Spectral sets and distinguished varieties in the symmetrized bidisc

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 $latex T = (T_1, T_2)$ be a pair of commuting operators on a Hilbert space. A set $latex X subseteq mathbb{C}^2$ is said to be a spectral set for $latex T$ if

(*)   $latex |p(T_1,T_2)| leq sup_{(x_1,x_2) in X}|p(x_1,x_2)|,$

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 $latex X$ contain the joint spectrum of $latex T$, and that (*) hold for all rational functions with poles off $latex X$. 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 $latex T = (T_1, ldots, T_d)$ of commuting operators, where $latex d$ does not have to be equal to $latex 2$. In particular, the case $latex d = 1$ is already very interesting, and there are some very hard results as well as open problems in this setting.

Let $latex overline{D}^2 = {(z_1,z_2) : |z_1|,|z_2| leq 1}$ be the closed unit bidisc. A very influential theorem of Ando is the following:

Theorem (T. Ando, 1963): Let $latex T = (T_1, T_2)$ be a commuting pair of operators on a Hilbert space such that $latex |T_1|, |T_2| leq 1$. Then $latex overline{D}^2$ is a spectral set for $latex T$, in other words: for every polynomial $latex p$, 

(**) $latex |p(T_1, T_2)| leq sup_{(z_1,z_2) in overline{D}^2}|p(z_1, z_2)| $ . 

Ando’s theorem has more to it (the full theorem also asserts that $latex overline{D}^2$ is a complete spectral set, which is equivalent to the fact that $latex T$ has a commuting unitary dilation), but this version is already quite deep, it turns out. Even when $latex T_1,T_2$ 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 $latex T = (T_1, T_2)$ be a pair of commuting operators on a finite dimensional Hilbert space. Then there is a one (complex)-dimensional algebraic variety $latex V subset overline{D}^2$ such that for every polynomial $latex p$,

(***)  $latex |p(T_1, T_2)| leq sup_{(z_1, z_2) in V} |p(z_1, z_2) |$.

This is surprising because, even the fact that the norm of $latex p(T_1,T_2)$ can be dominated by supping the scalar valued function $latex p$ on the bidisc (as in (**)) is not trivial, and this theorem tells us that we can dominate the norm of $latex p(T_1, T_2)$ by supping $latex p$ on the much much smaller set $latex V subset overline{D}^2$ (as in (***)).

Agler and McCarthy showed, furthermore, that if $latex T_1, T_2$ have no eignevalues of unit modulus, then the variety $latex V$ has the property that it exits the bidisc through the so-called distinguished boundary, that is the set $latex mathbb{T}^2 = {(z_1, z_2): |z_1| = |z_2| = 1}$. 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 $latex mathbb{C}^2$, called the symmetrized bidisc.

2. The symmetrized bidisc and $latex Gamma$-contractions

The symmetrized bidisc is the subset of $latex mathbb{C}^2$ defined as

$latex Gamma = {(s,p) = (z_1 + z_2, z_1 z_2) : |z_1|, |z_2|leq 1}.$

In other words, it is the image of the bidisc $latex overline{D}^2 = {(z_1,z_2) : |z_1|,|z_2| leq 1}$ under the symmetrization map $latex pi(z_1, z_2) = (z_1 + z_2, z_1 z_2)$. The coordinates of the symmetrized bidisc are denoted $latex s$ and $latex p$ to indicate that points in $latex Gamma$ come from taking products and sums of the coordinates of points in the bidisc. Agler and Young studied this domain extensively, mostly in the setting of $latex Gamma$-contractions, defined as follows.

Definition: A $latex Gamma$-contraction is a a pair $latex (S,P)$ of commuting operators for which $latex Gamma$ is a spectral set. 

A first major result obtained by these authors was that every $latex Gamma$ contraction has a commuting normal dilation that lives on the distinguished boundary of the symmetrized bidisc (the distinguished boundary of $latex Gamma$ 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 $latex (S,P)$ is a $latex Gamma$-contraction acting on a finite dimensional Hilbert space, then there is one (complex)-dimensional algebraic variety $latex W subset Gamma$ such that

$latex |q(S,P)| leq sup_{(s,p) in W}|q(s,p)|$

for every polynomial $latex q$. Thus, analogously to what happens in the bidisc, the norm of $latex q(S,P)$ is dominated by supping $latex q$ over the much smaller subset $latex W$.

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

One can explain this result quickly by waving hands: if $latex S$ and $latex P$ 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 $latex Gamma$-contractions, some of which is quite recent.

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

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.