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

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

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

(**) \|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 \overline{D}^2 is a complete spectral set, which is equivalent to the fact that T has a commuting unitary dilation), but this version is already quite deep, it turns out. Even when 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 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 V \subset \overline{D}^2 such that for every polynomial p,

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

Agler and McCarthy showed, furthermore, that if T_1, T_2 have no eignevalues of unit modulus, then the variety V has the property that it exits the bidisc through the so-called distinguished boundary, that is the set \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 \mathbb{C}^2, called the symmetrized bidisc.

2. The symmetrized bidisc and \Gamma-contractions

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

\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 \overline{D}^2 = \{(z_1,z_2) : |z_1|,|z_2| \leq 1\} under the symmetrization map \pi(z_1, z_2) = (z_1 + z_2, z_1 z_2). The coordinates of the symmetrized bidisc are denoted s and p to indicate that points in \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 \Gamma-contractions, defined as follows.

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

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

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

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

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

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

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

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