### “A toolkit for constructing dilations on Banach spaces”, by Fackler and Gluck

About a week ago an interesting preprint appeared on the arxiv: “A toolkit for constructing dilations on Banach spaces“, by Stephan Fackler and Jochen Gluck. I have been studying various aspects of dilations for some years, but I haven’t really given much thought to dilation theory in general classes of Banach spaces. This paper – which is very clearly organized and written – was very refreshing for me, and in it a very general framework for proving existence of dilations in classes of Banach spaces is presented. The paper also contains a nice overview of the literature, and I was surprised by learning also about old results in, and application of, dilation theory, which I was not aware of and perhaps I should have been. The purpose of this post is to record my first impression of this paper and to put down some links to the references, which I would like to study better at some point.

Recall, that by Sz.-Nagy’s dilation theorem, given contraction $T$ acting on a Hilbert space $H$, one can always construct a unitary $U$ acting on a Hilbert space $K \supseteq H$, such that

(*)        $T^n = P_H U^n \big|_H$   ,    $n=0,1,2,\ldots$

(Here $P_H$ denotes the orthogonal projection of $K$ onto $H$.) The operator $U$ is called a unitary dilation of $T$. This simple theorem is the starting point of a ton of developments in operator theory on Hilbert spaces.

In the setting of operator on Banach spaces, we say that that an operator $T$ acting on a Banach space $X$ has a dilation, if there exists a Banach space $Y$, an invertible isometry $U : Y \to Y$, and two contractions $J : X \to Y$ and $P : Y \to X$, such that

(**)         $T^n = P U^n J$   ,    $n=0,1,2,\ldots$

It is quite easy to see that if both $X$ and $Y$ are Hilbert spaces, then this boils down to the definition (*). Moreover, invertible isometry seems like the right generalization of unitary, and examining (**) for $n=0$, we see that $J$ must be isometric, and $JP$ is the projection onto $J(X)$. In this setting it is understood that we are looking for invertible isometric dilations, and no adjective is used alongside the word “dilation”. (Other kinds of dilations can also be considered, i.e., one can search for a positive dilation, etc.) Note that for an operator to have a dilation it must be a contraction, and we shall always understand that operators for which we seek a dilation are contractions.

One very simple thing I learned from this paper is that the existence of a dilation for every contraction in the setting of all Banach spaces is a ridiculously trivial matter: one just constructs $Y = \ell^\infty(\mathbb{Z}, X)$, (the bounded functions $\mathbb{Z} \to X$), defines

$J : X \to Y$     $J(x) = (\ldots, 0, 0, x, Tx, T^2x, \ldots)$,

(where the $x$ is in the $0$th place), one lets $U$ be the left shift, and $P : Y \to X$ be the projection onto $0$th summand. (A similar construction is given in the paper, using $Y = \ell^1(\mathbb{Z}, X)$.) The key point of this paper is that this might not be very helpful unless $Y$ shares with $X$ some regularity properies, such as being a Hilbert space, reflexivity, being an $L^p$ space on a finite measure space, etc. For example, if one wants to remain in the realm of Hilbert spaces, the above construction does not work, and one needs to proceed differently (the usual proof of the dilation theorem in Hilbert spaces (see Wikipedia) uses the existence of a square root; basic, but not trivial). In this post we will always understand that the $Y$ we seek is to be chosen from within a well defined class of Banach spaces.

The authors don’t concentrate on the problem of finding a dilation for a single operator. They treat a more general problem, and this generality is actually a key to their proof. They make the following definition:

Definition: Let $\mathcal{X}$ be a class of Banach spaces and let $X \in \mathcal{X}$. A set of bounded operators on $X$, say $\mathcal{T} \subseteq L(X)$, is said to have a simultaneous dilation (in $\mathcal{X}$), if there exists a $Y \in \mathcal{X}$ and a set of invertible isometries $\{U_T\}_{T \in \mathcal{T}}$, together with contractions $J : X \to Y$ and $P : Y \to X$, such that

$T_1 T_2 \cdots T_n = P U_{T_1} U_{T_2} \cdots U_{T_n} J$

for all $n=0,1,\ldots$ and all $T_1, \ldots, T_n \in \mathcal{T}$.

The main theorem is as follows (Theorem 2.9 in the paper):

Theorem: Suppose that $\mathcal{X}$ is a family of reflexive Banach spaces, that is closed under finite $\ell^p$ direct sums (for some fixed $p \in (1,\infty)$) and closed under ultra-products. If $\mathcal{T} \subseteq L(X)$ is a family of bounded operators on $X$ has simultaneous dilation in $\mathcal{X}$, then so does the weak-operator closure of the convex hull of $\mathcal{T}$

For example, the family of all unitaries on a Hilbert space have simultaneous dilation (trivially). Since the weak operator convex hull of unitaries contains all contractions, we find that all contractions on a Hilbert space have simultaneous dilation (here we used the Theorem in the case where $\mathcal{X}$ is the class of all Hilbert spaces, and $p = 2$).

The existence of a simultaneous dilation for all contractions on a Hilbert space is only epsilon harder than Sz.-Nagy’s dilation theorem, and is brought just to illustrate. A more interesting example is that positive invertible isometries on $L^p$ are weak-operator dense in the set of all positive contractions, we get that the set of all positive contractions on $L^p$ has simultaneous dilation. The paper doesn’t exhaust all the dilation possibilities that it opens up (I guess that is why it is called a “toolkit”), and the authors suggest that the methods could be used in other situations; for example, maybe it can be used to find $*$-endomorphic dilations to CP maps on C*-algebra.

Two very nice surprises were:

1. I learned of an application of N-dilations (see this link overview of the notion in the context of a single or commuting operators on Hilbert spaces). In fact, N-dilations seem to be essential for the proof. The authors prove that a family has simultaneous dilation if and only if it has simultaneous N-dilation for every N (this is similar to a Theorem 1.2 from this paper (in a slightly different setting), but curiously there the easy direction was the direct implication. I wonder if the reverse implication there could also be proved with ultra products…).
2. I found references to several earlier work regarding dilations (even N-dilations), unfortunately, a couple of them are in languages that I cannot read. In particular, I learned that the existence of dilations in the context of $L^p$ spaces allows to obtain pointwise ergodic theorems in $L^p$ spaces, as in this paper of Akcoglu and this paper of Akcoglu and Shucheston (I knew that Sz.-Nagy’s unitary dilation quickly reduces the mean ergodic theorem for contractions in $L^2$ to von Neumann’s mean ergodic theorem for unitaries, which is rather basic given the spectral theorem; however, the mean ergodic theorem for contractions in Hilbert spaces has a very elegant proof, it is not much different from von Neumann’s original proof, if I’m not mistaken. Pointwise ergodic theorems are harder, and $L^2$ is the easiest, so this is a far better application, even in the $p=2$ case, than what I was aware of).