Parametrizing Distinguished Varieties

Jim Agler ^{∗} U.C. San Diego

La Jolla, California 92093 John E. McCarthy ^{†} Washington University St. Louis, Missouri 63130

This paper is dedicated to Joseph Cima on the occasion of his 70^{th} birthday.

Abstract A distinguished variety is a variety that exits the bidisk through the distinguished boundary. We look at the moduli space for distinguished varieties of rank (2,2).

0 Introduction

In this paper, we shall be looking at a special class of bordered algebraic varieties that are contained in the bidisk 𝔻^{2} in ℂ^{2}.

Definition 0.1 A non-empty set V in ℂ^{2} is a distinguished variety if there is a polynomial p in ℂ[z, w] such that

V = {(z, w) ∈ 𝔻^{2} : p(z, w) = 0}

and such that

\overline{V} ∩ ∂(𝔻^{2}) = \overline{V} ∩ (∂𝔻)^{2}.  (0.2)

∗Partially supported by National Science Foundation Grant DMS 0400826

†Partially supported by National Science Foundation Grant DMS 0501079

Condition (0.2) means that the variety exits the bidisk through the distinguished boundary of the bidisk, the torus. We shall use ∂V to denote the set given by (0.2): topologically, it is the boundary of V within the zero set of p, rather than in all of ℂ^{2}.

In [1], the authors studied distinguished varieties, which we considered interesting because of the following two theorems:

Theorem 0.3 Let T_{1} and T_{2} be commuting contractive matrices, neither of which has eigenvalues of modulus 1. Then there is a distinguished variety V such that, for any polynomial p in two variables, the inequality

∥p(T_{1}, T_{2})∥ ≤ ∥p∥_{V}

holds.

Theorem 0.4 The uniqueness variety for a minimal extremal Pick problem on the bidisk contains a distinguished variety V that contains each of the nodes.

It is the goal of this paper to examine the geometry of distinguished varieties more closely, and in particular to parametrize the space of all distinguished varieties of rank (2, 2) (see Definition 0.5 below).

Notice that if V is a distinguished variety, for each z in the unit disk 𝔻, the number of points w satisfying (z, w) ∈ V is constant (except perhaps at a finite number of multiple points, where the w’s must be counted with multiplicity). So the following definition makes sense:

Definition 0.5 A distinguished variety is of rank (m, n) if there are generically m sheets above every first coordinate and n above every second coordinate

The principal result of this paper, Theorem 2.1, is a parametrization of distinguished varieties of rank (2, 2).

1 Structure theory

For positive integers m and n, let

U = ( \begin{matrix}A & B \\ C & D \end{matrix} ) : ℂ^{m} ⊕ ℂ^{n} → ℂ^{m} ⊕ ℂ^{n}  (1.1)

be an (m + n)-by-(m + n) unitary matrix. Let

Ψ(z) = A + zB(I − zD)^{−1}C  (1.2)

be the m-by-m matrix valued function defined on the unit disk 𝔻 by the entries of U. This is called the transfer function of U. Let

U′ = ( \begin{matrix}D^{∗} & B^{∗} \\ C^{∗} & A^{∗} \end{matrix} ) : ℂ^{n} ⊕ ℂ^{m} → ℂ^{n} ⊕ ℂ^{m},

and let

Ψ′(w) = D^{∗} + wB^{∗}(I − wA^{∗})^{−1}C^{∗}.

Because U^{∗}U = I, a calculation yields

I − Ψ(z)^{∗}Ψ(z) = (1 − |z|^{2}) C^{∗}(I − z̄D^{∗})^{−1}(I − zD)^{−1}C,  (1.3)

so Ψ(z) is a rational matrix-valued function that is unitary on the unit circle. Such functions are called rational matrix inner functions, and it is well-known that all rational matrix inner functions have the form (1.2) for some unitary matrix decomposed as in (1.1) — see e.g. [2] for a proof. The set

V = {(z, w) ∈ 𝔻^{2} : det(Ψ(z) − wI) = 0}  (1.4)
= {(z, w) ∈ 𝔻^{2} : det(Ψ′(w) − zI) = 0}  (1.5)
= \{(z, w) ∈ 𝔻^{2} : det ( \begin{matrix}A − wI & zB \\ C & zD − I \end{matrix} ) = 0\} (1.6)

is a distinguished variety, because when |z| = 1, the eigenvalues of Ψ(z) are unimodular (and a similar statement holds for Ψ′). The converse was proved in [1]: all distinguished varieties of rank (m, n) can be represented in this way.

So the moduli space for distinguished varieties of rank (m, n) is a quotient of the space of (m + n)-by-(m + n) unitaries. Let us write U_{n}^{m} to denote the set of (m + n)-by-(m + n) unitaries decomposed as in (1.1). The following result is well-known.

Proposition 1.7 Let U and U_{1} be in U_{n}^{m}, with respective decompositions

U = ( \begin{matrix}A & B \\ C & D \end{matrix} ) U_{1} = ( \begin{matrix}A_{1} & B_{1} \\ C_{1} & D_{1} \end{matrix} ) .

Then they give rise to the same transfer function iff and only if there is an n-by-n unitary W such that

( \begin{matrix}A_{1} & B_{1} \\ C_{1} & D_{1} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & W^{∗} \end{matrix} ) ( \begin{matrix}A & B \\ C & D \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & W \end{matrix} ) .  (1.8)

Proof: By looking at the coefficients of powers of z in the transfer function, we see that U and U_{1} have the same transfer function if and only if

A = A_{1}
BD^{n}C = B_{1}D_{1}^{n}C_{1} ∀ n ∈ ℕ.  (1.9)

Equation (1.8) is equivalent to

A_{1} = A
B_{1} = BW
C_{1} = W^{∗}C
D_{1} = W^{∗}DW.

Clearly these equations imply (1.9).

To see the converse, note that the fact that U and U_{1} are unitaries and A = A_{1} means BB^{∗} = B_{1}B_{1}^{∗}. If B is invertible, define W to be B^{−1}B_{1}. This is unitary since B and B_{1} have the same absolute values, and then the equations BC = B_{1}C_{1} and BDC = B_{1}D_{1}C_{1} yield (1.8).

If B is not invertible, then A has norm one. Decompose

( \begin{matrix}A & B \\ C & D \end{matrix} )  =  ( \begin{matrix}( \begin{matrix}A′ & 0 \\ 0 & A′′ \end{matrix} ) & ( \begin{matrix}0 \\ B′′ \end{matrix} ) \\ ( 0 C′′ ) & D \end{matrix} ) ,

and apply the same argument to B′′ and C′′. □

Remark 1.10 Note that W is unique unless ∥A∥ = 1.

2 Parametrizing distinguished varieties of rank (2, 2)

In this section, we address the question of when two different unitaries in U_{2}^{2} give rise to the same distinguished variety . From the previous section we see that this is equivalent to asking when two rational matrix inner functions are isospectral.

Theorem 2.1 Let U, Ψ and V be as in (1.1), (1.2) and (1.4), with U in U_{2}^{2}. Let

U_{0} = ( \begin{matrix}A_{0} & B_{0} \\ C_{0} & D_{0} \end{matrix} )

be another unitary in U_{2}^{2}. Then U and U_{0} give rise to the same distinguished variety iff

(i) A and A_{0} have the same eigenvalues. (ii) D and D_{0} have the same eigenvalues. (iii) BC and B_{0}C_{0} have the same trace.

Proof: For simplicity in the proof we will assume that det(A) ≠ 0 and that A and D both have two eigenvalues. (We can attain the remaining cases as a limit of these).

Let

Q(z, w) = det ( \begin{matrix}A − wI & zB \\ C & zD − I \end{matrix} )  (2.2)

= \frac{det D}{det A^{∗}} det ( \begin{matrix}D^{∗} − z & wB^{∗} \\ C^{∗} & wA^{∗} − I \end{matrix} )  (2.3)
= p_{2}(z)w^{2} + p_{1}(z)w + p_{0}(z)  (2.4)
= q_{2}(w)z^{2} + q_{1}(w)z + q_{0}(w),  (2.5)

where p_{i} and q_{j} are polynomials of degree at most 2. As V is the zero set of Q, it is sufficient to prove that conditions (i) — (iii) completely determine Q. Let μ_{1} and μ_{2} be the eigenvalues of D and λ_{1} and λ_{2} be the eigenvalues of A.

We have

p_{2}(z) = det(zD − I)
= (zμ_{1} − 1)(zμ_{2} − 1),

so is determined by (ii), the eigenvalues of D. Similarly q_{2}(w) is determined by (i), the eigenvalues of A.

From (2.3) we see that the coefficient of z^{2} in Q is (det D/ det A^{∗}). Dividing (2.4) by p_{2}, we get

det(Ψ(z) − wI) = w^{2} + \frac{p_{1}(z)}{p_{2}(z)}w + \frac{p_{0}(z)}{p_{2}(z)}.  (2.6)

As Ψ is a matrix inner function, we must have that the last term in (2.6), which is the product of the eigenvalues of Ψ, is inner. Therefore

p_{0}(z) = e^{iθ} (z − \overline{μ_{1}})(z − \overline{μ_{2}}).

where

e^{iθ} = (det D/ det A^{∗}).

It remains to determine p_{1}.

Lemma 2.7 With notation as above, let

det(Ψ(z) − wI) = w^{2} − a_{1}(z)w + a_{0}(z).  (2.8)

Then

a_{1}(z) = a_{0}(z)\overline{a_{1}(\frac{1}{\overline{z}})}.  (2.9)

Proof: For any fixed z in 𝔻, there are two w’s with (z, w) in V. The function a_{0}(z) is the product of these w’s, and a_{1}(z) is their sum. Labelling them (locally) as w_{1}(z) and w_{2}(z), the right-hand side of (2.9) is

(w_{1}(z)w_{2}(z)) (\overline{w_{1}(\frac{1}{\overline{z}})} + \overline{w_{2}(\frac{1}{\overline{z}})}).

When the modulus of z is 1, because the variety is distinguished, the righthand side of (2.9) equals the left-hand side. By analytic continuation, they must be equal everywhere. □

Applying the lemma to −p_{1}/p_{2} and p_{0}/p_{2}, we get

p_{1}(z) = e^{iθ}z^{2}\overline{p_{1}(\frac{1}{\overline{z}})}.  (2.10)

Writing

p_{1}(z) = b_{2}z^{2} + b_{1}z + b_{0},

(2.10) gives the two equations

e^{iθ}\overline{b_{2}} = b_{0}
e^{iθ}\overline{b_{1}} = b_{1}.

Comparing (2.4) and (2.5), the coefficient of z^{2}w gives us b_{2} (since we know q_{2}), and hence we also know b_{0}.

Finally, if we know the coefficient of zw in the power series expansion of (2.6), we will know

b_{1}p_{2}(0) − b_{0}p′_{2}(0),

and be done. But

Ψ(z) − wI = A − wI + zBC + O(z^{2}),

so the coefficient of zw is −tr(BC), which is given by (iii). □

3 Open Problems

Two distinguished varieties are geometrically equivalent if there is a biholomorphic bijection between them.

Question 3.1 When do two unitaries give rise to geometrically equivalent distinguished varieties ?

Notice that all distinguished varieties of rank (1, n) or (m, 1) are geometrically equivalent, since they are all biholomorphic to the unit disk.

Question 3.2 When are two distinguished varieties of rank (2, 2) geometrically equivalent?

Question 3.3 What is the generalization of Theorem 2.1 to distinguished varieties of rank (2, 3) or (3, 3)?

W. Rudin showed that smoothly bounded planar domains are geometrically equivalent to distinguished varieties iff their connectivity is 0 or 1 [3].

Question 3.4 Which distinguished varieties are geometrically equivalent to planar domains?

Question 3.5 How can one read the topology of a distinguished variety from a unitary that determines it as in Section 1?

References

[1] J. Agler and J.E. McCarthy. Distinguished varieties. Acta Math. To appear.

[2] J. Agler and J.E. McCarthy. Pick Interpolation and Hilbert Function Spaces. American Mathematical Society, Providence, 2002.

[3] W. Rudin. Pairs of inner functions on finite Riemann surfaces. Trans. Amer. Math. Soc., 140:423–434, 1969.