Subnormal operators and Quadrature domains

John E. McCarthy ^{∗} Liming Yang ^{†} Washington University, St. Louis, Missouri 63130, U.S.A.

September 19 1994

0 Introduction

The object of this paper is to classify rationally cyclic subnormal operators with finite rank self-commutators, and to show their correspondence with quadrature domains. We establish a connection between subnormal operators, the understanding of which has dramatically increased since the appearance of Thomson’s break-through techniques for exhibiting bounded point evaluations [21], and quadrature domains, which have been successfully studied by techniques of Riemann surfaces, complex analysis and potential theory [1], [11] and [19]. In particular, we show that any quadrature domain gives rise to a rationally cyclic subnormal operator with finite rank self-commutator; that any such operator comes from a quadrature domain; and that the unitary equivalence classes of such operators are parametrized by a certain number of points in Ω (depending on the exact rank of the self-commutator) and certain real parameters depending on the connectivity of Ω.

A bounded linear operator S on a Hilbert space H is called subnormal if it has a normal extension, i.e. there exists a normal operator N on a superspace of H which leaves H invariant and such that N|_{H} = S. The self-commutator of S is

[S^{∗}, S] = S^{∗}S − SS^{∗}.

The operator S is called cyclic if there exists a vector ξ in H such that {p(S)ξ : p a polynomial} is dense in H, and is called rationally cyclic if there exists ξ so that {r(S)ξ : r a rational function with poles off σ(S)} is dense in H. An operator is called irreducible if it commutes with no non-trivial projection.

∗Partially supported by National Science Foundation grant DMS 9301508

†Partially supported by National Science Foundation grant DMS 9401234

Any rationally cyclic subnormal operator is unitarily equivalent to some S_{K,μ}, where K is a compact subset of ℂ, μ is a finite positive Borel measure supported on K, and S_{K,μ} denotes multiplication by the coordinate function on the space R^{2}(K, μ), the closure of the rational functions with poles off K in the space L^{2}(μ) (see [5] for this and other basic facts about subnormal operators). The minimal normal extension of S_{K,μ} is N_{μ}, multiplication by the coordinate function in L^{2}(μ). As different K’s can give rise to the same S_{K,μ}, we shall assume that K is minimal, so K = σ(S_{K,μ}).

In [17], R. Olin et al. classified all cyclic subnormal operators with finite rank self commutators, by first classifying the irreducible cyclic subnormal operators with this property, and then showing when a finite direct sum of such operators is again cyclic (it will clearly always have finite rank self-commutator). An alternative proof of their theorem, using the theory of quadrature domains (see below), appears in [16]. For the irreducible case, their theorem is as follows, where 𝔻 is the unit disk and H^{∞} the space of bounded analytic functions in 𝔻:

Theorem 0.1 (Olin, Thomson and Trent) Let S be an irreducible cyclic subnormal operator. Then S has finite rank self-commutator if and only if there is a rational function r, bounded in 𝔻 and a weak-star generator of H^{∞}, and a measure ν that is the sum of Haar measure on the boundary of 𝔻 and a finite number of point masses in 𝔻, such that S is unitarily equivalent to r(S_{_{𝔻̄},ν}).

Let A denote Lebesgue area measure in the plane. A domain Ω in ℂ is called a quadrature domain if there exist points z_{1}, …, z_{N} in Ω and constants a_{m,n} such that for every function f analytic in Ω and area-integrable, the identity

∫_{Ω} f(z)dA = ∑_{n=1}^{N} ∑_{m=0}^{N_{n}} a_{m,n}f^{(m)}(z_{n})

holds. The number ∑_{n=1}^{N}(N_{n} + 1) is called the order of the quadrature domain. Let us note the following facts about quadrature domains: (Q1) Let Ω be a bounded open set in ℂ. Ω is a quadrature domain if and only if there is

a function R, called the Schwarz function, meromorphic in Ω and continuously extendable to each point of ∂Ω, such that R(z) = z̄ on ∂Ω [1].

(Q2) The equation R(z) = z̄ has at most finitely many solutions inside Ω [12]. Let us denote these points by E(Ω).

(Q3) The boundary of a quadrature domain is an irreducible algebraic curve, except for possibly finitely many points [11]. (Removing any of the points of E(Ω) yields a new domain that satisfies the same quadrature identity. For simplicity we shall assume that any quadrature domain contains all these isolated points, which can be achieved by replacing Ω by Int(\overline{Ω}) [12]).

(Q4) A bounded simply connected domain is a quadrature domain if and only if it is the conformal image of the unit disk under a rational function [1].

Fact (Q4) means that Theorem 0.1 can be rephrased to say that S_{K,μ} is irreducible, cyclic and has finite rank self-commutator if and only if G, the interior of K, is a quadrature domain, K is polynomially convex, and μ is mutually absolutely continuous with respect to the sum of harmonic measure for G plus an at most finite number of point masses at points of G.

For a connected open set G, harmonic measure is defined in the following way: pick a point a in G. For any continuous real-valued function f on the boundary of G, define a function f̂ on G by

f̂(z) = sup{g(z) : g is subharmonic on G, and lim sup_{z→ζ} g(z) ≤ f(ζ), ∀ζ ∈ ∂G}.

The functional f ↦ f̂(a) is continuous on C_{ℝ}(∂G), and therefore comes from a measure, ω_{a}, which is called harmonic measure for G at a. Whilst a different choice of a will yield a different measure, the two measures will be boundedly mutually absolutely continuous, so, by an abuse of language, we refer to harmonic measure for G without specifying a point a.

Our main result is that if “cyclic” is replaced by “rationally cyclic”, the theorem as paraphrased above is true with the polynomial convexity condition dropped, viz.:

Theorem 1.12 Let S_{K,μ} be a rationally cyclic subnormal operator with spectrum K. Let Ω be the interior of K. Then S_{K,μ} is irreducible and has finite rank self-commutator if and only if the following conditions are satisfied:

(i) K = \overline{Ω}. (ii) Ω is a quadrature domain. (iii) μ|_{∂Ω} is absolutely continuous with respect to harmonic measure for Ω, which we will denote by ω, and

∫_{∂Ω} log(\frac{dμ}{dω})dω > −∞.

(iv) μ|_{Ω} is either zero or a finite sum of point masses.

In [24] an attempt is made to classify all subnormal operators with finite rank selfcommutator. However, for any quadrature domain that is not simply connected, none of the family of rationally cyclic subnormal operators associated to this domain are unitarily equivalent to the model operators constructed in [24].

The reason Theorem 1.12 requires so much more work to establish in the rationally cyclic case than in the cyclic case is that rational approximation in the mean is much less well understood than polynomial approximation. Alternatively, one can view it as a reflection of the fact that simply connected quadrature domains are just rational images of the disk, and therefore easier to work with than general quadrature domains.

In Section 2 we classify unitary equivalence classes and similarity classes of rationally cyclic subnormal operators with finite rank self-commutators. In Section 3 we discuss connections between the operator theoretic properties of a rationally cyclic subnormal operator with finite rank self-commutator and geometric properties of the quadrature domain that is the interior of its spectrum.

1 Irreducible rationally cyclic subnormals with finite rank self-commutators

Let us fix a compact set K in ℂ and a probability measure μ on K, and write S_{μ} for S_{K,μ}. Throughout this section we shall assume that S_{μ} is irreducible, σ(S_{μ}) = K, and (except in Lemma 1.4) that [S_{μ}^{∗}, S_{μ}] is a finite rank operator.

First, we need the following definitions: a point ζ is called a bounded point evaluation for R^{2}(K, μ) if there exists a constant C_{ζ} satisfying

|r(ζ)| ≤ C_{ζ} ∥r∥_{R^{2}(K,μ)}

for every r in Rat(K), the rational functions with poles off K. If ζ is a bounded point evaluation, it makes sense to talk about the value of f(ζ) for any function f in R^{2}(K, μ), because evaluation at ζ, defined a priori only for rational functions, extends by continuity to all of R^{2}(K, μ). The point ζ is called an analytic bounded point evaluation if it is in the interior of the bounded point evaluations, and if, for every f ∈ R^{2}(K, μ), the function z ↦ f(z) is analytic in a neighbourhood of ζ.

Lemma 1.1 There exists a polynomial p such that z̄p is in R^{2}(K, μ).

Proof: The operator S_{μ} has the normal extension N_{μ} on L^{2}(μ); let P be the projection from L^{2}(μ) onto R^{2}(K, μ). Let K = ker[S_{μ}^{∗}, S_{μ}]. It is straightforward to verify that K = {ξ ∈ R^{2}(K, μ) : N_{μ}^{∗}ξ ∈ R^{2}(K, μ)}, and hence K is S_{μ}-invariant, and of finite codimension by hypothesis.

A finite codimensional invariant subspace must have the form clos(pR^{2}(K, μ)) for some polynomial p; indeed p can be taken to be the minimal polynomial of the compression of S_{μ} to K^{⊥}, so p has the same degree as rank of the self-commutator, and all its zeroes lie in the set of bounded point evaluations of R^{2}(K, μ). (This idea first appears in [10]). □

We therefore have that z̄p = f ∈ R^{2}(K, μ). If we knew that K were the closure of its interior, and that f were continuous on K and analytic on the interior of K, we could then conclude that f/p were a meromorphic function, continuous on K, and agreeing with z̄ on the boundary, and hence from (Q1) that the interior of K were a quadrature domain. It is the purpose of the next series of lemmata to establish that this is indeed the case.

Lemma 1.2 \overline{IntK} = K.

Proof: Since K \ sptμ ⊂ σ(S_{μ}) \ σ_{e}(S_{μ}) ⊂ IntK, we have

K = IntK ∪ sptμ.

It suffices to show that sptμ ⊂ \overline{IntK}. Let ϕ be a smooth function with sptϕ ∩ \overline{IntK} empty. Let g ⊥ R^{2}(K, μ) such that |g| > 0 a.e. μ (for a proof that such a g exists, see [5, Lemma V.17.10]). Then for each λ ∈ K^{c}, we have

∫ \frac{z̄ − λ̄}{z − λ}p(z)g(z)dμ(z) = 0.

Since the function ∫ \frac{z̄ − λ̄}{z − λ}pgdμ is a continuous function of λ except at atoms of μ, we see that

∫ \frac{z̄ − λ̄}{z − λ}p(z)g(z)dμ(z) = 0

a.e. off IntK with respect to area measure. Since

ϕ(λ) = \frac{1}{π} ∫ \frac{z̄ − λ̄}{z − λ} ¯∂^{2}ϕ(z)dA(z),

we get that

∫ ϕ(λ)p(λ)g(λ)dμ(λ) = \frac{1}{π} ∫ ¯∂^{2}ϕ(z)dA(z) ∫ \frac{λ̄ − z̄}{λ − z}p(λ)g(λ)dμ(λ)
=  0

Therefore, p(z) = 0 μ-a.e. off \overline{IntK}. Since S_{μ} is irreducible we conclude that sptμ ⊂ \overline{IntK}. □

The analytic bounded point evaluations of R^{2}(K, μ), call them G, are dense in \overline{IntK} [6, Thm. 2.1], and hence in K. If we already knew that G were connected (which it turns out to be), then R(∂G), the uniform closure of Rat(∂G), would equal C(∂G) [9, Cor. VIII.8.4], and the following lemma would be trivial.

Lemma 1.3 Let

R_{0}(∂G) = cl{ϕ̂p + r : ϕ is a bounded Borel function on ∂G and r ∈ R(∂G)},

where ϕ̂(λ) = ∫ \frac{1}{z − λ}ϕ(z)dA(z) is the Cauchy transform of ϕ. Then

R_{0}(∂G) = C(∂G).

Proof: Suppose that ν ⊥ R_{0}(∂G). Then ν ⊥ R(∂G), so

ν̂(λ) = 0

a.e. with respect to area measure off ∂G. On the other hand, it follows from

∫ ϕ̂pdν = 0

that

∫ ϕ ˆ (pν)dA = 0,

and so p̂ν = 0 a.e. on ∂G with respect to area measure. But

p̂ν = pν̂,

so ν̂ = 0 a.e. on ℂ with respect to area measure. Therefore ν = 0, and so R_{0}(∂G) = C(∂G). □

As the following lemma may perhaps be of use elsewhere, let us note that it does not require that S_{μ} have finite rank self-commutator.

Lemma 1.4 Let ϕ be a bounded Borel function supported on ∂G \ ∂K. Then ϕ̂ is in R^{2}(K, μ).

Proof: Let O_{n} = \{z ∈ Int(K) : dist(z, ∂K) < \frac{1}{n}\}, and ϕ_{n} = ϕ|_{ℂ\O_{n}}. Because the area of O_{n} tends to zero, ϕ̂_{n} tends to ϕ̂ uniformly, so it is sufficient to show that each ϕ̂_{n} is in R^{2}(K, μ).

The Cauchy transform of any bounded function is continuous (this can be seen by approximating the kernel by continous kernels), so the lemma will follow if we can show that any function h, that is continuous on ℂ_{∞} and analytic off ∂G ∩ (Int(K) \ O_{n}), is in R^{2}(K, μ). To show this, we need to use the techniques Thomson developed in [21] and the modifications by Conway and Elias in [6], and for the purpose of this lemma we shall assume that the reader is familiar with these two papers.

Let g be an arbitrary function in R^{2}(K, μ)^{⊥}. Fix k so that 2^{−k} < \frac{1}{2n}. Cover the plane with closed squares of side 2^{−k} and disjoint interiors, and let {Q_{p}} be an enumeration of the \frac{5}{4} enlargement of the interiors of these squares (so \{Q_{p}\} forms an open cover of the plane, and no point is contained in more than 4 different Q_{p}’s). Let {ψ_{p}} be a C^{1} partition of unity subordinate to {Q_{p}}, with ∥¯∂ψ_{p}∥_{∞} ≤ 80 · 2^{k}. Let h_{p} = T_{ψ_{p}}h, where T_{ψ_{p}} is the Vitushkin localization operator. Note that h_{p} is zero unless ∂G ∩ (Int(K) \ O_{n}) ∩ Q_{p} is non-empty, and h = ∑ h_{p}. Let

ω_{k}(h) = sup\{|h(z) − h(λ)| : |z − λ| ≤ \sqrt{2} 2^{−k}\}.

Then the following hold, for some constant C independent of k:

(i) ∥h_{p}∥_{∞} ≤ Cω_{k}(h).

(ii) |h′_{p}(∞)| ≤ Cω_{k}(h)2^{−k}.

(iii) |β(h_{p}, z_{p})| ≤ Cω_{k}(h)2^{−2k}, where z_{p} is the center of Q_{p}.

Let m be the number of Q_{p}’s intersecting ∂G ∩ (Int(K) \ O_{n}), and let ν_{1} = gμ. For each p for which ∂G ∩ (Int(K) \ O_{n}) ∩ Q_{p} is non-empty, choose w_{p} in this set. As w_{p} is not an analytic bounded point evaluation for R^{2}(K, μ), it must be a light point for |ν̂_{1}| [6, Thm. 3.16]. Now apply Lemma 3.12 of [6], with s = 1, a = w_{p}, δ = 2^{−k}, ε = \frac{1}{m},

α = \frac{2^{k}h′_{p}(∞)}{Cω_{k}(h)},

and

β = \frac{2^{2k}β(h_{p}, w_{p})}{Cω_{k}(h)}.

This yields a function f_{p}, continuous on ℂ_{∞} and analytic off ∂G ∩ (Int(K) \ O_{n}), with the properties that

(i) h_{p} − Cω_{k}(h)f_{p} has a triple zero at infinity, and (ii) | ∫ f_{p}gdμ| < \frac{1}{m}. A standard Vitushkin scheme argument now gives that, for some universal constant C_{1},

∑ |h_{p} − Cω_{k}(h)f_{p}| ≤ C_{1}Cω_{k}(h),

so approximating h by ∑ Cω_{k}(h)f_{p} we get that

| ∫ hgdμ| ≤ 2C_{1}Cω_{k}(h).

Now let k tend to infinity, and we get that

∫ hgdμ = 0.

As g was arbitrary, we conclude that h is in R^{2}(K, μ) as desired. □

We now revert to the assumptions at the beginning of the section, and take p to be as in Lemma 1.1.

Lemma 1.5 Let ϕ be a bounded Borel function on ∂G. Then ϕ̂p ∈ R^{2}(K, μ).

Proof: We can write ϕ as ϕ_{1} + ϕ_{2}, where the first summand is supported by ∂K, and the second summand is carried by ∂G \ ∂K. By the previous lemma, ϕ̂_{2}, and therefore also pϕ̂_{2}, is in R^{2}(K, μ). By [22, Thm 4.1], ϕ̂_{1} is in z̄R(K) + R(K); as (z̄R(K) + R(K))p is contained in R^{2}(K, μ), we also have that pϕ̂_{1} is in R^{2}(K, μ). □

For each point λ in G, let k_{λ} denote the kernel function for R^{2}(K, μ), i.e. that function for which

∫ r(z)k̄_{λ}(z)dμ(z) = r(λ)

for all r in Rat(K).

Lemma 1.6 Let {λ_{n}} be a sequence in G that converges to some point λ_{0} in ∂G. Suppose that \frac{k_{λn}}{∥k_{λn}∥} converges to zero weakly. Then for every disk Δ centered at λ_{0} , we have

lim_{λ_{n}→λ_{0}} ∫_{Δ} \frac{|k_{λ_{n}}|^{2}}{∥k_{λ_{n}}∥^{2}} dμ = 1.

Proof: Let ϵ > 0. Using Lemma 1.3, we can choose a rational function r with poles off ∂G and a bounded Borel function ϕ carried by ∂G such that (ϕ̂p+r)(λ_{0}) = 1, ∥ϕ̂p+r∥_{∞} ≤ 1+ϵ, and |ϕ̂p + r| < ϵ on ∂G \ Δ. Let U be a neighborhood of ∂G that contains no poles of r and such that

|ϕ̂p + r| < 2ϵ, on U \ Δ.

We wish to establish that, for λ near λ_{0},

ϕ̂(λ)p(λ) + r(λ) ≈ ∫_{U}(ϕ̂p + r)\frac{|k_{λ}|^{2}}{∥k_{λ}∥^{2}} dμ,

which is the content of Equation (1.10) below.

Note first that, although we know from Lemma 1.5 that ϕ̂p is in R^{2}(K, μ), we have not yet established that its value at any λ in G is necessarily ⟨ϕ̂p, k_{λ}⟩. In other words, ϕ̂p is a priori a continuous function on ℂ_{∞}, that agrees μ-a.e. with some g in R^{2}(K, μ). There is a natural extension of g to G given by g̃(λ) = ⟨g, k_{λ}⟩; we want to show that g̃(λ) = ϕ̂(λ)p(λ). This is so because

\frac{ϕ̂(z)p(z) − ϕ̂(λ)p(λ)}{z − λ} = \frac{ϕ̂(z) − ϕ̂(λ)}{z − λ}p(z) + \frac{p(z) − p(λ)}{z − λ}ϕ̂(λ)
= [∫_{∂G} \frac{1}{w − z} \frac{ϕ(w)}{w − λ}dA(w)] p(z) + \frac{p(z) − p(λ)}{z − λ}ϕ̂(λ)

and both these latter terms are in R^{2}(K, μ) by Lemma 1.5. Therefore

⟨ϕ̂p, k_{λ}⟩ = ⟨(z − λ)\frac{ϕ̂(z)p(z) − ϕ̂(λ)p(λ)}{z − λ} + ϕ̂(λ)p(λ), k_{λ}⟩
= ϕ̂(λ)p(λ),

as desired. It follows, moreover, that

ϕ̂(λ)p(λ) = ⟨ϕ̂p\frac{k_{λ}}{∥k_{λ}∥^{2}} , k_{λ}⟩.

Let P be the orthogonal projection from L^{2}(μ) onto R^{2}(K, μ), and A_{1} be multiplication by the characteristic function of U^{c} followed by P. Then A_{1} is compact, because if f_{n} is any sequence tending weakly to zero, it will tend to zero uniformly on U^{c}, and so A_{1}f_{n} will tend to zero in norm. We have

ϕ̂(λ)p(λ) = ∫_{U} ϕ̂p\frac{|k_{λ}|^{2}}{∥k_{λ}∥^{2}} dμ + ⟨A_{1} \frac{k_{λ}}{∥k_{λ}∥}, \frac{k_{λ}}{∥k_{λ}∥}⟩  (1.7)

We want a similar equation for r(λ). Let r = s/t, where s and t are polynomials with no common zeroes. Let Q be the orthogonal projection from R^{2}(K, μ) onto tR^{2}(K, μ) (note I−Q is finite rank), and B be the left inverse of M_{t} (multiplication by t) for which M_{t}B = Q. We have

⟨\frac{k_{λ}}{∥k_{λ}∥}, (M_{s}B)^{⋆} \frac{k_{λ}}{∥k_{λ}∥}⟩ = s(λ)⟨\frac{k_{λ}}{∥k_{λ}∥}, B^{⋆} \frac{k_{λ}}{∥k_{λ}∥}⟩
= s(λ)⟨(I − Q)\frac{k_{λ}}{∥k_{λ}∥}, B^{⋆} \frac{k_{λ}}{∥k_{λ}∥}⟩ + s(λ)⟨Q\frac{k_{λ}}{∥k_{λ}∥}, B^{⋆} \frac{k_{λ}}{∥k_{λ}∥}⟩.

As B(I − Q) = 0, and

⟨Q\frac{k_{λ}}{∥k_{λ}∥}, B^{⋆} \frac{k_{λ}}{∥k_{λ}∥}⟩ = \frac{1}{∥k_{λ}∥^{2}} [BQk_{λ}](λ)
= \frac{1}{∥k_{λ}∥^{2}} \frac{1}{t(λ)}[Qk_{λ}](λ),

we have

⟨\frac{k_{λ}}{∥k_{λ}∥}, (M_{s}B)^{⋆} \frac{k_{λ}}{∥k_{λ}∥}⟩ = \frac{1}{∥k_{λ}∥^{2}} r(λ)⟨Qk_{λ}, k_{λ}⟩
= r(λ)[1 − ⟨(I − Q)\frac{k_{λ}}{∥k_{λ}∥}, \frac{k_{λ}}{∥k_{λ}∥}⟩],

so

r(λ) = ⟨\frac{k_{λ}}{∥k_{λ}∥}, (M_{s}B)^{⋆} \frac{k_{λ}}{∥k_{λ}∥}⟩ + r(λ)⟨(I − Q)\frac{k_{λ}}{∥k_{λ}∥}, \frac{k_{λ}}{∥k_{λ}∥}⟩  (1.8)

Now

⟨\frac{k_{λ}}{∥k_{λ}∥}, (M_{s}B)^{⋆} \frac{k_{λ}}{∥k_{λ}∥}⟩ = ⟨M_{χ_{U}} M_{s}B \frac{k_{λ}}{∥k_{λ}∥}, \frac{k_{λ}}{∥k_{λ}∥}⟩ + ⟨M_{χ_{U^{c}}}M_{s}B \frac{k_{λ}}{∥k_{λ}∥}, \frac{k_{λ}}{∥k_{λ}∥}⟩  (1.9)

where χ_{U} is the characteristic function of U. The operator A_{2} = M_{χ_{U^{c}}}M_{s}B is compact, and the first term on the right-hand side of (1.9) can be written

⟨M_{χ_{U}} M_{s}B \frac{k_{λ}}{∥k_{λ}∥}, \frac{k_{λ}}{∥k_{λ}∥}⟩ = ∫_{U} s(z)\frac{(Bk_{λ})(z)}{∥k_{λ}∥} \frac{\overline{k_{λ}(z)}}{∥k_{λ}∥}dμ(z)
= \frac{1}{∥k_{λ}∥^{2}} ∫_{U} \frac{s(z)}{t(z)}[t(z)(Bk_{λ})(z)]\overline{k_{λ}(z)}dμ(z)
= \frac{1}{∥k_{λ}∥^{2}} ∫_{U} \frac{s(z)}{t(z)}[(Qk_{λ})(z)]\overline{k_{λ}(z)}dμ(z)
= ∫_{U} \frac{s}{t} \frac{|k_{λ}|^{2}}{∥k_{λ}∥^{2}} dμ − ∫_{U} \frac{s}{t}[(I − Q)\frac{k_{λ}}{∥k_{λ}∥}]\overline{\frac{k_{λ}}{∥k_{λ}∥}}dμ
= ∫_{U} \frac{s}{t} \frac{|k_{λ}|^{2}}{∥k_{λ}∥^{2}} dμ − ⟨M_{(s/t)χ_{U}} (I − Q)\frac{k_{λ}}{∥k_{λ}∥}, \frac{k_{λ}}{∥k_{λ}∥}⟩.

As A_{3} = M_{(s/t)χ_{U}} (I − Q) is compact, writing A_{4} = A_{1} + A_{2} + A_{3} and using equations (1.7) and (1.8), we get that

ϕ̂(λ)p(λ) + r(λ) = ∫_{U}(ϕ̂p + r)\frac{|k_{λ}|^{2}}{∥k_{λ}∥^{2}} dμ + \{⟨A_{4} \frac{k_{λ}}{∥k_{λ}∥}, \frac{k_{λ}}{∥k_{λ}∥}⟩ + r(λ)⟨(I − Q)\frac{k_{λ}}{∥k_{λ}∥}, \frac{k_{λ}}{∥k_{λ}∥}⟩\}

(1.10)

The terms in braces are just a compact perturbation (with an unimportant non-linearity in the second one). By the way ϕ and r were chosen, and using equation (1.10), we get

|ϕ̂(λ_{n})p(λ_{n}) + r(λ_{n})| ≤ ∫_{Δ} |ϕ̂p + r|\frac{|k_{λ_{n}}|^{2}}{∥k_{λ_{n}}∥^{2}} dμ + ∫_{U\Δ} |ϕ̂p + r|\frac{|k_{λ_{n}}|^{2}}{∥k_{λ_{n}}∥^{2}} dμ

+∥A_{4} \frac{k_{λ_{n}}}{∥k_{λ_{n}}∥}∥ + |r(λ_{n})|∥(I − Q)\frac{k_{λ_{n}}}{∥k_{λ_{n}}∥}∥

≤ (1 + ϵ) ∫_{Δ} \frac{|k_{λ_{n}}|^{2}}{∥k_{λ_{n}}∥^{2}} dμ + 2ϵ

+∥A_{4} \frac{k_{λ_{n}}}{∥k_{λ_{n}}∥}∥ + |r(λ_{n})|∥(I − Q)\frac{k_{λ_{n}}}{∥k_{λ_{n}}∥}∥.

Letting λ_{n} tend to λ_{0}, we get

1 ≤ (1 + ϵ) lim_{λ_{n}→λ_{0}} ∫_{Δ} \frac{|k_{λ_{n}}|^{2}}{∥k_{λ_{n}}∥^{2}} dμ + 2ϵ.

As ϵ was arbitrary, the lemma is proved. □

We are finally able to show that f = z̄p extends continuously to K, and hence that each component of the interior of K is a quadrature domain.

Lemma 1.11 Define f by

f(λ) = \{ \begin{matrix}∫ z̄p(z)\overline{k_{λ}(z)}dμ(z) & λ ∈ G \\ λ̄p(λ) & λ ∈ ∂G \end{matrix}

Then f is continuous on \overline{G}.

Proof: Let λ_{0} ∈ ∂G and {λ_{n}} ⊂ G be a sequence converging to λ_{0}.

Case 1. Suppose that λ_{0} ∈ intK ∩ ∂G. It follows from [6, Thm 5.1] that λ_{0} is not a bounded point evaluation for R^{2}(K, μ), so ∥k_{λ_{n}}∥ must tend to infinity (for else some subsequence would have a weak limit, which would be a kernel function for λ_{0}). For every rational function r with poles off K,

⟨r, \frac{k_{λ_{n}}}{∥k_{λ_{n}}∥}⟩ = \frac{r(λ_{n})}{∥k_{λ_{n}}∥} → 0,

so \frac{k_{λn}}{∥k_{λn}∥} converges to zero weakly.

For every ϵ > 0, there is a δ > 0 such that when λ is in 𝔻(λ_{0}, δ) ∩ sptμ, we have

|λ̄p(λ) − λ̄_{0}p(λ_{0})| < ϵ.

For λ in G,

f(λ) = ∫ z̄p\frac{|k_{λ}|^{2}}{∥k_{λ}∥^{2}} dμ,

so

|f(λ_{n}) − λ̄_{0}p(λ_{0})| = | ∫ (λ̄p(λ) − λ̄_{0}p(λ_{0}))\frac{|k_{λ_{n}}|^{2}}{∥k_{λ_{n}}∥^{2}} dμ(λ)|
≤ ϵ ∫_{𝔻(λ_{0},δ)} \frac{|k_{λ_{n}}|^{2}}{∥k_{λ_{n}}∥^{2}} dμ + 2∥z̄p∥_{∞} ∫_{𝔻(λ_{0},δ)^{c}} \frac{|k_{λ_{n}}|^{2}}{∥k_{λ_{n}}∥^{2}} dμ.

Therefore by Lemma 1.6

lim sup_{n→∞} |f(λ_{n}) − λ̄_{0}p(λ_{0})| ≤ ϵ.

As ϵ is arbitrary, we have

lim_{n→∞} f(λ_{n}) = λ̄_{0}p(λ_{0}).

Case 2. Suppose that λ_{0} ∈ ∂K and λ_{0} is not a bounded point evaluation for R^{2}(K, μ). Using the same proof as in Case 1, we again get that

lim_{n→∞} f(λ_{n}) = λ̄_{0}p(λ_{0}).

Case 3. Suppose that λ_{0} ∈ ∂K and λ_{0} is a bounded point evaluation for R^{2}(K, μ). Let k_{0} ∈ R^{2}(K, μ) be such that

r(λ_{0}) = ⟨r, k_{0}⟩

when r is a rational function with poles off K. Choose w_{n} ∈ K^{c} such that w_{n} → λ_{0}. Then

f_{n}(z) = (z − λ_{0})\frac{z̄ − w̄_{n}}{z − w_{n}} p(z)

is in clos((z − λ_{0})R^{2}(K, μ)). By the Lebesgue dominated convergence theorem, f_{n} converges to (z̄ − λ̄_{0})p(z) in L^{2}(μ), so (z̄ − λ̄_{0})p is in clos((z − λ_{0})R^{2}(K, μ)). Therefore

⟨z̄p, k_{0}⟩ = ⟨(z̄ − λ̄_{0})p, k_{0}⟩ + λ̄_{0}p(λ_{0})
= λ̄_{0}p(λ_{0}).

Now suppose that a is any limit point of {f(λ_{n})}. There exists a subsequence {λ_{n_{l}}} such that f(λ_{n_{l}}) converges to a. If ∥k_{λ_{n_{l}}}∥ converges to infinity, then using the same argument as in Case 1 we get that a = λ̄_{0}p(λ_{0}). So suppose that there exists a subsequence of k_{n_{l}} that converges weakly; then it must converge to k_{0} because lim r(λ_{n_{l}}) = r(λ_{0}) for r in Rat(K). By relabelling if necessary, we may assume that k_{n_{l}} → k_{0} weakly. Then

a = lim f(λ_{n_{l}}) = lim⟨z̄p, k_{λ_{n_{l}}}⟩ = ⟨z̄p, k_{0}⟩ = λ̄_{0}p(λ_{0}).

So f is continuous on \overline{G} and analytic on G as desired. □ We can now prove the main theorem:

Theorem 1.12 Let S_{μ} be a rationally cyclic subnormal operator with spectrum K. Let Ω be the interior of K. Then S_{μ} is irreducible and has finite rank self-commutator if and only if the following conditions are satisfied:

(i) K = \overline{Ω}. (ii) Ω is a quadrature domain. (iii) μ|_{∂Ω} is absolutely continuous with respect to harmonic measure for Ω, which we will denote by ω, and

∫_{∂Ω} log(\frac{dμ}{dω})dω > −∞.

(iv) μ|_{Ω} is either zero or a finite sum of point masses.

Proof: Sufficiency: Suppose conditions (i) through (iv) are satisfied. By (Q1) there is a meromorphic function R on Ω that is continuous on \overline{Ω} and equals z̄ on ∂Ω. Write R as f/q_{1}, where q_{1} is a polynomial, and f is holomorphic on Ω. Because ∂K is an algebraic curve (Q3), A(K) = R(K), so f is in R(K).

Let q_{2} be a polynomial with zeroes at the atoms of μ. Then for any g in R^{2}(K, μ),

z̄q_{1}(z)q_{2}(z)g(z) = f(z)q_{2}(z)g(z) μ − a.e.

and the right-hand side is in R^{2}(K, μ). Therefore q_{1}q_{2}R^{2}(K, μ) is contained in the kernel of [S_{μ}^{∗}, S_{μ}], so the rank of [S_{μ}^{∗}, S_{μ}] is at most the degree of q_{1}q_{2}, and therefore finite.

Because K is finitely connected, R(K) is a hypo-Dirichlet algebra [23] ( R(K) is hypo-Dirichlet means the uniform closure of Re(R(K)) is of finite codimension in C_{ℝ}(∂K)). It follows then from condition (iii) that the analytic bounded point evaluations of R^{2}(K, μ) are Ω, and that R^{2}(K, μ) is irreducible [2].

Necessity: Condition (i) follows from Lemma 1.2.

Let G be the set of analytic bounded point evaluations of R^{2}(K, μ). Let p be as in Lemma 1.1. By Lemma 1.11, there is a continuous analytic function f that equals z̄p(z) on ∂G. It follows that G can only have finitely many components, because any component that did not contain a zero of p would have the holomorphic function f/p equal to z̄ on the boundary, which is impossible because the two harmonic functions f/p and z̄ would then have to agree on the interior also.

As K = ∪_{i=1}^{N}\overline{G_{i}}, where each G_{i} is connected and bounded by an algebraic curve, strong interpolation holds for R^{2}(K, μ) in the sense of Conway and Elias [6], and by [6, Thm. 4.2], because S_{μ} is irreducible, G can in fact have only one component, so G is a quadrature domain. The complement of G cannot contain isolated points, so by (Q3) G = int(\overline{G}) = Ω.

As R(z) = f(z)/p(z) = z̄ μ-a.e., the support of μ must be contained in ∂Ω ∪ E(Ω), which yields (iv). If μ_{s} is the singular part of μ|_{∂Ω} with respect to Ω, then R^{2}(K, μ) = R^{2}(μ − μ_{s}) ⊕ L^{2}(μ_{s}) [15], so irreducibility forces μ|_{∂Ω} to be absolutely continuous. Finally, by [2, Thm 5.1], the points of Ω are analytic bounded point evaluations for R^{2}(μ|_{∂Ω}) (and hence for R^{2}(K, μ)) if and only if

∫_{∂G} log(\frac{dμ}{dω})dω > −∞.

□

2 Equivalence of rationally cyclic subnormals with finite rank self-commutators

Using Theorem 1.12 and the work of Ahern and Sarason on hypo-Dirichlet algebras [2], we can classify rationally cyclic subnormal operators with finite rank self-commutators into unitary equivalence classes. First, observe that any pure rationally cyclic subnormal operator with finite rank self-commutator (pure means having no normal summand) can be written as a finite direct sum ⊕_{i=1}^{n}S_{i} where each S_{i} is an irreducible rationally cyclic subnormal operator with finite rank self-commutator and, for distinct indices i and j, the spectra of S_{i} and S_{j} intersect in at most a finite set (this follows from Bezout’s theorem that the intersection of two distinct ireducible algebraic curves is a finite set, so if σ(S_{i}) intersected σ(S_{j}) in an infinite set, their interiors would have to intersect, and then the direct sum S_{i} ⊕ S_{j} would not be rationally cyclic). Moreover, any such sum will yield a pure rationally cyclic subnormal operator with finite rank self-commutator.

Furthermore, two subnormal operators are unitarily equivalent, similar, or quasi-similar, if and only if their pure parts are, respectively, unitarily equivalent, similar, or quasi-similar, and their normal parts are unitarily equivalent [5, Prop II.13.7] (two operators S and T, are said to be quasisimilar if there exist bounded operators with dense ranges and no kernels, X and Y, such that SX = XT and Y S = T Y). So for simplicity we shall only discuss equivalence of irreducible rationally cyclic subnormal operators with finite rank selfcommutators, as the non-irreducible case can be got from this by taking direct sums and matching the summands.

Let us recall some facts about hypo-Dirichlet algebras. Let us fix a piecewise-smoothly bounded planar domain Ω, of connectivity t + 1, and let K denote its closure. Let ω be harmonic measure for Ω. In each bounded component of Ω^{c} pick a point z_{i}, i = 1, …, t. A function g in R^{2}(K, ω) is called inner if there are real numbers α_{1}, …, α_{t} such that

|g(z)| = |z − z_{1}|^{α_{1}} … |z − z_{t}|^{α_{t}}  (2.1)

for ω-a.e. z. A function h is called outer if {rh : r ∈ Rat(K)} is dense in R^{2}(K, ω). We need the following two facts: (HD1) If v is a non-negative function in L^{2}(ω), and ∫ log(v)dω > −∞, then there is a function f in R^{2}(K, ω) with |f| = v a.e. [2]. (HD2) Any function f in R^{2}(K, ω) can be written as the product gh where g is inner and h is outer. If, in addition, all the α_{i}’s in 2.1 are chosen in the range [0, 1), then these factors are unique (up to a constant unimodular multiple) ([2] and [3]).

Theorem 2.2 Let Ω = int(\overline{Ω}) be a quadrature domain, K be the closure of Ω, and assume the connectivity of Ω is t + 1. Let n be the order of Ω, and let E(Ω) be as defined in (Q2). Let s be an integer greater than or equal to n.

(i) The set of unitary equivalence classes of irreducible rationally cyclic subnormal operators with spectrum K and rank s self-commutator is parametrized by

[0, 1)^{t} × {Ω × (0, ∞)}^{s−n} × {ℝ^{+}}^{E(Ω)}.

(ii) All irreducible rationally cyclic subnormal operators with spectrum K and finite-rank self-commutator are similar, so the spectrum is a complete similarity and quasi-similarity invariant for pure rationally cyclic subnormal operators with finite-rank self-commutators.

Proof: (i) Let S_{μ} be such an operator. In the notation of the previous section, the rank of the self-commutator of S is the degree of p, which is the sum of the order of Ω and the number of atoms of μ in Ω \ E(Ω). Let S_{ν} be another such operator. For S_{μ} and S_{ν} to be unitarily equivalent, they must be mutually absolutely continuous, and there must be a function ψ in R^{2}(K, ν), a rationally cyclic vector for S_{ν}, with |ψ|^{2} = \frac{dμ}{dν} ν−a.e. [5, Prop II.13.1].

First we match them up on the boundary. By (HD1), there are functions f_{1} and f_{2} in R^{2}(K, ω) with |f_{1}|^{2} = \frac{dμ}{dω} and |f_{2}|^{2} = \frac{dν}{dω} ω−a.e.. On ∂Ω, we have |ψ| = |f_{1}/f_{2}|; by (HD2) we can choose such a ψ to be rationally cyclic if and only if the inner factors of f_{1} and f_{2} agree up to integer multiples of (z − z_{i}). If this is the case, the modulus of ψ will be determined completely by its boundary values (because log |ψ| will be harmonic [2]), so for S_{μ} and S_{ν} to be unitarily equivalent, at each atom λ we must have μ(λ) = |ψ(λ)|^{2}ν(λ).

Thus to choose a unitary equivalence class, we first make t choices on the t inner boundary components, and that will give all possibilities for measures μ supported on ∂K that give rise to irreducible S_{μ}’s with rank n self-commutator. We can then choose atoms at points of E(Ω) without increasing the rank of the self-commutator, so can assign any nonnegative weights to these points. Finally, we must choose s−n atoms at points of Ω\E(Ω), and assign strictly positive weights to these atoms.

(ii) Let S_{μ} be as before. Because the atoms are a compact subset of the set of analytic bounded point evaluations, there is a constant C such that

∫_{K} |r|^{2}dμ ≤ C ∫_{∂Ω} |r|^{2}dμ ≤ C ∫_{K} |r|^{2}dμ

for all rational r. Thus the atoms do not affect the similarity class and can be dropped. On the boundary, the inner factor f_{1} is bounded and bounded away from zero, so also does not affect the similarity class. Therefore S_{μ} is similar to S_{ω}, so any two irreducible subnormal operators with finite rank self-commutator and spectrum K are similar.

Conversely, similar and quasi-similar subnormal operators have the same spectrum [4], so there is a bijective correspondence between similarity (and quasi-similarity) classes of pure rationally cyclic subnormal operators with finite-rank self-commutator and finite unions of non-intersecting quadrature domains. □

In the special case that Ω is simply connected, by (Q4) it is the conformal image of the unit disk under a rational map r. Let ϕ be the inverse of r; then it is easy to see that the Schwarz function on Ω is given by

R(z) = \overline{r(\frac{1}{\overline{ϕ(z)}})}.

Therefore the order of Ω, which is equal to the number of poles of R in Ω counted according to multiplicity, is equal to the degree of the rational function r. Moreover E(Ω) is the set of points z in Ω for which r(1/\overline{ϕ(z)}) = z. We thus have another proof of Theorem 2 of [17], where they prove that, in our language, if Ω is a simply connected quadrature domain, and μ is harmonic measure on ∂Ω plus a finite number of point masses in Ω, then the rank of the self-commutator of S_{μ} is equal to the degree of the rational map mapping the disk conformally onto Ω plus the number of atoms of μ in Ω \ E(Ω).

3 Geometric connections between subnormal operators and quadrature domains

As we have established a correspondence between quadrature domains and rationally cyclic subnormal operators with finite rank self-commutators, it is of interest to see how these two areas illuminate one another. We should point out that very recently M. Putinar has found another connection between operator theory and quadrature domains by studying hyponormal operators with rank one self-commutators [18].

Two pure subnormal operators S_{1} and S_{2} are unitarily equivalent if and only if there is a unitary operator from the range of [S_{1}^{⋆}, S_{1}] := L_{1} to the range of [S_{2}^{⋆}, S_{2}] := L_{2} that simultaneously intertwines both S_{1}^{⋆}|_{L_{1}} with S_{2}^{⋆}|_{L_{2}}, and [S_{1}^{⋆}, S_{1}] |_{L_{1}} with [S_{2}^{⋆}, S_{2}] |_{L_{2}} - see e.g. [13]. So each bounded quadrature domain of order n gives rise to two n × n matrices; conversely, given two n × n matrices, if the first can be extended to a bounded rationally cyclic subnormal operator, with non-atomic spectral measure, and the second to its self-commutator, so that the original n-dimensional Hilbert space is the range of the self-commutator, then this pair corresponds to a bounded quadrature domain, and modulo the U(n) equivalence of such pairs, this is a bijection. In theory, therefore, this gives a parametrization of all bounded quadrature domains.

The first theorem about quadrature domains was proved by B. Epstein in 1962, who proved that if Ω is a simply connected quadrature domain of order 1, then it is a disk [7]; later Epstein and Schiffer showed that the simple connectivity hypothesis was unnecessary [8]. In 1973, B. Morrel showed that the only pure subnormal operators with rank one self commutators were translations and dilations of the unilateral shift [14]; using Theorem 1.12 it can be seen that Morrel’s theorem implies that of Epstein and Schiffer (in the plane; their theorem also works in higher dimensions).

The geometry of quadrature domains has been studied in great depth by B. Gustafsson and M. Sakai - for a bibliography, and a very readable account of quadrature domains, see H. Shapiro’s book [20]. Let us mention here specifically the papers [12] and [19] where they obtain inequalities on some geometric features of quadrature domains. To discuss these, we first need some definitions.

Let Ω = int(\overline{Ω}) be a quadrature domain, of connectivity t + 1 and order n. Let W be a plane domain, bounded by finitely many smooth curves and conformally equivalent to Ω, and let ϕ : W → Ω be a conformal map. Let C the set of cusp points of Ω, and D be the set of double points of Ω, defined respectively as

C = {z ∈ ∂Ω : z = ϕ(ζ) for some ζ ∈ ∂W with ϕ′(ζ) = 0}
D = {z ∈ ∂Ω : z = ϕ(ζ_{1}) = ϕ(ζ_{2}) for two different ζ_{j} ∈ ∂W }

E := E(Ω) has already been defined in Q(2); let c, d, e denote the cardinalities of C, D, E, respectively. Then the following inequalities hold:

Theorem 3.1 (Gustafsson, [12])

t + c + 2d + e ≤ (n − 1)^{2}.

Theorem 3.2 (Sakai, [19])

c + e ≥ t + n − 1.

(Sakai actually proves an equality, by assigning indices to points of C and E and summing appropriately).

Let us describe how to find these numbers operator theoretically. Let K = \overline{Ω}. Then n is the minimum, over all rationally cyclic subnormal operators with spectrum K, of the rank of the self-commutator. The number e is the maximum, over all rationally cyclic subnormal operators with spectrum K and rank n self-commutators, of the number of atoms in the spectral measure of the minimal normal extension.

Once one knows the spectrum, the numbers t, c and d can all be read off fairly easily from looking at it. One could do it analytically by noting that t is the dimension of the space of unitary equivalence classes of rationally cyclic subnormal operators with finite rank self-commutators, spectrum K, and non-atomic spectral measures. The number d is the difference between the number of components of K^{c} (which could be calculated, e.g., by looking at the spectra of restrictions of S to invariant subspaces) and the number of components of Ω^{c}. To check if a point z in ∂K is in C, one can, by passing to some (S − λ)^{−1} if necessary, assume that z is in the outer boundary of K. One can then find (S − z)^{3/2}; this will have Fredholm index −2 arbitrarily close to z if and only if z is in D or is an inward pointing cusp; (S − z)^{3} will not have Fredholm index less than or equal to −2 close to z if and only if z is an outward pointing cusp.

These are probably not useful ways of finding c or d. However we can prove a simple version of Gustafsson’s theorem involving t, e and n.

Proposition 3.3

t + e ≤ (n − 1)^{2}.

Proof: Let μ be the sum of harmonic measure for Ω and a point mass at each point of E. By the previous section, the rank of the self-commutator of S_{μ} is n, so by Section 1 there is a polynomial p of degree n with z̄p in R^{2}(K, μ).

Now let us calculate the codimension of R^{2}(K, μ)+\overline{R^{2}(K, μ)} in L^{2}(μ) (where the bar denotes complex conjugation). This will be t plus the number of atoms of μ, so t+e. However, because p is of degree n, it is clear that the closed linear span of {z^{i}z̄^{j} : 1 ≤ i, j ≤ n − 1} and R^{2}(K, μ) + \overline{R^{2}(K, μ)} is a reducing subspace for N_{μ}, so is all of L^{2}(μ). Therefore the codimension is at most (n − 1)^{2}, so t + e ≤ (n − 1)^{2}. □

As not all subnormal operators with self-commutator even of rank two are rationally cyclic, it seems that they are not a very sharp tool for proving inequalities like Theorems 3.1 and 3.2. However, they may be able to provide other insights. For example, the essential spectrum depends only on the 2n^{2} entries in S^{∗}|_{L} and [S^{⋆}, S]|_{L}, and an explicit formula for this in terms of these entries would give the algebraic curve describing ∂Ω. We hope that other connections can be found.

References

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