### Spectral sets and distinguished varieties in the symmetrized bidisc

#### by Orr Shalit

In this post I will write about a new paper, “Spectral sets and distinguished varieties in the symmetrized bidisc“, that Sourav Pal and I posted on the arxiv, and give the background to understand what we do in that paper.

#### 1. Spectral sets and the work of Agler and McCarthy

Let be a pair of commuting operators on a Hilbert space. A set is said to be a **spectral set** for if

(*)

for every polynomial in two variables. Having such an inequality is non-trivial, as it gives control of the norm of an operator – something that is, in general, quite hard to obtain – in terms of supremum of a polynomial function on a compact set, which is a problem of different nature (note that I do not claim that it is easy).

**Remark:** The definition I gave above is not the usual one: usually one also includes in the definition the requirement that contain the joint spectrum of , and that (*) hold for all rational functions with poles off . For objects touched upon in this post, the usual definition is equivalent to the simpler one given above.

**Remark:** Generally, spectral sets are studied for tuple of commuting operators, where does not have to be equal to . In particular, the case is already very interesting, and there are some very hard results as well as open problems in this setting.

Let be the closed unit bidisc. A very influential theorem of Ando is the following:

**Theorem (T. Ando, 1963):** *Let be a commuting pair of operators on a Hilbert space such that . Then is a spectral set for , in other words: for every polynomial , *

*(**) . *

Ando’s theorem has more to it (the full theorem also asserts that is a **complete spectral set**, which is equivalent to the fact that has a commuting unitary dilation), but this version is already quite deep, it turns out. Even when are operators on a **finite dimensional** Hilbert space it is quite striking, and there is no proof of this theorem that stays within the realm of linear algebra (this in contrast to the case of a single operator, see this (expository) paper by Eliahu Levy and me).

In 2005 the paper “Distinguished varieties” by Jim Agler and John McCarthy was published, and added a fresh perspective on this topic. They proved the following surprising result.

**Theorem (Agler-McCarthy, 2005): ***Let be a pair of commuting operators on a finite dimensional Hilbert space. Then there is a one (complex)-dimensional algebraic variety such that for every polynomial ,*

(***) .

This is surprising because, even the fact that the norm of can be dominated by supping the scalar valued function on the bidisc (as in (**)) is not trivial, and this theorem tells us that we can dominate the norm of by supping on the much much smaller set (as in (***)).

Agler and McCarthy showed, furthermore, that if have no eignevalues of unit modulus, then the variety has the property that it exits the bidisc through the so-called **distinguished boundary**, that is the set . Such a variety is said to be a **distinguished variety**** in the bidisc**. They later went on and gave a complete characterization of all distinguished varieties in the bidisc.

A research objective of Sourav Pal and myself is to study spectral-set-related phenomena in various complex domains, and to ultimately understand the effect that the complex geometry of a domain has on the affiliated operator theory. In our new paper we make a first modest step, by looking for an analogoue of Agler and McCarthy’s result in a different subset of , called the **symmetrized bidisc**.

#### 2. The symmetrized bidisc and -contractions

The **symmetrized bidisc** is the subset of defined as

In other words, it is the image of the bidisc under the **symmetrization map** . The coordinates of the symmetrized bidisc are denoted and to indicate that points in come from taking **p**roducts and **s**ums of the coordinates of points in the bidisc. Agler and Young studied this domain extensively, mostly in the setting of -contractions, defined as follows.

**Definition:** *A -contraction is a a pair of commuting operators for which is a spectral set. *

A first major result obtained by these authors was that every contraction has a commuting normal dilation that lives on the distinguished boundary of the symmetrized bidisc (the distinguished boundary of is the symmetrization of the distinguished boundary of the bidisc). As a consequence, any pair that has the symmetrized bidisc as a spectral set has the symmetrized bidisc as a **complete spectral set** (see this paper, and the papers that reference it).

#### 3. The new results and what next

What Sourav and I showed in the new paper was that if is a -contraction acting on a **finite dimensional** Hilbert space, then there is one (complex)-dimensional algebraic variety such that

for every polynomial . Thus, analogously to what happens in the bidisc, the norm of is dominated by supping over the much smaller subset .

It is important to note that this result does not directly follow from Agler and McCarthy’s result, since not every -contraction is the symmetrization of a pair of commuting contractions.

One can explain this result quickly by waving hands: if and operate on a finite dimensional space, then they must satisfy some algebraic relation, thus they can be considered as one operator, or – very roughly – as a “one dimensional operator curve”. Making the argument precise requires an understanding of the model theory of -contractions, some of which is quite recent.

We also show that the variety is in many cases a distinguished variety, and we give a partial characterization of the distinguished varieties in .

These results are a first step in our program to understand spectral-set-phenomena in various complex domains. What we would like to understand better next is pairs of commuting contractions that have the unit ball as a spectral set.

Having read the post, I fail to realize why it is natural to require both members of a $\Gamma$-contraction, on the outset, to be contractions. After all, the sum of two contractions need not be a contraction, neither is $z_1+z_2$ necessarily a member of the unit disc when $z_1$ and $z_2$ are.

Thanks Eliahu. That was a typo, indeed it is not natural, and would imply that scalar pairs in are not -contractions. I fixed it.