Algebraic pairs of isometries

Jim Agler ^{∗} U.C. San Diego La Jolla, CA 92093 Greg Knese U.C. Irvine Irvine CA 92697

John E. McCarthy ^{†} Washington University St. Louis, MO 63130

March 31, 2009

Abstract We consider pairs of commuting isometries that are annihilated by a polynomial. We show that the polynomial must be inner toral, which is a geometric condition on its zero set. We show that cyclic pairs of commuting isometries are nearly unitarily equivalent if they are annihilated by the same minimal polynomial.

0 Introduction

Isometries form one of the best-understood classes of operators on Hilbert spaces. By the von Neumann-Wold decomposition, every isometry is the direct sum of a unitary operator and a vector-valued shift. The non-unitary part of the isometry is called the pure part.

Pairs of commuting isometries are more complicated. If the first isometry is pure, it can be modeled as a vector-valued shift, multiplication by the coordinate function on H^{2} ⊗ L, where L is a Hilbert space of the appropriate dimension, and H^{2} is the Hardy space. The second isometry then becomes multiplication by an operator-valued inner function on L, i.e. an analytic operator-valued function on the unit disk 𝔻 whose boundary values are isometric a.e. [8, 18].

∗Partially supported by National Science Foundation Grant DMS 0400826

†Partially supported by National Science Foundation Grant DMS 0501079

Although this description is very powerful, it leaves open many questions. The purpose of this note is to study a restricted class of pairs of commuting isometries V = (V_{1}, V_{2}), namely ones that satisfy an algebraic relation: q(V) = 0 for some polynomial q of two variables. We shall call such a pair an algebraic isopair, and we shall say that an isopair is pure if both isometries are pure. Pure algebraic isopairs turn out to have a rich structure.

It is easy to find an algebraic isopair annihilated by the polynomial z^{2}−w^{2}, but a moment’s thought shows that none can be annihilated by z^{2} − 2w^{2}. The polynomial 1 − zw can annihilate an isopair, but only if this is a pair of unitaries whose joint spectrum is contained in

𝕋^{2} ∩ {(z, w) : 1 − zw = 0}.

(Throughout the paper, we shall use the notation that 𝔻 is the open unit disk {z : |z| < 1}, 𝕋 is the unit circle {z : |z| = 1}, and 𝔼 is the exterior of the closed disk {z : |z| > 1}.) No pure isopair is annihilated by 1 − zw.

What polynomials q can be the minimal annihilating polynomial for some pure isopair?

Theorem 1.20: Let V = (V_{1}, V_{2}) be a pure algebraic isopair on a Hilbert space H. Then there exists a square-free inner toral polynomial q that annihilates V. Moreover, if p is any polynomial that annihilates V, then q divides p.

A polynomial q is called an inner toral polynomial if its zero set lies in 𝔻^{2} ∪𝕋^{2} ∪𝔼^{2}; the zero set of an inner toral polynomial is called a distinguished variety. We discuss these in Section 1.

Theorem 1.20 gives a way to construct algebraic isopairs. Start with an inner toral polynomial q; put a nice measure μ on Z_{q} ∩ 𝕋^{2}; construct the Hardy space H^{2}(μ) that is the closure in L^{2}(μ) of the polynomials; and look at the pair of operators on H^{2}(μ) given by multiplication by the coordinate functions. In a way that will be made precise in Section 3, this construction in some sense gives you all cyclic algebraic isopairs.

However, they also arise in another setting. In [6, 20], it is shown that on every finitely connected planar domain R there is a pair of inner functions (u_{1}, u_{2}) that map the domain conformally onto some distinguished variety intersected with the bidisk. If ν is a measure on ∂R that is a log-integrable weight times harmonic measure, one can form a Hardy space H^{2}(ν) (provided every component in the complement of R has interior, this is just the closure in L^{2}(ν) of all functions analytic in a neighborhood of \overline{R}). Multiplication by u_{1} and u_{2} on H^{2}(ν) then give a pure cyclic algebraic isopair.

In Section 2, we show that a q-isopair (an isopair annihilated by q ∈ ℂ[z, w]) can almost be broken up into a direct sum of isopairs corresponding to each of the irreducible factors of q. Specifically, we have:

Theorem 2.1: Let V = (V_{1}, V_{2}) be a pure algebraic isopair with minimal polynomial q, and let q_{1}, q_{2}, …, q_{N} be the (distinct) irreducible factors of q. If both V_{1} and V_{2} have finite dimensional cokernels, then V has a finite codimension invariant subspace K on which

V |_{K}= W_{1} ⊕ W_{2} ⊕ ⋯ ⊕ W_{N}

where W_{j} is a q_{j}-isopair, j = 1, …, N.

The restriction to K is essential. Our main result says that any two pure cyclic algebraic isopairs are nearly unitarily equivalent if and only if they have the same minimal polynomial. “Nearly” means after restricting to a finite codimensional invariant subspace. So we say that two pairs are nearly unitarily equivalent if and only if each one is unitarily equivalent to the other restricted to a finite codimensional invariant subspace. We say a pair is nearly cyclic if, when restricted to a finite codimensional invariant subspace, it becomes cyclic. We have:

Theorem 3.3 Any two nearly cyclic pure isopairs are nearly unitarily equivalent if and only if they have the same minimal polynomial.

In Section 4, we find a function-theoretic consequence of the operator theory. Given a polynomial q, one can ask when Y = Z_{q} ∩𝕋^{2} is polynomially convex. Apart from the trivial case of when q has factors of (z − e^{iθ}) or (w − e^{iθ}), the answer is that Y fails to be polynomially convex if and only if q has an inner toral factor.

Theorem 4.1 Let q be a polynomial in two variables with no linear factors. Then Y = Z_{q} ∩ 𝕋^{2} is polynomially convex if and only if q has no inner toral factor.

1 Inner isopairs

Definition 1.1 An isopair is a pair V = (V_{1}, V_{2}) of commuting isometries. An algebraic isopair is an isopair that satisfies a polynomial p ∈ ℂ[z, w]:

p(V) = 0

in which case V may be called a p-isopair.

Definition 1.2 An isopair V is pure if

⋂_{m≥0} V_{1}^{m}H = {0} = ⋂_{n≥0} V_{2}^{n}H.

Suppose V = (V_{1}, V_{2}) is an isopair with V_{1} pure. Let k be the dimension of the cokernel of V_{1} (which is the Fredholm index of V_{1}^{∗}, and which we will call the multiplicity of V_{1}). Then standard model theory for isometries, as described for example in [8] or [18], says that V can be modeled on the Hilbert space H^{2} ⊗ L, where L is a Hilbert space of dimension k, and H^{2} is the Hardy space. There is a B(L) valued inner function Φ so that V is unitarily equivalent to the pair (M_{z}, M_{Φ}), where M_{z} is multiplication by the independent variable (times I_{L}) and M_{Φ} is multiplication by the operatorvalued function Φ. If V_{1} is of finite multiplicity, k is finite and Φ is matrixvalued. If, in addition, V_{2} is of finite multiplicity, then Φ is a matrix-valued rational inner function, i.e. an analytic matrix-valued function, each of whose entries is rational with poles outside the closed unit disk, and such that everywhere on the unit circle the matrix is unitary. Finally, if V_{2} is also pure, this means that Φ is not the direct sum of a constant unitary and another inner function. We shall say in this case that the function Φ is pure.

Example 1.3 A simple example is the pair V = (V_{1}, V_{2}) = (M_{z^{2}}, M_{z^{3}}) on the classical Hardy space H^{2}. In this case V satisfies V_{1}^{3} − V_{2}^{2} = 0. This pair is unitarily equivalent to the pair (M_{z}, M_{Φ}) on H^{2} ⊗ ℂ^{2} where

Φ(z) = (\begin{matrix}0 & z^{2} \\ z & 0 \end{matrix}) .

The unitary equivalence U comes from mapping f = f_{1}(z^{2}) + zf_{2}(z^{2}) ∈ H^{2} to (f_{1}(z), f_{2}(z))^{t} ∈ H^{2} ⊗ ℂ^{2}, where here we are dividing f into its even and odd parts. It is easy to check that this Hilbert space isomorphism intertwines the two operator pairs:

UM_{z^{2}}(f_{1}(z^{2}) + zf_{2}(z^{2})) = U(z^{2}f_{1}(z^{2}) + z^{3}f_{2}(z^{2})) = (\begin{matrix}zf_{1}(z) \\ zf_{2}(z) \end{matrix}) = M_{z} (\begin{matrix}f_{1}(z) \\ f_{2}(z) \end{matrix})

(and similarly for M_{z^{3}} on H^{2} and M_{Φ} on H^{2} ⊗ ℂ^{2}).

Pure isopairs cannot be annihilated by arbitrary polynomials. We shall show below (Theorem 1.20) that there is a minimal annihilating polynomial for any algebraic isopair. This minimal polynomial must be inner toral. To define this, let us first establish the notation that 𝔻 is the open unit disk, 𝕋 is the unit circle, and 𝔼 is the exterior of the closed unit disk in the plane.

Definition 1.4 A distinguished variety is an algebraic set A in ℂ^{2} such that

A ⊆ 𝔻^{2} ∪ 𝕋^{2} ∪ 𝔼^{2}.

A polynomial is called inner toral if its zero set is a distinguished variety.

The terminology “inner toral” (and explanation for it) is from [3]. The idea behind the name “distinguished variety” is that the variety exits the bidisk through the distinguished boundary. There is a close connection between pure algebraic isopairs and distinguished varieties. One theorem along these lines, proved first in [1] and then, by a different method, in [13], is:

Theorem 1.5 Let A be a distinguished variety. Then there is a pure rational matrix-valued inner function Φ so that, if

det(Φ(z) − wI) = \frac{q(z, w)}{p(z, w)},  (1.6)

then A is the zero-set of q. Moreover, if Φ is any pure rational matrix-valued inner function, and the polynomial q is defined by (1.6), then the zero set of q is a distinguished variety.

A sort of converse to Theorem 1.5 is that the minimal annihilating polynomial of any pure isopair is inner toral.

Definition 1.7 V = (V_{1}, V_{2}) is an inner isopair if V is a pure isopair satisfying

q(V) = 0

where q ∈ ℂ[z, w] is inner toral.

Theorem 1.8 Every pure algebraic isopair is inner.

To prove Theorem 1.8, we shall need some preliminary results. First, we shall establish some basic facts about cyclic isopairs.

Definition 1.9 An isopair V is cyclic if there exists f ∈ H such that

ℂ[V]f := {p(V)f : p ∈ ℂ[z, w]}

is dense in H.

Definition 1.10 We say a polynomial p ∈ ℂ[z, w] has degree (n, m) if it has degree n in z and m in w.

Lemma 1.11 If V is a cyclic isopair satisfying p(V) = 0 where p has degree (n, m), then for all (α, β) ∈ 𝔻^{2},

dim[ker(V_{1} − αI)^{∗} ∩ ker(V_{2} − βI)^{∗}] ≤ 1
dim ker(V_{1} − αI)^{∗} ≤ m
dim ker(V_{2} − βI)^{∗} ≤ n.

Proof: Applying appropriate Möbius transformations to V_{1} and V_{2}, we can assume without loss of generality that (α, β) = (0, 0). Let f be a cyclic vector.

Suppose g ∈ ker V_{1}^{∗} ∩ ker V_{2}^{∗}. Then

⟨Q(V)f, g⟩ = ⟨Q(0, 0)f, g⟩

for any Q ∈ ℂ[z, w]. If dim(ker V_{1}^{∗} ∩ker V_{2}^{∗}) > 1 then we could find a nonzero vector in ker V_{1}^{∗} ∩ ker V_{2}^{∗} perpendicular to f. This would contradict cyclicity. So, dim ker V_{1}^{∗} ∩ ker V_{2}^{∗} ≤ 1.

If dim ker V_{1}^{∗} > m then we can choose g ∈ ker V_{1}^{∗} perpendicular to V_{2}^{j}f for j = 0, 1, …, m − 1. Now observe that

0 = p(V)^{∗}g = p(0, V_{2})^{∗}g.

Let Q ∈ ℂ[z, w] and write

Q(0, w) = s(w)p(0, w) + r(w)

where r has degree less than m (by the Euclidean algorithm). Then,

Q(V)^{∗}g  = Q(0, V_{2})^{∗}g
= s(V_{2})^{∗}p(0, V_{2})^{∗}g + r(V_{2})^{∗}g

= r(V_{2})^{∗}g and so

⟨Q(V)f, g⟩ = ⟨r(V_{2})f, g⟩ = 0

since g is perpendicular to V_{2}^{j}f for j = 0, 1, …, m − 1. This contradicts cyclicity (Q was arbitrary). So, dim ker V_{1}^{∗} ≤ m.

Similarly, dim ker V_{2}^{∗} ≤ n. □ Let mult denote the multiplicity of an isometry:

multV_{i} = dim ker V_{i}^{∗}.

We shall say that an isopair V = (V_{1}, V_{2}) has finite multiplicity if both V_{1} and V_{2} do.

Lemma 1.12 Let V = (V_{1}, V_{2}) be a cyclic pure algebraic isopair and suppose V is annihilated by an irreducible polynomial q ∈ ℂ[z, w]. Then

(1) q is inner toral,

(2)

deg q = (multV_{2}, multV_{1}), and

(3) q divides any polynomial p that satisfies p(V) = 0.

Proof: By Lemma 1.11, we have

deg q ≥ (multV_{2}, multV_{1})  (1.13)

(in each component separately). Now, V has a model as a pair of multiplication operators (M_{z}, M_{Φ}) on H^{2} ⊗ ℂ^{k} where k = multV_{1}. Since V_{2} has finite multiplicity (by Lemma 1.11), Φ must be a rational matrix valued inner function.

Let

f(z, w_{1}, w_{2}) = \frac{q(z, w_{1}) − q(z, w_{2})}{w_{1} − w_{2}}.

Letting

Q(z, w) = f(zI, wI, Φ(z)) = (q(zI, wI) − q(zI, Φ(z))) (wI − Φ(z))^{−1}, (1.14) we see that

Q(z, w)(wI − Φ(z)) = q(z, w)I.  (1.15)

Now, Q is not identically zero (else q would be also), and Q has lower degree in w than q. So, the nonzero entries of the matrix polynomial Q cannot vanish identically on Z_{q}, the zero-set of q.

As

(wI − Φ(z)) Q(z, w) = q(z, w)I
= 0  on Z_{q},

we have

w Q(z, w) = Φ(z) Q(z, w) on Z_{q}.

So if p ∈ ℂ[z, w] annihilates V, then

p(zI, Φ(z))Q(z, w) = p(z, w)Q(z, w) = 0  on Z_{q}.  (1.16)

As q is irreducible, and Q does not vanish identically on Z_{q}, (1.16) shows that q divides p, proving (3).

Now consider d ∈ ℂ(z)[w], given by

d(z, w) = det(wI − Φ(z)).

By Cayley-Hamilton, d(z, Φ(z)) ≡ 0, and therefore the numerator of d annihilates V = (M_{z}, M_{Φ}). By the above, q divides the numerator of d and since the degree of the numerator of d is k = multV_{1}, we see that k is greater than or equal to the degree of q in w. The reverse inequality is in (1.13), so these two numbers are equal. Interchanging the roles of V_{1} and V_{2}, we may conclude

deg q = (multV_{2}, multV_{1})

and this proves the second claim of the proposition. Also, the zero set of d is inner toral and this implies Z_{q} is inner toral, since Z_{q} ⊂ Z_{d}. This proves the first claim. □

If V is a cyclic isopair, then in particular it is a cyclic subnormal pair, and so has another nice representation. For μ any compactly supported measure in ℂ^{2}, let P^{2}(μ) denote the closure of the polynomials in L^{2}(μ). Then we have the following representation; see [5, 15] and references therein for details.

Theorem 1.17 Let V be a cyclic isopair on the Hilbert space H, with cyclic vector u. Then there is a positive Borel measure μ on 𝕋^{2} and a unitary operator U from H onto P^{2}(μ) that maps u to the constant function 1, and such that U intertwines V with the pair (M_{z}, M_{w}) of multiplication by the coordinate functions.

Theorem 1.17 makes it easy to prove that the minimal polynomial of a pure algebraic isopair is square-free.

Lemma 1.18 Suppose V is a pure p-isopair, and the irreducible factors of p are p_{i}, each with multiplicity t_{i}:

p = ∏ p_{i}^{t_{i}}.

Let q = ∏ p_{i}. Then q(V) = 0.

Proof: Choose some vector u. Let

K = \overline{ℂ[V]u}

and let T = V |_{K}. By Theorem 1.17, T is unitarily equivalent to (M_{z}, M_{w}) on some P^{2}(μ). As p(T) = 0, we must have that p vanishes on the support of μ. Therefore so does q, and so

∥q(V)u∥^{2} = ∫ |q|^{2}dμ = 0.

As u was arbitrary, we must have that q(V) = 0. □

Lemma 1.19 Suppose V is a pure q-isopair where q ∈ ℂ[z, w] is a product of distinct irreducible factors and V is not annihilated by any factor of q. Then, q is inner toral and divides any polynomial that annihilates V.

Proof: First, we claim that any irreducible factor of q is inner toral. Let q_{0} be an irreducible factor and write q = q_{0}q_{1}. Then, u := q_{1}(V)u_{0} ≠ 0 for some u_{0} in our Hilbert space. Consider the cyclic subspace K generated by u

K := \overline{ℂ[V]u} = ∨\{g(V)u : g ∈ ℂ[z, w]\},

where ∨ denotes the closed linear span. Let T = V |_{K} be the pure q_{0}-isopair obtained by restricting V to the invariant subspace K. By Lemma 1.12, q_{0} must be inner toral. As q_{0} was an arbitrary irreducible factor, all factors of q are inner toral, and this implies q is inner toral.

Also, if g(V) = 0 for some g ∈ ℂ[z, w], then g(T) = 0 and by Lemma 1.12 this implies q_{0} divides g. As q_{0} was an arbitrary irreducible factor of q, we see that q divides g. This proves the second claim of the lemma. □

Putting together what we have proved, we get the following theorem, which contains Theorem 1.8.

Theorem 1.20 Let V be a pure algebraic isopair. Then there exists a squarefree inner toral polynomial q that annihilates V. Moreover, if p is any polynomial that annihilates V, then q divides p.

We shall call the polynomial q the minimal polynomial of V.

2 Decomposition of algebraic isopairs

In this section we show how algebraic isopairs are nearly a direct sum of algebraic isopairs annihilated by irreducible polynomials.

Theorem 2.1 Let V be a pure algebraic isopair with minimal polynomial q, and let q_{1}, q_{2}, …, q_{N} be the (distinct) irreducible factors of q. If V has finite multiplicity, then V has a finite codimension invariant subspace K on which

V |_{K}= W_{1} ⊕ W_{2} ⊕ ⋯ ⊕ W_{N}

where W_{j} is a q_{j}-isopair, j = 1, …, N.

As a quick example, consider the reducible algebraic set z^{2} = w^{2}. We can define a Hilbert space by defining

||p||^{2} = ∫_{0}^{2π} |p(e^{iθ}, e^{iθ})|^{2}dθ + ∫_{0}^{2π} |p(e^{iθ}, −e^{iθ})|^{2}dθ

for each p ∈ ℂ[z, w] and then completing this to an H^{2} space. The pair (M_{z}, M_{w}) will be a (z^{2} − w^{2})-isopair and the decomposition from the above proposition consists of letting K be the functions that vanish at (0, 0), and dividing this Hilbert space into the functions that are a multiple of z − w and those that are a multiple of z + w. K^{⊥} is the constant functions.

Before we prove the theorem we need the following.

Lemma 2.2 Suppose p ∈ ℂ[z, w] is inner toral and reducible p = p_{1}p_{2}. If V is a pure p-isopair, then

ran p_{1}(V) ⊥ ran p_{2}(V)

(ran denotes the range).

Proof: Since p_{1} is inner toral, it is a fact that p_{1} is symmetric in the sense that

z^{n}w^{m}\overline{p_{1}(1/z̄, 1/w̄)} = μp_{1}(z, w)

where μ is a unimodular constant and (n, m) is the degree of p_{1} (see [13]). In fact, we may assume μ = 1 by replacing p_{1} with an appropriate constant multiple. Hence, if we write

p_{1}(z, w) = ∑_{j=0}^{n} ∑_{k=0}^{m} a_{jk}z^{j}w^{k}

it follows that \overline{a_{jk}} = a_{(n−j)(m−k)}. This can be used to deduce

p_{1}(V)^{∗}V_{1}^{n}V_{2}^{m} = p_{1}(V)

since V_{1}, V_{2} are isometries.

Then,

p_{1}(V)^{∗}p_{2}(V) = p_{1}(V)^{∗}(V_{1}^{n}V_{2}^{m})^{∗}V_{1}^{n}V_{2}^{m}p_{2}(V)
= (V_{1}^{n}V_{2}^{m})^{∗}p_{1}(V)^{∗}V_{1}^{n}V_{2}^{m}p_{2}(V)
= (V_{1}^{n}V_{2}^{m})^{∗}p_{1}(V)p_{2}(V)
= (V_{1}^{n}V_{2}^{m})^{∗}p(V) = 0

So, for any f, g ∈ H

⟨p_{2}(V)f, p_{1}(V)g⟩ = ⟨p_{1}(V)^{∗}p_{2}(V)f, g⟩ = 0.

Hence,

ran p_{1}(V) ⊥ ran p_{2}(V)
□

Proof of Theorem 2.1: Let p = q_{2}q_{3} ⋯ q_{N} (where q_{1}, q_{2}, …, q_{N} come from the statement of the proposition). We will show that V has a finite codimension invariant subspace on which V is the direct sum of a q_{1}-isopair and a p-isopair. The proposition will then follow by induction.

By Lemma 2.2, ran q_{1}(V) and ran p(V) are orthogonal. Let K′ = (ran q_{1}(V)+ ran p(V))^{⊥}. The assumption that V has finite multiplicity implies that K′ is finite dimensional as follows. First note that as in the previous lemma we may assume that p and q_{1} are symmetric, so that we have the formulas:

p(V)^{∗} = V_{1}^{∗n}V_{2}^{∗m}p(V) q_{1}(V)^{∗} = V_{1}^{∗j}V_{2}^{∗k}q_{1}(V)

where the degree of p is (n, m) and the degree of q_{1} is (j, k). If f ∈ K′, then 0 = p(V)^{∗}f = V_{1}^{∗n}V_{2}^{∗m}p(V)f and 0 = q_{1}(V)^{∗}f = V_{1}^{∗j}V_{2}^{∗k}q_{1}(V). Since V is assumed to have finite multiplicity, the kernels of V_{1}^{∗n}V_{2}^{∗m} and V_{1}^{∗j}V_{2}^{∗k} are both finite dimensional. Hence, the ranges of p(V) |_{K′} and q_{1}(V) |_{K′} are both finite dimensional (since they map into kerV_{1}^{∗n}V_{2}^{∗m} and kerV_{1}^{∗j}V_{2}^{∗k} respectively).

Now, since q_{1} and p are relatively prime, there exist nonzero polynomials A, B ∈ ℂ[z, w], C ∈ ℂ[z] such that

A(z, w)q_{1}(z, w) + B(z, w)p(z, w) = C(z)

(let C be the resultant of q_{1} and p). Substituting V

A(V)q_{1}(V) + B(V)p(V) = C(V_{1})

it is then apparent that C(V_{1}) |_{K′} has finite dimensional range. If K′ were infinite dimensional then C(V_{1}) would have nontrivial kernel. This is impossible (a pure isometry cannot have eigenvalues for instance), so K′ is finite dimensional.

It is clear that K_{1} = ran p(V) and K_{2} = ran q_{1}(V) are mutually orthogonal invariant subspaces for V. Also, V |_{K_{1}} is a q_{1}-isopair and V |_{K_{2}} is a pisopair. Since K_{1} ⊕ K_{2} has finite codimension, the proposition is proved with K = K_{1} ⊕ K_{2}. □

3 Nearly cyclic isopairs

Definition 3.1 An isopair V is nearly cyclic if there is a vector u such that

\overline{ℂ[V]u} = ∨\{g(V)u : g ∈ ℂ[z, w]\}

is of finite codimension.

For example, the pair (M_{z^{2}}, M_{z^{3}}) on H^{2}(𝕋), is not cyclic, because for any f ∈ H^{2}(𝕋), we can find a function g orthogonal to ℂ[z^{2}, z^{3}]f. Namely, write

f(z) = a + bz + higher order terms

and define g(z) = −b̄ + āz. Then, ⟨f, g⟩ = 0 and since g is linear it is orthogonal to multiples of z^{2}. On the other hand, the pair (M_{z^{2}}, M_{z^{3}}) is nearly cyclic:

\overline{ℂ[M_{z^{2}}, M_{z^{3}}]1} = H^{2} ⊖ ℂ\{z\}.

Definition 3.2 Two isopairs V = (V_{1}, V_{2}) on H and V′ = (V′_{1}, V′_{2}) on H′ are nearly unitarily equivalent if there is a finite codimension V -invariant subspace K of H, a finite codimensional V′-invariant subspace K′ of H′, and unitary operators U : H → K′ and U′ : H′ → K′ such that

UV_{r}U^{∗} = V′_{r}|_{K′}  r = 1, 2
U′V′_{r}U′^{∗} = V_{r}|_{K}  r = 1, 2.

(In words, each one is unitarily equivalent to the other restricted to a finite codimensional invariant subspace).

The principal result of this section is:

Theorem 3.3 Any two nearly cyclic pure isopairs are nearly unitarily equivalent if and only if they have the same minimal polynomial.

For the rest of this section, fix some square-free inner toral polynomial q. The necessity is obvious, as restricting an algebraic isopair to a finite

codimensional invariant subspace does not change the minimal polynomial. Let us give an overhead view of the proof of sufficiency in Theorem 3.3. To show any two nearly cyclic pure algebraic isopairs with the same annihilating polynomial are nearly equivalent it suffices to show (1) that any such nearly cyclic isopair has a finite dimensional extension to a cyclic isopair (and therefore the nearly cyclic isopairs can be extended and restricted to cyclic ones) and (2) that all such cyclic isopairs are nearly unitarily equivalent to one particular choice of a nearly cyclic isopair W. The idea is if V is a given nearly cyclic isopair then we have the following diagram

W → some cyclic → V → some cyclic → W

where the arrows denote restrictions to finite codimensional invariant subspaces.

The study of cyclic isopairs and the construction of W requires us to lift many of our questions to a finite Riemann surface that desingularizes Z_{q} ∩𝔻^{2}.

Let Ω = Z_{q} ∩ 𝔻^{2}. As described in [2], there is a finite Riemann surface S and a holomap h from S onto Ω (a holomap is a proper holomorphic map that is one-to-one and non-singular except on finitely many points). We shall let A(S) denote the algebra of functions that are holomorphic on S and continuous up to the boundary, and we shall let A_{h}(S) be the finite codimensional subalgebra that is the closure of polynomials in h = (h_{1}, h_{2}). Note that A_{h}(S) can be described by a finite number of (homogeneous) linear relations on derivatives of elements of A(S) at a finite number of specified points. This implies that we can find an element of g ∈ A_{h}(S) that multiplies A(S) into A_{h}(S):

gA(S) ⊂ A_{h}(S).

We simply must choose g to vanish to sufficient order at a finite number of points.

Let ω be harmonic measure on S at some point z_{0}, fixed hereinafter. Let A^{2}(ω) be the closure in L^{2}(ω) of A(S) and let W be M_{h} = (M_{h_{1}}, M_{h_{2}}) on A^{2}(ω). Our eventual goal is to show that every cyclic isopair is nearly unitarily equivalent to W.

In addition, we can elaborate on the structure of cyclic isopairs via h. Let V be a cyclic q-isopair. We may model V as multiplication by coordinate functions (M_{z}, M_{w}) on P^{2}(μ) for some measure μ on Z_{q} ∩ 𝕋^{2}. Let ν = h^{∗}(μ) be the pull-back of μ to X := ∂S (i.e. ν(E) = μ(h(E)) for any Borel subset of X). Let A^{2}(ν) be the closure in L^{2}(ν) of A(S), and let A_{h}^{2}(ν) be the closure of A_{h}(S) in L^{2}(ν).

Then V is unitarily equivalent to M_{h} = (M_{h_{1}}, M_{h_{2}}) on A_{h}^{2}(ν). As A_{h}^{2}(ν) is of finite codimension in A^{2}(ν), V has a finite-dimensional extension to an isopair V^{S} that is unitarily equivalent to M_{h} on A^{2}(ν). Let us record these observations and a few more.

Lemma 3.4 If V is a cyclic pure q-isopair, then there exists a positive Borel measure ν on X such that V is unitarily equivalent to M_{h} on A_{h}^{2}(ν). Furthermore, the measures ν and ω are mutually absolutely continuous and satisfy

∫ log \frac{dν}{dω} dω > −∞.

Proof: By a result of J. Wermer [21], the algebra A(S) is a hypo-Dirichlet algebra on X = ∂S , i.e. the real parts of functions in A(S) form a finite codimensional subspace of C_{ℝ}(X). As V is pure, μ is non-atomic (an atom would yield a bounded point evaluation on Z_{q} ∩ 𝕋^{2} and the corresponding evaluation kernel would be an eigenvector for V^{∗}). Hence ν is non-atomic and V^{S} is also pure. (Note that a finite dimensional extension of a pure pair can only fail to be pure if the extension has a unitary summand and hence an eigenvalue. However, in this case we can multiply elements of A^{2}(ν) by some g ∈ A_{h}(S) and produce an element of A_{h}^{2}(ν). An eigenvector f of V^{S} would then produce an eigenvector gf of V, contradicting purity of V.)

Therefore A^{2}(ν) ≠ L^{2}(ν), and A^{2}(ν) has no L^{2} summand. Next, we claim that ν is absolutely continuous with respect to ω, using an argument from [14]. Indeed, let us write ν_{a} and ν_{s} for the absolutely continuous and singular parts of ν. Let E be an F_{σ} set such that ν_{a}(E) = 0 and ν_{s}(X \ E) = 0. By Forelli’s lemma ([7, II.7.3], applicable here because of Corollary 1 to Theorem 3.1 of [4]), there is a sequence f_{n} in A(S) with ∥f_{n}∥_{X} ≤ 1, and such that f_{n}(x) tends to 0 for every x in E, and to 1 for ω-a.e. x. Some subsequence of f_{n} converges weak-* to a function g. By the dominated convergence theorem, ∫ gdω = 1, so g = 1 ω-a.e. Again using dominated convergence, we see that for every h in L^{2}(ν_{s}) we have ∫ ghdν = 0. Therefore 1 − g, which is in A^{2}(ν) ∩ L^{∞}(ν), agrees with the characterictic function of E ν-a.e., so

A^{2}(ν) = A^{2}(ν_{a}) ⊕ A^{2}(ν_{s}).

By the Kolmogorov-Krein theorem [7, V.8.1], A^{2}(ν_{s}) = L^{2}(ν_{s}), so we conclude that ν_{s} is null, as claimed.

As A^{2}(ν) ≠ L^{2}(ν), it follows from the work of P. Ahern and D. Sarason on hypo-Dirichlet algebras [4, Corollary to Thm. 10.1], that

∫ log \frac{dν}{dω} dω > −∞.  (3.5)
□

The next proposition allows us to dispense with dealing with nearly cyclic isopairs.

Proposition 3.6 Any nearly cyclic pure q-isopair is unitarily equivalent to a cyclic pure q-isopair restricted to a finite codimensional invariant subspace.

Proof: Let V be a nearly cyclic pure q-isopair on the Hilbert space H, let K be a finite codimensional invariant subspace on which V is cyclic, and let F = H ⊖ K (a finite dimensional subspace). Since V |_{K} may be modeled as (M_{z}, M_{w}) on P^{2}(μ) for some measure μ supported on Z_{q} ∩ 𝕋^{2}, we shall simply identify K = P^{2}(μ) and V |_{K}= (M_{z}, M_{w}). Then, the pair V can be written in block form as

V = \begin{matrix}P^{2}(μ) & F \\ \begin{matrix}P^{2}(μ) \\ F \end{matrix} & (\begin{matrix}(M_{z}, M_{w}) & (B_{1}, B_{2}) \\ 0 & (A_{1}, A_{2}) \end{matrix}) \end{matrix}

where (A_{1}, A_{2}) is a pair of commuting contractions on the finite dimensional space F that (by purity) have no unimodular eigenvalues.

Let u : 𝔻 → 𝔻 be a finite Blaschke product that annihilates A_{1}:

u(A_{1}) = 0.

If we apply such a u to V_{1} (we can do this since u’s power series is absolutely convergent in \overline{𝔻}) we get an isometry

u(V_{1}) = \begin{matrix}P^{2}(μ) & F \\ \begin{matrix}P^{2}(μ) \\ F \end{matrix} & (\begin{matrix}M_{u} & u(B_{1}) \\ 0 & 0 \end{matrix}) \end{matrix}

where M_{u} is multiplication by u on P^{2}(μ). In particular, the range, say L, of u(V_{1}) is contained in P^{2}(μ). Therefore, u(V_{1}) : H → L can be thought of as a Hilbert space isomorphism that intertwines V on H and (M_{z}, M_{w}) on L ⊂ P^{2}(μ). This proves V can be modeled as a restriction of (M_{z}, M_{w}) on P^{2}(μ) (i.e. a cyclic pair) to an invariant subspace (i.e. L). The key thing left to prove is that L has finite codimension in P^{2}(μ). For this it suffices to prove uP^{2}(μ) has finite codimension in P^{2}(μ) since uP^{2}(μ) ⊂ L.

To see this, it helps to use the A^{2}(ν) model described above (in Lemma 3.4). Namely, we need to prove u(h_{1})A_{h}^{2}(ν) has finite codimension in A_{h}^{2}(ν). Since A_{h}^{2}(ν) has finite codimension in A^{2}(ν), it suffices to prove u(h_{1})A^{2}(ν) has finite codimension in A^{2}(ν). This follows from the fact that u(z) vanishes finitely often among (z, w) ∈ Z_{q} (and therefore u(h_{1}) has finitely many zeros on S) and so any analytic function f in A(S) which vanishes to higher order at u(h_{1})’s zeros than u(h_{1}), is a multiple of u(h_{1}): f/(u(h_{1})) ∈ A(S). □

Thus, any nearly cyclic has an finite dimensional extension to a cyclic and by definition a finite codimension restriction to a cyclic. So, now we let V be a pure cyclic q-isopair, which we think of as M_{h} on A_{h}^{2}(ν) and we let V^{S} be the extension of V to A^{2}(ν). The following lemma is really a chain of lemmas, since we prefer to introduce ideas from references as we need them. Recall that W refers to M_{h} on A^{2}(ω).

Lemma 3.7 With notation as above, V is unitarily equivalent to W restricted to a finite codimensional invariant subspace.

Proof: Suppose we can find a function f in A^{2}(ω) that is nearly outer, in the sense that the invariant subspace it generates,

[f] := \overline{A(S) f},

is of finite codimension, and such that |f|^{2} = \frac{dν}{dω}. Let

[f]_{W} := \overline{ℂ[W]f} = \overline{A_{h}(S) f},

which will be of finite codimension in [f].

Then the map g · f ↦ g extends to a unitary between [f]_{W} in A^{2}(ω) and A_{h}^{2}(ν) that intertwines W |_{[f]} and V. By Lemma 3.9, which we prove below, such a nearly outer function exists. □

When does a nearly outer function exist with a given log-integrable modulus? Let L be the codimension of Re(A(S)) in C_{ℝ}(X).

Ahern and Sarason proved that any log-integrable positive function can be written as |f|^{2} for some f in A^{2}(ω), and they conjectured that f can be chosen so that [f] is of codimension no more than the codimension of ℜ(A(S)) in C_{ℝ}(X) [4]. This conjecture is still open, though it has been proved in the planar case by G. Tumarkin and S.Ya. Khavinson [19].

However, using results of S.Ya. Khavinson [11, 12] for general finite Riemann surfaces, which generalize results of D. Khavinson for the planar case [9, 10], we can prove that f can be chosen with [f] of finite codimension.

The idea is that when we write down the Green integral (or Poisson integral) of a measure, the resulting harmonic function’s conjugate function will in general be multi-valued. To fix this, we need to worry about the periods on a homology basis K_{r} for S. There are L = 2h + n − 1 such curves, where S has h handles and n boundary components. Choose L disjoint arcs Δ_{j} in ∂S, and positive measures ν_{j} supported on each arc, so that each ν_{j} is boundedly absolutely continuous with respect to ω (in fact we may simply take ν_{j} to be harmonic measure restricted to Δ_{j}). Khavinson shows that after shrinking Δ_{j} if necessary, the matrix A of periods of the harmonic conjugate of the Green integrals of the ν_{j} along the curves K_{r} is non-singular [11], and hence may be used to “correct” the periods of other functions (while at the same time we have some control over what is happening on the boundary).

The Green kernel is defined by

P (z, ζ) = \frac{1}{2π} \frac{\frac{∂}{∂n_{ζ}} G(z, ζ)}{\frac{∂}{∂n_{ζ}} G(z_{0}, ζ)} z ∈ S, ζ ∈ ∂S,

where G(z, ζ) is the Green’s function with pole at z, and n_{ζ} is the outward normal. The Green integral of a measure ν is then

∫_{∂S} P (z, ζ)dν(ζ).

(Note we are using z, ζ to refer to points of S, while these letters are typically reserved for uniformizers on Riemann surfaces.)

Let

ω_{j}(z) = ∫_{Δ_{j}} P (z, ζ)dν_{j}(ζ).

Let us explicitly define:

Definition 3.8 A function f ∈ A^{2}(ω) is nearly outer if

[f] := \overline{A(S)f}

has finite codimension in A^{2}(ω).

Lemma 3.9 Any log-integrable function w on ∂S is the modulus squared of a nearly outer function.

Proof: First, we write down the Green integral of 1/2 log w

∫_{∂S} P (z, ζ)\frac{1}{2} log w(ζ)dω(ζ).

There exist real constants λ_{j} such that

h(z) = ∫_{∂S} P (z, ζ) [\frac{1}{2} log w(ζ)dω(ζ) − ∑_{j=1}^{L} λ_{j}dν_{j}(ζ)]  (3.6)

has a single-valued harmonic conjugate ∗h, and h has boundary values given by

h(ζ) = \begin{cases}\frac{1}{2} log w(ζ) & ζ ∈ ∂S \ ∪Δ_{j} \\ \frac{1}{2} log w(ζ) − λ_{j} \frac{dν_{j}}{dω} (ζ) & ζ ∈ Δ_{j} \end{cases}

i.e.

hdω = (1/2) log wdω − ∑_{j} λ_{j}dν_{j}.  (3.7)

The function

g(z) = exp(h(z) + i ∗ h(z))

is outer in the sense that

log |g(z_{0})| = \frac{1}{2π} ∫_{∂S} log |g(ζ)|dω(ζ)

(by (3.6) and (3.7) since P (z_{0}, ζ) = \frac{1}{2π}).

It follows from [4] (Theorem 7.1 and the discussion following Theorem 9.1) that it is also outer in the sense that [g] is all of A^{2}(ν). If we can find a finite Blaschke product F whose modulus on the boundary is

log |F (ζ)| = \{ \begin{matrix}0 & ζ ∈ ∂S \ ∪Δ_{j} \\ λ_{j} \frac{dν_{j}}{dω} (ζ) & ζ ∈ Δ_{j}, \end{matrix} ,  (3.10)

then f = F g will be a nearly outer function (g is outer and any function that vanishes on the zeros of F will be divisible by F) satisfying |f|^{2} = w a.e. on ∂S. We shall prove F exists in Lemma 3.11. □

Following Schiffer-Spencer [16], Khavinson defines a basis for the space of abelian differentials of the first kind via the Green’s function. Specifically, dZ_{j} is defined using the local expression

Z′_{j}(z) = −\frac{1}{π} ∫_{K_{j}} \frac{∂^{2}G(z, ζ)}{∂z∂ζ}dζ

or

Im Z_{j}(z) = 2i\frac{1}{2π} ∫_{K_{j}} \frac{∂G(z, ζ)}{∂ζ}dζ

where again K_{1}, …, K_{L} form a canonical homology basis for S. It should be noted that Im Z_{j} is single valued on S \ K_{j} (and hence single valued everywhere when K_{j} is a boundary cycle), but has a jump across K_{j} when K_{j} is a cycle corresponding to a handle.

Lemma 3.11 Given real constants λ_{j}, j = 1, …, L, there exists a bounded holomorphic function F : S → ℂ with finitely many zeros in S and boundary modulus satisfying

log |F (ζ)| = \begin{cases}0 & ζ ∈ ∂S \ ∪Δ_{j} \\ λ_{j} \frac{dν_{j}}{dω} (ζ) & ζ ∈ Δ_{j} \end{cases}  (3.12)

Proof: Khavinson shows that for each point α in S there is a function B(z; α) that has a single zero at α, and such that

log |B(z; α)| = \{ \begin{matrix}0 & z ∈ ∂S \ ∪Δ_{j} \\ c_{j} \frac{dν_{j}}{dω} & z ∈ Δ_{j} \end{matrix} .  (3.13)

Moreover, one finds the c_{j} (real constants) by using the (invertible) period matrix A and the abelian differentials dZ_{j} by the formula

( c_{1} ; ⋮ ; c_{L} ) = −2πA^{−1} ( Im Z_{1}(α) ; ⋮ ; Im Z_{L}(α) ) .

Also, for each (d_{1}, …, d_{L})^{t} ∈ ℤ^{L}, if we set

(c_{1} ; ⋮ ; c_{L}) = −2πA^{−1} (d_{1} ; ⋮ ; d_{L})

then there is a holomorphic function B on S with no zeros and boundary modulus values satisfying equation (3.13).

Therefore to prove the lemma it suffices to show that we can take a finite positive integer combination of vectors of the form (Im Z_{1}(α), …, Im Z_{L}(α))^{t} and obtain every element of ℝ^{L}/ℤ^{L}, for then we could find a combination satisfying

−\frac{1}{2π} A (λ_{1} ; ⋮ ; λ_{L}) = ∑_{j=1}^{k} (Im Z_{1}(α_{j}) ; ⋮ ; Im Z_{L}(α_{j})) mod ℤ^{L}

and upon modifying it by an element (d_{1}, …, d_{L})^{t} ∈ ℤ^{L}, we would obtain a finite Blaschke product with zeros at the α_{1}, …, α_{k} and the desired boundary modulus values. For this it suffices to prove the following claim.

Claim: As α_{1}, …, α_{L} vary over S, the vectors

( ∑_{r=1}^{L} Im Z_{1}(α_{r}) ; ⋮ ; ∑_{r=1}^{L} Im Z_{L}(α_{r}) )  (3.14)

have interior in ℝ^{L}.

Indeed, given the claim, it follows that there is some finite N so that sums of N vectors of the form (3.14) (i.e. with LN points α_{r}) form a ball large enough that it covers an entire cell of ℝ^{L}/ℤ^{L}, and so a Blaschke product of degree LN will satisfy (3.12).

Proof of claim: There is no harm in assuming our argument takes place inside some coordinate neighborhood. Consider the derivative of (3.14) with respect to α_{1}, …, α_{L}. If the L-by-L matrix (Z′_{j}(α_{r})) is of full rank, then for some choice of unimodular τ_{r} the real matrix (Im (τ_{r}Z′_{j}(α_{r}))) is invertible (a linear algebra exercise). This proves the claim in this case because if we replace α_{r} by τ_{r}α_{r} in (3.14) and take derivatives we get a nonsingular Jacobian matrix.

Otherwise, there are real numbers c_{j}, not all zero, so that, for every 1 ≤ r ≤ L, we have

∑_{j=1}^{L} c_{j}Z′_{j}(α_{r}) = 0.  (3.15)

But (3.15) means that the differential ∑ c_{j}dZ_{j}, which extends by reflection to the double of S, vanishes at L points in S and L more on the reflection. The double has genus L, and so a differential of the first kind must have exactly 2L − 2 zeroes (see [16] (3.5.1)). As the dZ_{j}’s are linearly independent, this forces all the c_{j}’s to be zero, a contradiction. □

This chain of lemmas completes the proof of Lemma 3.7, which says that any cyclic V is a finite codimensional restriction of W.

Proof of Theorem 3.3. In light of Lemma 3.7, it remains to show that V, a cyclic q-isopair, can be restricted to a finite codimensional subspace to become unitarily equivalent to W. Again we view V as M_{h} on A_{h}^{2}(ν). Now A_{h}(S) has finite codimension in A(S) and is defined by a finite number of linear relations on derivatives (at a finite number of points) of elements of A(S) (see [2]). In particular, any element of A^{2}(ν) that vanishes to high enough order at these finite points will be inside A_{h}^{2}(ν). This can be accomplished by multiplying A^{2}(ν) by an appropriate finite Blaschke product F. Then, F A^{2}(ν) ⊂ A_{h}^{2}(ν) and the operator M_{h} restricted to F A^{2}(ν) is unitarily equivalent to M_{h} on A^{2}(σ) where σ = |F |^{2}ν. This proves V has a finite codimension restriction to M_{h} on A^{2}(σ).

So it suffices to show that one can find a unitary equivalence between M_{h} on A^{2}(σ) restricted to an invariant subspace of finite codimension and M_{h} on A^{2}(ω). But this can be done by finding a nearly outer function f with modulus |f|^{2} = \frac{dω}{dσ} just as in Lemma 3.9. □

Remark 3.16 In the proof of Theorem 3.3, we did not strongly use the fact that V and W are isopairs. We could more generally look at nearly cyclic pure subnormal pairs whose spectral measures were supported on the boundary of some hyperbolic algebraic set.

We can translate Theorem 3.3 into the matrix models and get the following result.

Corollary 3.17 Suppose (M_{z}, M_{Φ}) and (M_{z}, M_{Ψ}) are two nearly cyclic qisopairs on H^{2}(𝕋) ⊗ ℂ^{k} (where as usual Φ and Ψ are k × k matrix valued rational inner functions on 𝔻). Then there exists a matrix valued rational inner function F such that

Φ(z) = F (z)Ψ(z)F (z)^{−1}  (3.18)

Note that in general an expression like F ΨF^{−1} need not be holomorphic in the disk.

Proof: By Theorem 3.3, (M_{z}, M_{Φ}) has a restriction to a finite codimension invariant subspace K that is unitarily equivalent to (M_{z}, M_{Ψ}). Since K is shift invariant and of finite codimension, it is of the form F (H^{2}⊗ℂ^{k}) where F is a rational matrix valued inner function. However, since K is invariant under M_{Φ}, we see that ΦF (H^{2} ⊗ ℂ^{k}) ⊂ F (H^{2} ⊗ ℂ^{k}). This implies G = F^{−1}ΦF is holomorphic and also rational and inner. It is not hard to show that (M_{z}, M_{Φ}) on F (H^{2} ⊗ ℂ^{k}) is unitarily equivalent to the pair (M_{z}, M_{G}) on H^{2} ⊗ ℂ^{k} (i.e. the map F g ↦ g is the required Hilbert space isomorphism that intertwines the operators). Therefore, (M_{z}, M_{G}) and (M_{z}, M_{Ψ}) are unitarily equivalent. This can only occur if G and Ψ are unitarily equivalent. This proves (3.18) after replacing F with an appropriate unitary multiple. □

4 Convex hulls

The operator-theoretic ideas of Sections 1 and 3 allow us to prove a result in function theory, Theorem 4.1 below. E.L. Stout has proved a similar result for irreducible analytic subvarieties (private communication).

The central issue is what one can say about the intersection of an algebraic set A with the two-torus in ℂ^{2}. One immediate distinction is whether A∩𝕋^{2} is large in the sense that no polynomial can vanish on A∩𝕋^{2} without vanishing identically on A; if this holds we call the set A toral. (In two dimensions, as we are here, this just means that A ∩ 𝕋^{2} is infinite, though the definition makes sense in higher dimensions.) However, the two curves

A_{1} = {(z, w) : z = w}

and

A_{2} = {(z, w) : zw = 1}

are both toral, yet X_{1} = A_{1} ∩𝕋^{2} and X_{2} = A_{2} ∩𝕋^{2} are qualitatively different. The first bounds an analytic disk in 𝔻^{2}; the second does not. Theorem 4.1 says that one way to understand this is to observe that A_{1} is a distinguished variety, and A_{2} is not.

We wish to exclude curves that contain horizontal or vertical planes (i.e. zero sets of polynomials z − ζ_{1} or w − ζ_{2}). If ζ_{r} is unimodular, such a zero set fills a disk in the boundary of 𝔻^{2}, and the polynomial hull of the intersection of this disk with the torus is the closed disk. So, we exclude linear factors to make the statement of the theorem concise; however one could drop this restriction and conclude that the set X is not polynomially convex if and only if q has a factor that is either inner toral or of the form (z − e^{iθ}) or (w − e^{iθ}).

Recall that a polynomial q is inner toral if Z_{q} ⊆ 𝔻^{2} ∪ 𝕋^{2} ∪ 𝔼^{2}.

Theorem 4.1 Let q be a polynomial in two variables with no linear factors. Then Y = Z_{q} ∩ 𝕋^{2} is polynomially convex if and only if q has no inner toral factor.

Proof: (i) First assume that Y is not polynomially convex, so there is some point ζ in the polynomial hull of Y that is not in Y. As every point of 𝕋^{2} is a peak point for A(𝔻^{2}), we cannot have ζ in 𝕋^{2}. There exists a complex measure λ supported on Y so that

p(ζ) = ∫_{Y} p dλ  (4.2)

for all polynomials p. Let dμ = |dλ| be the total variation of λ. Then by (4.2), for every polynomial p we have

|p(ζ)| ≤ C [∫_{Y} |p|^{2}dμ]^{1/2} .  (4.3)

(A point ζ satisfying inequality (4.3) for all polynomials p is called a bounded point evaluation for P^{2}(μ).)

Claim: ζ ∈ 𝔻^{2}.

Else, some component, say ζ_{1}, is unimodular. Applying (4.3) to polynomials of the form

p(z, w) = (\frac{z + ζ_{1}}{2} )^{n} r(w)

and letting n tend to infinity, we would get

|r(ζ_{2})| ≤ C [∫_{Y ∩{z=ζ_{1}}} |r|^{2}dμ]^{1/2} .  (4.4)

As ζ_{2} is in 𝔻, (4.4) asserts that the measure μ|_{Y ∩{z=ζ_{1}}} is a measure on the circle that has a bounded point evaluation inside the disk. By Szegő’s theorem [17], this means that Y ∩ {z = ζ_{1}} must be the whole circle ζ_{1} × 𝕋, and so q must have (z − ζ_{1}) as a factor, contrary to assumption. ⊲

Let V_{1} be (M_{z}, M_{w}) on P^{2}(μ), and let V be the pure part of V_{1}. By Lemma 4.5, V is non-zero. So, by Theorem 1.20, there is some square-free inner toral polynomial p such that p(V) = 0, and p = 0 μ_{a}-a.e. As both p and q vanish on the support of μ_{a}, which is an infinite set, they must share a common factor, which is an inner toral factor of q.

(ii) Suppose that q has a factor p that is inner toral. Then Z_{p} ∩𝔻^{2}, which is non-empty by Theorem 1.5, is contained in the polynomial hull of Z_{p} ∩ 𝕋^{2}, and hence of Y. □

Lemma 4.5 Suppose μ is a measure on Z_{q} ∩ 𝕋^{2}, and P^{2}(μ) has a bounded point evaluation ζ in 𝔻^{2}. Then the isopair (M_{z}, M_{w}) on P^{2}(μ) has a non-zero pure part, and this is unitarily equivalent to (M_{z}, M_{w}) on P^{2}(μ_{a}), where μ_{a} is the part of μ that is absolutely continuous with respect to arc-length on 𝕋^{2}.

Proof: Using the notation of Section 3, every point in h^{−1}(ζ) (which can be more than one point if ζ is a point of multiplicity of q) is a bounded point evaluation for A_{h}^{2}(ν), and therefore for A^{2}(ν). Just as in the proof of Lemma 3.4, the Kolmogorov-Krein theorem says that

A^{2}(ν) = A^{2}(ν_{a}) ⊕ L^{2}(ν_{s}),

where ν_{a} and ν_{s} are, respectively, the absolutely continuous and singular parts of ν with respect to harmonic measure ω. Therefore, every point in S is a bounded point evaluation for A^{2}(ν_{a}). Pushing back down to Ω = Z_{q} ∩ 𝕋^{2} again, we find that every point is a bounded point evaluation for P^{2}(μ_{a}) where μ_{a} = h_{∗}(ν_{a}).

To see that (M_{z}, M_{w}) on P^{2}(μ_{a}) is pure, assume that one of the isometries, M_{z} say, has a unitary part. Then there is some function f of norm one in P^{2}(μ_{a}) such that

∥M_{z}^{∗n}f∥ = ∥P z̄^{n}f∥ = ∥f∥  (4.6)

for all n, where P is the projection from L^{2}(μ_{a}) onto P^{2}(μ_{a}). From (4.6) we get that

M_{z}^{∗n}f = P z̄^{n}f = z̄^{n}f ∀n.  (4.7)

Choose some bounded point evaluation ζ = (ζ_{1}, ζ_{2}) in 𝔻^{2} of P^{2}(μ_{a}) such that f(ζ) ≠ 0. (Such a point must exist, for otherwise f ∘ h would be in A_{h}^{2}(ν_{a}) and vanish at every point of S, and so by [4] again would be identically zero.) Let k_{ζ} be the kernel function at ζ, i.e. the unique function in P^{2}(μ_{a}) satisfying

p(ζ) = ⟨p, k_{ζ}⟩ ∀ polynomials p.

Then

f(ζ) = ⟨z^{n}z̄^{n}f, k_{ζ}⟩
= ⟨z̄^{n}f, P z̄^{n}k_{ζ}⟩

Therefore

∥P z̄^{n}k_{ζ}∥ ≥ |f(ζ)| ∀n.

But

P z̄^{n}k_{ζ} = ζ̄_{1}^{n}k_{ζ},

and this must tend to zero as n goes to infinity. □

5 Non-cyclic algebraic isopairs

We do not understand algebraic isopairs that are not nearly cyclic.

Let us say that an isopair V is essentially k-cyclic if there are k vectors u_{1}, …, u_{k} so that

∨{p_{1}(V)u_{1}, …, p_{k}(V)u_{k} : p_{1}, …, p_{k} ∈ ℂ[z, w]}

is of finite codimension, and if no set of k − 1 vectors suffices.

Question 5.1 Suppose V and V′ are both essentially k-cyclic isopairs with the same minimal polynomial. Are they nearly unitarily equivalent?

Question 5.2 Are all essentially k-cyclic algebraic isopairs nearly equivalent to a direct sum of k cyclic algebraic isopairs?

References

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