### Topics in Operator Theory, Lecture 3: Dilations of commuting operators

#### by Orr Shalit

We continue in this lecture to consider dilation theory of contractions. In the theory of Sz.-Nagy and Foias, the main route proceeded from the existence of the minimal isometric and unitary dilations, to the study of how such dilations look like, and to use them to extract information about an operator from its dilations. The only application we saw until now was von Neumann’s inequality, which is not a trivial fact, but let’s admit it: somewhat rinky dinky. But after a deeper look is taken into the structure of the minimal unitary dilation, the way for more significant applications opens. One of these applications is a functional calculus (for non-selfadjoints) that extends the holomorphic functional calculus. Another application is an affirmative solution to the invariant subspace problem for certain classes of operators. The main parts of this theory are laid down in the book Harmonic Analysis of Operators On Hilbert Space.

We will not follow that route. Rather, we will see what dilation theory can help us to understand regarding tuples of commuting operators (which is also treated to some extent in the book). Surprisingly, this will lead to a truly nifty application in function theory.

#### 1. Unitary extension of families of isometries

**Definition:** Let be commuting contractions. A commuting tuple of operators is said to be a * commuting isometric/unitary extension/dilation* of if all s are isometric/unitary and they are all either (i) extensions of the respective s (in the case of extension); or (ii) satisfy

(in the case of dilation).

**Theorem 1: ***Every tuple of commuting isometries has a unitary extension. *

**Proof: **Let be commuting isometries acting on . Let be the unitary extension of (that we constructed in the previous post), acting on a Hilbert space . We did not delve on this matter, but the is in fact a minimal unitary dilation, in the sense that

.

Now we shall define operators on with the following properties:

- are a commuting family,
- are all isometries, and,
- For , if was already an isometry, then so is .

Once we construct the above family, the proof would almost be complete: If are all unitaries, then we’ll be done. If, say, is not a unitary, then we repeat the above construction, first constructing the minimal unitary extension of , and then extending so that form commuting isometries, such that whenever is a unitary, is also a unitary. In this way, we have a commuting family extending such that at least are unitaries. Continuing this way, the proof will be complete.

It remains to define the operators , and to show that they have the desired properties. There is really no freedom in the definition, since we must have , and so (keeping in mind that we require on ) we must define

(*) ,

keeping in mind that elements of the form span . To see that this map preserves inner product (and hence well-defines an isometry), we take and , and check

.

Thus, for every , the definition (*) extends to a well defined isometry on , which clearly extends (by considering the case). Moreover, if is a unitary, then the range of is dense in , so, being isometric with dense range, is a unitary.

Finally,

,

and since elements of the form are total in , it follows that . This completes the proof.

**Exercise A: **Show that the unitary dilation constructed in the above proof is minimal.

**Exercise B: **Prove that a unitary dilation of an isometry (or isometries) is an extension.

#### 2. Ando’s theorem

If are any contractions, then we know from Exercise C in the previous post that there are isometric dilations that simultaneously dilate any noncommutative monomial. However, if the are assumed to be commuting, there is no reason that the isometries that you constructed in your solution to Exercise C will also commute (recall the construction and try to understand where this fails). In fact, we will soon see that in general, when , a -tuple of commuting contractions does not have an isometric (nor a unitary) dilation.

The case is special.

**Theorem 2 (Ando’s isometric dilation theorem): ***Every pair of commuting contractions has an isometric dilation (in fact, an isometric co-extension). *

**Proof: **The proof is “not deep”, and just boils down to finding a sufficently clever construction. Let be two commuting contractions. We define the isometric dilation as follows. We begin by defining , and

We begin by defining isometric dilation and by

,

where . Each is an isometric dilation – in fact, a co-extension – of , but they do not necessarily commute:

while

.

For to be equal to we need that . There is really no reason for equality to hold here, the fact that and commute does not help at all (think of a simple example where equality fails).

However, we have the following (writing for ):

and by commutativity and symmetry, this clearly equals . We can therefore find a unitary such that

for all (this unitary is guaranteed to exist thanks to the zeros that we stuffed in the definition; think about it). Letting , we now define and . It is now easy to check that is still a co-extension of and also that and commute. This concludes the proof.

**Corollary (Ando’s unitary dilation theorem): ***Every pair of commuting contractions has a unitary dilation. *

**Corollary (Ando’s inequality):** * for every polynomial and every pair of commuting contractions . *

**Proof:** For the proof, one proceeds as usual, with the difference that now one needs the spectral theorem for commuting normals, rather than the spectral theorem for single operators. Alternatively, one can use the theory of commutative C*-algebras.

**Exercise C: **Define the notion of “minimal dilation” for isometric and unitary dilations of commuting tuples. Prove that if a tuple has a dilation, then it has a minimal dilation. Prove that the minimal isometric/unitary dilation of a pair of contractions is **not unique**.

**Theorem 3 (commutant lifting theorem): ***Let be a contraction, and let be the minimal isometric dilation of (which, we know, is a co-extension). For every commuting with , there exists that commutes with , is a co-extension of , and satisfies . *

**Proof: **WLOG, . Let be an isometric co-extension of , given by Ando’s theorem. Then is an isometric co-extension of , so by Exercise C in the previous post, and , where and is the minimal isometric co-extension of . We can therefore write, with respect to the decomposition ,

and

.

We claim that this does the job. It is a co-extension of because is, and so , whence . All that remains to prove is that commutes with . One can see this simply by multiplying the above matrices and looking at the entry. Alternatively, being a direct sum, we have . So

.

The proof is done.

#### 3. Counter example

The following example is due to Kaijser and Varopoulos.

**Example: **Let be given by

Let .

Using your either your brain or your favourite computer algebra software (or both), you should check the following facts:

- for all ,
- for all ,
- ,
- and finally:

and so . It follows that do not have a unitary dilation (otherwise, they would satisfy a von Neumann inequality), and therefore they also cannot have an isometric dilation.

This raises the question: let be a tuple of commuting contractions, and suppose that they satisfy a von Neumann type inequality:

.

Does it follow that must have a unitary dilation? In other words, is the failure of a von Neumann type inequality the only obstruction to the existence of dilations? We leave this question for now. In the next lecture, we will apply the dilation theory that we have developed thus far to function theory.

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