Arveson proved the following dilation result: every commuting row contraction dilates to a canonical operator $\tilde T = M^{\alpha} \oplus U$ where $M = (M_{z_1}, \cdots, M_{z_n})$ is the tuple of coordinate multipliers on Drury-Arveson space (symmetric Fock space) and $U$ is a spherical unitary.

Besides these results which fall into the class that you seem interested in there are many directions that dilation theory has taken. For instance, dilating relations other than commutation and dilating representations of non-selfadjoint algebras, to name only two.