QUASISIMILARITY OF RATIONALLY CYCLIC SUBNORMAL OPERATORS

John E. McCarthy

Abstract. We give a new proof of Raphael’s theorem that quasisimilar rationally cyclic subnormal operators have naturally isomorphic commutants. We also classify all subnormal operators quasisimilar to R_{K,μ}, when R(K) is a hypo-Dirichlet algebra, and μ is supported on the boundary of K.

Introduction

Two bounded linear operators on a Hilbert space, S and T, are said to be quasisimilar if there exist quasi-affinities X and Y (i.e. bounded operators with dense ranges and no kernels) such that SX = XT and Y S = T Y. Quasisimilarity is an interesting equivalence relation for subnormal operators, because, on the one hand, it preserves such properties as the spectrum [4] and the essential spectrum [14], and on the other, it provides a large class of equivalent operators which, in certain cases, are amenable to a nice classification.

If K is a compact subset of ℂ, Rat(K) will denote the rational functions with poles off K, and an operator S on H is called Rat(K)-cyclic if the spectrum of S is contained in K, and there is some vector ξ in H, such that {r(S)ξ|r ∈ Rat(K)} is dense in H. Any Rat(K)-cyclic subnormal operator is unitarily equivalent to multiplication by the independent variable on R^{2}(K, μ), the closure of Rat(K) in L^{2}(μ), for some finite, positive Borel measure supported on K [6,p.146]. We shall call this latter operator R_{K,μ}, and μ will be called the symbol of R_{K,μ}.

If f is in R^{2}(K, μ)∩L^{∞}(μ), M_{f}, the operator of multiplication by f on R^{2}(K, μ), is clearly in the commutant of R_{K,μ}, and, by a theorem of Yoshino [15], anything in the commutant is of this form. In [12], Raphael proved that the commutants of quasisimilar rationally cyclic (i.e. Rat(K)-cyclic for some K) subnormal operators are isometrically isomorphic and weak-star homeomorphic. In Section 1 we give a

Typeset by AMS-TEX 2 JOHN E. MCCARTHY

new proof of this theorem, which is shorter, and, we think, simpler. We would like to thank John Conway for bringing [12] to our attention.

In [3], Clary classified all subnormal operators quasisimilar to the unilateral shift. This was extended by Hastings [10] to isometries of finite cyclic multiplicity. In Section 2 we extend this in a different direction, namely to classifying all subnormal operators quasisimilar to R_{K,μ}, when R(K) is a hypo-Dirichlet algebra, and μ is supported on the boundary of K. R(K) denotes the uniform closure of Rat(K) in C(K), and R(K) is called a hypo-Dirichlet algebra if (i) the uniform closure of ℜ(R(K)) is of finite codimension in ℜ(C(∂K)), and (ii) the linear span of {log|r| : r, r^{−1} ∈ R(K)} is dense in ℜ(C(∂K)). If ℜ(R(K)) is actually dense in ℜ(C(∂K)), R(K) is called a Dirichlet algebra. If K is finitely connected then R(K) is hypo-Dirichlet [2], but this is not necessary (although the interior of K must always be finitely connected).

By “measure” we shall always mean a compactly supported finite Borel measure on ℂ. Two measures will be called equivalent if they are mutually absolutely continuous. If K is polynomially convex, R^{2}(K, μ) is just the closure of the polynomials in L^{2}(μ), which we shall write as P^{2}(μ), and in this case we shall write S_{μ} for R_{K,μ}. The (bounded linear) operators on a Hilbert space H will be written B(H). The weak-star closure of Rat(K) in L^{∞}(μ) will be written R^{∞}(K, μ). Note that R^{∞}(K, μ) is contained in R^{2}(K, μ) ∩ L^{∞}(μ).

Section 1

We first need a structure theorem to translate from operator theory to function theory. The result we need is from [5], where it is proved for the polynomial case, but the rational case works exactly the same way.

Proposition 1.1. Suppose R_{1} and R_{2} are rationally cyclic subnormal operators. Then they are quasisimilar if and only if there exist measures μ_{1} and μ_{2}, a compact set K, an R_{K,μ_{1}}-cyclic vector ϕ in R^{2}(K, μ_{1}) ∩ L^{∞}(μ_{1}), and constants C_{1} and C_{2}, such that R_{i} is unitarily equivalent to R_{K,μ_{i}}, (i = 1, 2), and

∫ |rϕ|^{2}dμ_{1} ≤ C_{1} ∫ |r|^{2}dμ_{2} ≤ C_{2} ∫ |r|^{2}dμ_{1}

QUASISIMILARITY OF RATIONALLY CYCLIC SUBNORMAL OPERATORS 3

for all r in Rat(K).

We next prove two lemmata, which are just special cases of Theorem 1.4. The first says that if R_{K,μ} and R_{K,ν} are actually similar, and ν is absolutely continuous with respect to μ, then their commutants are naturally isomorphic.

Lemma 1.2. Suppose μ and ν are measures supported on K, ν is absolutely continuous with respect to μ, and

∫ |r|^{2}dμ ≤ C_{1} ∫ |r|^{2}dν ≤ C_{2} ∫ |r|^{2}dμ

for all r in Rat(K). Then the identity map on Rat(K) extends to an isometric isomorphism and weak-star homeomorphism from R^{2}(K, μ)∩L^{∞}(μ) onto R^{2}(K, ν)∩ L^{∞}(ν).

Proof. By continuity, the map X : r ↦ r, for r in Rat(K), extends to a bounded invertible operator from R^{2}(K, μ) to R^{2}(K, ν), which intertwines R_{K,μ} and R_{K,ν}. Let f be in R^{2}(K, μ) ∩ L^{∞}(μ). Then there is a sequence r_{n} in Rat(K) which converges to f in L^{2}(μ), and, by passing to a subsequence if necessary, we can assume r_{n} tends to f μ-almost everywhere. Xf is the L^{2}(ν) limit of Xr_{n} = r_{n}, and because ν is absolutely continuous with respect to μ, r_{n} converges ν-almost everywhere, so Xf = f, ν-almost everywhere. Therefore

∥Xf∥_{L^{∞}(ν)} ≤ ∥f∥_{L^{∞}(μ)}.

Moreover XM_{f}X^{−1} = M_{Xf}, so

∥f∥_{L^{∞}(μ)} = ∥M_{f}∥_{B(R^{2}(K,μ))}
= ∥X^{−1}M_{Xf}X∥_{B(R^{2}(K,μ))}
≤ ∥X∥∥X^{−1}∥∥M_{Xf}∥_{B(R^{2}(K,ν))}
≤ ∥X∥∥X^{−1}∥∥Xf∥_{L^{∞}(ν)}.

Therefore X̃ : R^{2}(K, μ) ∩ L^{∞}(μ) → R^{2}(K, ν) ∩ L^{∞}(ν), the restriction of X, is bounded below in the L^{∞}(μ) − L^{∞}(ν) norms, and is multiplicative (since it’s just a restriction map), so it must be an isometry. X̃ is surjective because X is invertible. Finally, X̃ is weak-star continuous, since ν is absolutely continuous with respect to

4 JOHN E. MCCARTHY

μ, and so by the Krein-Smulian theorem and the Banach-Alaoglu theorem, it is a weak-star homeomorphism. ////

The second lemma requires only that R_{K,μ} and R_{K,ν} be quasisimilar, but adds the hypothesis that μ and ν be mutually absolutely continuous. Lemma 1.3. Suppose μ and ν are mutually absolutely continuous measures, ϕ is an essentially bounded Rat(K)-cyclic vector for R_{K,μ}, and that for some constants C_{1} and C_{2},

∫ |rϕ|^{2}dμ ≤ C_{1} ∫ |r|^{2}dν ≤ C_{2} ∫ |r|^{2}dμ

for all r in Rat(K). Then the identity map on Rat(K) extends to an isometric isomorphism and weak-star homeomorphism from R^{2}(K, μ)∩L^{∞}(μ) onto R^{2}(K, ν)∩ L^{∞}(ν). Proof. As in the previous lemma, the map X from R^{2}(K, μ) to R^{2}(K, ν), which leaves Rat(K) fixed, restricts to a map X̃ from R^{2}(K, μ) ∩ L^{∞}(μ) to R^{2}(K, ν) ∩ L^{∞}(ν) which is the identity map (point-wise) on functions, and therefore is isometric, multiplicative and weak-star continuous. It only remains to show that it is surjective.

To this end, define two auxiliary maps W : R^{2}(K, ν) → R^{2}(K, |ϕ|^{2}μ), and U : R^{2}(K, |ϕ|^{2}μ) → R^{2}(K, μ), by W (r) = r and U(r) = ϕr for all r in Rat(K), and extending them by continuity. Again, W̃ X̃ maps R^{2}(K, μ) ∩ L^{∞}(μ) into R^{2}(K, |ϕ|^{2}μ) ∩ L^{∞}(|ϕ|^{2}μ) by the identity map on functions, and if W̃ X̃ is surjective, so is X̃. But U is invertible, and UR_{K,|ϕ|^{2}μ}U^{−1} = R_{K,μ}, so if g is in R^{2}(K, |ϕ|^{2}μ) ∩ L^{∞}(|ϕ|^{2}μ), then UM_{g}U^{−1} = M_{g} is in the commutant of R_{K,μ}, so g is in R^{2}(K, μ) ∩ L^{∞}(μ), and hence in the range of W̃ X̃. Therefore X̃ is surjective, as desired. ////

The general case of the theorem now follows easily. Theorem 1.4. Suppose R_{1} and R_{2} are quasisimilar rationally cyclic subnormal operators. Then their commutants are naturally isometrically isomorphic and weakstar homeomorphic. Proof. By Proposition 1.1, we can find measures μ_{1}, μ̃_{1}, μ_{2}, and constants C_{1}, C_{2}, such that R_{1} is unitarily equivalent to both R_{K,μ_{1}} and R_{K,μ̃_{1}}, R_{2} is unitarily equivalent to R_{K,μ_{2}}, and

∫ |r|^{2}dμ̃_{1} ≤ C_{1} ∫ |r|^{2}dμ_{2} ≤ C_{2} ∫ |r|^{2}dμ_{1}

for all r in Rat(K).

Let σ = (μ_{1} + μ_{2}), and τ = (μ̃_{1} + μ_{2}). Then

∫ |r|^{2}dσ ≤ (1 + \frac{C_{2}}{C_{1}} ) ∫ |r|^{2}dμ_{1} ≤ (1 + \frac{C_{2}}{C_{1}} ) ∫ |r|^{2}dσ,

so R_{K,σ} is similar to R_{K,μ_{1}}, and, by Lemma 1.2, the identity map on Rat(K) extends to an isometric isomorphism and weak-star homeomorphism from R^{2}(K, σ)∩L^{∞}(σ) onto R^{2}(K, μ_{1}) ∩ L^{∞}(μ_{1}). Likewise, R_{K,τ} is similar to R_{K,μ_{2}}, and so R^{2}(K, τ) ∩ L^{∞}(τ) is mapped isometrically and weak-star homeomorphically onto R^{2}(K, μ_{2}) ∩ L^{∞}(μ_{2}).

R_{K,μ_{1}} is unitarily equivalent to R_{K,μ̃_{1}}, so μ_{1} and μ̃_{1} are equivalent [6, p.217], so σ and τ are equivalent. Because quasisimilarity is clearly preserved under similarity transformations, R_{K,σ} is quasisimilar to R_{K,τ}. Applying Proposition 1.1 to R_{K,σ} and R_{K,τ}, we get unitarily equivalent R_{K,σ̃} and R_{K,τ̃} satisfying the hypotheses of Lemma 1.3, and so we can finally conclude that all of the six operators R_{K,μ_{2}}, R_{K,τ} , R_{K,τ̃}, R_{K,σ̃}, R_{K,σ}, R_{K,μ_{1}} have isometrically isomorphic and weak-star homeomorphic commutants, under maps that leave R^{∞} fixed. ////

To show both steps are needed in the above proof, here is a simple example of two similar, cyclic subnormal operators with mutually singular symbols: Let λ_{n} be Lebesgue measure on the circle of centre 0 and radius 1 − 2^{−n}, normalized to have total mass 2^{−n}. Let μ_{1} be ∑_{m=0}^{∞} λ_{2m+1}, and μ_{2} be ∑_{m=0}^{∞} λ_{2m+2}. Then for any polynomial p,

∫ |p|^{2}dμ_{1} ≤ 2 ∫ |p|^{2}dμ_{2} ≤ 4 ∫ |p|^{2}dμ_{1}.

So S_{μ_{1}} is similar to S_{μ_{2}}, and μ_{1} and μ_{2} are mutually singular.

Section 2

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

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

The functional f ↦ f̂(a) is continuous on C_{ℜ}(∂U), and therefore comes from a measure, ω_{a}, which is called harmonic measure for U 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 U without specifying a point a. Harmonic measure for a compact set K, with interior components U_{n}, is ω = ∑ 2^{−n}ω_{n}, where ω_{n} is harmonic measure for U_{n}. For a fixed compact set K, μ_{a} and μ_{s} will mean, respectively, the absolutely continuous and singular parts of μ with respect to ω, and μ_{an} and μ_{sn} will mean the absolutely continuous and singular parts with respect to ω_{n}.

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∥_{2,μ}

for every r in Rat(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.

A (Gleason) part of R(K) is a maximal set P in ℂ such that for x and y in P, if e_{x} and e_{y} denote the functionals of evaluation at x and y respectively, then ∥e_{x} −e_{y}∥_{R(K)^{∗}} < 2. By [7, p.146] representing measures for points in different parts are mutually singular.

Let us recall some facts about hypo-Dirichlet algebras. These facts, and their proofs, all come from [1].

If the codimension of ℜ(R(K)) in ℜ(C(∂K)) is t, then there are t measures η_{1}, …, η_{t} which annihilate ℜ(R(K)) and are absolutely continuous with respect to harmonic measure for K. There are t invertible functions F_{1}, …, F_{t} in R(K) such that ∫ log|F_{j}|dη_{k} = δ_{jk}. A function h in R^{2}(K, ω) is called inner if, for some

QUASISIMILARITY OF RATIONALLY CYCLIC SUBNORMAL OPERATORS 7

real numbers α_{1}, …, α_{t}, |h| = |F_{1}|^{α_{1}} … |F_{t}|^{α_{t}} ω-almost everywhere. If v in L^{2}(μ) satisfies ∫ log|v|dω > −∞, then there exists an inner function h and a function g that is a Rat(K)-cyclic vector for R_{K,ω}, with |v| = |g||h| ω-almost everywhere.

We are now able to prove our main theorem.

Theorem 2.1. Let K be a compact planar set, such that R(K) is a hypo-Dirichlet algebra. Let μ be a measure supported on the boundary of K, and let ν be a measure supported on K. Then R_{K,ν} is quasisimilar to R_{K,μ} if and only if

(a) (ν|_{∂K})_{s} is mutually absolutely continuous with respect to μ_{s};

(b) If

∫_{∂U_{n}} log \frac{dμ}{dω_{n}} = −∞

then

∫_{∂U_{n}} log \frac{dν}{dω_{n}} = −∞,

ν_{an} is equivalent to μ_{an}, and ν(U_{n}) = 0;

(c) If

∫_{∂U_{n}} log \frac{dμ}{dω_{n}} > −∞  (1)

then

∫_{∂U_{n}} log \frac{dν}{dω_{n}} > −∞.  (2)

Proof. (i) Necessity.

By hypothesis, there exists a Rat(K)-cyclic vector ϕ for R_{K,μ}, and a Rat(K)- cyclic vector ψ for R_{K,ν}, such that for all r in Rat(K)

∫ |rϕ|^{2}dμ ≤ ∫ |r|^{2}dν
∫ |rψ|^{2}dν ≤ ∫ |r|^{2}dμ.  (3)

Fix some interior component U_{n}. Every representing measure for a point of U_{n} that is supported on ∂K is absolutely continuous with respect to ω_{n} [1,3.1], so we can apply Forelli’s lemma [7,p.42], which guarantees the existence of a sequence r_{k} in Rat(K), with ∥r_{k}∥_{∞} ≤ 1, and with r_{k} tending to 1 ω_{n}-almost everywhere, and to zero (μ + ν|_{∂K} + ω)_{sn}-almost everywhere. Therefore r_{k} converges ω weakstar to the characteristic function of cl(U_{n}), so on the interior of K r_{k} converges

8 JOHN E. MCCARTHY

pointwise to this characteristic function. Thus we can conclude that r_{k} actually converges (μ+ν+ω) weak-star to the characteristic function χ_{E_{n}} of a set E_{n}, where χ_{E_{n}} = 1 (ω_{n} + ν|_{U_{n}}) a.e., and χ_{E_{n}} = 0 (μ + ν|_{ℂ\U_{n}} + ω)_{sn} a.e. Multiplying r by χ_{E_{n}} in inequalities (3), they become

∫_{cl(U_{n})} |rϕ|^{2}dμ_{n} ≤ ∫_{cl(U_{n})} |r|^{2}dν_{n}
∫_{cl(U_{n})} |rψ|^{2}dν_{n} ≤ ∫_{cl(U_{n})} |r|^{2}dμ_{n},  (3′)

where μ_{n} = μ_{an} and ν_{n} = ν_{an} + ν|_{U_{n}}.

Let ν|_{∂K} be called σ. As 1 − χ_{(∪E_{n})} is also in the (μ + ν + ω) weak-star closure of the rationals, inequalities (3) also restrict to the singular parts of the measures on the boundary:

∫_{∂K} |rϕ|^{2}dμ_{s} ≤ ∫_{∂K} |r|^{2}dσ
∫_{∂K} |rψ|^{2}dσ ≤ ∫_{∂K} |r|^{2}dμ_{s}.  (3′′)

By the Kolmogoroff-Krein theorem [7, p.135], no point of the interior of K is a bounded point evaluation for R^{2}(K, μ_{s}). But bounded point evaluations are preserved under quasisimilarity [11], so no point in the interior of K is a bounded point evaluation for R^{2}(K, σ) either. But if a point ζ is a positive distance from the support of a measure μ, it is a bounded point evaluation for R^{2}(K, μ) if and only if \frac{1}{z−ζ} is not in R^{2}(K, μ). Therefore R^{2}(K, μ_{s}) = R^{2}(∂K, μ_{s}), and R^{2}(K, σ) = R^{2}(∂K, σ).

But by [1,3.1] every point in ∂K has a unique representing measure for R(K), and hence for R(∂K), so every point in ∂K is a peak point for R(∂K), and thus by Bishop’s peak point criterion [7, p.54], R(∂K) = C(∂K). Therefore R^{2}(K, μ_{s}) = L^{2}(μ_{s}), R^{2}(K, σ) = L^{2}(σ), and inequalities (3′′) imply that the normal operators N_{μ_{s}} (multiplication by the independent variable on L^{2}(μ_{s})) and N_{σ} are quasisimilar. This forces them to be unitarily equivalent [6, p.82], and so μ_{s} and σ are mutually absolutely continuous. Thus (a) holds.

The Szegö theorem for hypo-Dirichlet algebras [1, 10.1] proves that in case (b), no point of U_{n} is a bounded point evaluation for R^{2}(K, μ_{n}), so by the above argument, μ_{n} and ν_{n} are equivalent. Moreover, since points of U_{n} are not bounded point evaluations for R^{2}(K, ν_{n}),

∫_{∂U_{n}} log \frac{dν}{dω_{n}} dω_{n} = −∞,

and so (b) holds.

Part (c) would follow from the Szegö theorem and the preservation under quasisimilarity of bounded point evaluations if ν had no mass on U_{n}. Since it may, however, have mass on the interior, we use Riemann maps to localise it to the boundary of each component of the complement of cl(U_{n}). We must be somewhat careful, though, as these boundaries need not be Jordan curves, so there are technical difficulties in extending the map to the boundary.

Assume U_{n} satisfies (1). Let V be a component of ℂ \ cl(U_{n}). Let W be the complement of cl(V). Then W is simply connected (in the Riemann sphere), contains U_{n}, and ∂W = ∂V ⊆ ∂U_{n}. By inverting, if necessary, we can assume without loss of generality that V is the unbounded component of ℂ \ cl(U_{n}), and hence that cl(W) is compact in ℂ. (This assumption is purely for convenience, allowing us to quote results directly. The same conclusions will hold for bounded V.)

Let g be the Riemann map from W onto the unit disk, and let m be harmonic measure for W. Since cl(W) is simply connected, R(cl(W)) is a Dirichlet algebra, so by [6, p.356] there is a function G in R^{2}(cl(W), m) with |G| = 1 m-almost everywhere, and ∫ Gdm_{b} = g(b) for all b in W (i.e. G is the extension of g to the boundary).

Since G is bounded, by [1,4.1] there exist rational functions ρ_{k} in Rat(cl(W)), with ∥ρ_{k}∥_{∞} ≤ 1, which tend to G m-almost everywhere, and so, by the dominated convergence theorem, ρ_{k}(b) tends to g(b) for all b in W. So if Ĝ is the function that equals G on ∂W and g on W, then, because ω_{n}|_{∂W} is absolutely continuous with respect to m [9], ρ_{k} tends to Ĝ in L^{1}(ω_{n}). Therefore Ĝ is in R^{1}(cl(W), ω_{n})∩L^{∞}(ω_{n}), which, by [1,4.1], is just R^{∞}(cl(W), ω_{n}). Since μ_{n} is equivalent to ω_{n}, Ĝ is in R^{∞}(cl(W), μ_{n}), and hence, by Theorem 1.4, in R^{∞}(cl(W), ν_{n}).

Therefore, inequalities (3′) remain true if r is replaced by r(Ĝ)^{k}. Letting k tend to infinity kills off the interior contribution, because |Ĝ| is less than one on the

10 JOHN E. MCCARTHY

interior, so

∫_{∂V} |rϕ|^{2}dμ_{n} ≤ ∫_{∂V} |r|^{2}dν_{n}

and

∫_{∂V} |rψ|^{2}dν_{n} ≤ ∫_{∂V} |r|^{2}dμ_{n}.  (4)

Summing inequalities (4) over all V in the complement of U_{n}, we finally get that R_{K,μ_{n}} is quasisimilar to R_{K,ν_{n}|_{∂Un}}, so preservation of bounded point evaluations implies that (2) holds.

(ii) Sufficiency

We must show that the normal parts of R_{K,μ} and R_{K,ν} are unitarily equivalent, and that the pure parts are quasisimilar [6, p.223]. Note first that for different components U_{n} and U_{m}, the harmonic measures ω_{n} and ω_{m} are singular. This is because, by [8, 8.3] (applicable because of [1]), the parts of R(K) are connected, and, as remarked above, any point in ∂K has a unique representing measure, so lies in a part of its own. Therefore each U_{n} constitutes a part, and so ω_{n} and ω_{m} are singular.

Now the normal parts correspond to the measures in cases (a) and (b), and by hypothesis these are mutually absolutely continuous, so the normal parts are unitarily equivalent. By applying Forelli’s lemma again, it is therefore sufficient to prove R_{K,ν_{n}} is quasisimilar to R_{K,μ_{n}} in case (c). We shall do this by showing that if ν is a measure on cl(U_{n}), and ν|_{∂U_{n}} is absolutely continuous with respect to ω_{n} and satisfies (2), then R_{K,ν} is quasisimilar to R_{K,ω_{n}}.

By the remarks preceding the theorem, there is an inner function h and a generator g so that \frac{dν}{dω_{n}} = |gh| ω_{n}-almost everywhere. Let A : R^{2}(K, ω_{n}) → hR^{2}(K, ω_{n}) be given by Ar = hr. Because h is inner, its modulus on ∂K is essentially bounded away from zero, and is also bounded above. Therefore A is invertible.

Define B : R^{2}(K, ν) → hR^{2}(K, ω_{n}) by Br = ghr. B has dense range, and because

∫ |ghr|^{2}dω_{n} = ∫_{∂U_{n}} |r|^{2}dν ≤ ∫_{cl(U_{n})} |r|^{2}dν,

B is bounded. Let X = A^{−1}B. Then X is a quasi-affinity satisfying R_{K,ω_{n}}X = XR_{K,ν}.

QUASISIMILARITY OF RATIONALLY CYCLIC SUBNORMAL OPERATORS 11

To get the other intertwining operator, let ν̂ be the sweep of ν, i.e. that measure on ∂U_{n} satisfying

∫_{cl(U_{n})} f̂dν = ∫_{∂U_{n}} fdν̂

(where f̂ is as defined at the beginning of Section 2). Then ν̂ is absolutely continuous with respect to ω_{n} [6, p.335], and

∫_{∂U_{n}} log \frac{dν̂}{dω_{n}} dω_{n} ≥ ∫_{∂U_{n}} log \frac{dν}{dω_{n}} dω_{n} > −∞,

so there exists an inner function h̃ and a generator g̃ satisfying \frac{dν̂}{dω_{n}} = |h̃g̃| ω_{n}- almost everywhere. Define C : R^{2}(K, |g̃|^{2}dω_{n}) → R^{2}(K, ω_{n}) by Cr = g̃r. C is unitary. Define D : R^{2}(K, |g̃|^{2}dω_{n}) → h̃R^{2}(K, |g̃|^{2}dω_{n}) by Dr = h̃r. D is invertible. Define E : R^{2}(K, ν̂) → h̃R^{2}(K, |g̃|^{2}dω_{n}) by Er = h̃r. E is unitary. Define F : R^{2}(K, ν̂) → R^{2}(K, ν) by F r = r. F is a quasi-affinity.

Now, put Y = F E^{−1}DC^{−1}. Y is a quasi-affinity, and Y R_{K,ω_{n}} = R_{K,ν}Y. Therefore R_{K,ν} is quasisimilar to R_{K,ω_{n}}, as required. ////

It is a result of Trent [13] that the set of analytic bounded point evaluations of R^{2}(K, μ) is precisely the spectrum of R_{K,μ} less the essential spectrum. Thus, Theorem 2.1 could be restated:

Suppose R(K) is a hypo-Dirichlet algebra, and μ is supported on ∂K. Then R_{K,ν} is quasisimilar to R_{K,μ} if and only if the two operators have the same spectrum and essential spectrum, and their normal parts are unitarily equivalent.

This is not true in general, however. Here is an example of a family of pure, cyclic subnormal operators, which have the same spectral pictures, but are not quasisimilar.

Example 2.2. For α a positive real number, let μ_{α} be the weighted area measure (1 − r)^{α}rdrdθ on the unit disk. Suppose that for two numbers α < γ, S_{μ_{α}} were quasisimilar to S_{μ_{γ}}, so that there were a quasi-affinity X intertwining them. If X1 were ϕ, then we would have

∫ |p|^{2}|ϕ|^{2}(1 − r)^{α}rdrdθ ≤ C ∫ |p|^{2}(1 − r)^{γ}rdrdθ.  (5)

Suppose ϕ(z) = ∑ a_{k}z^{k}. Then letting p(z) = z^{n} in (5), we would get, in terms of the beta function,

∑_{k=0}^{∞} |a_{k}|^{2}β(2n + 2k + 2, α + 1) ≤ Cβ(2n + 2, γ + 1).

But this is impossible, since the first non-zero element on the left-hand side behaves asymptotically like n^{−(α+1)}, and the right-hand side is O(n^{−(γ+1)}).

QUASISIMILARITY OF RATIONALLY CYCLIC SUBNORMAL OPERATORS 13

Acknowledgements

This work was done while the author was a graduate student at the University of California at Berkeley. The author would like to thank his advisor, Professor Donald Sarason, for his patience and support.

References

1 P. Ahern and D. Sarason “The H^{p} spaces of a class of function algebras,” Acta Mathematica 117 [1967] 123-163 2 P. Ahern and D. Sarason “On some hypo-Dirichlet algebras of analytic functions,” Amer. Jour. Math. 89 [1967] 932-941 3 W.S. Clary “Quasi-similarity and subnormal operators,” Ph.D. Thesis, University of Michigan, 1973 4 W.S. Clary “Equality of spectra of quasi-similar hyponormal operators,” Proc. Amer. Mat. Soc. 53 [1975] 88-90 5 J.B. Conway “On quasisimilarity for subnormal operators,” Illinois Jour. Math. 24 [1980] 689-702 6 J.B. Conway “Subnormal Operators,” Pitman, Boston, 1981 7 T.W. Gamelin “Uniform Algebras,” Prentice-Hall, Englewood Cliffs, 1969 8 T.W. Gamelin “Embedding Riemann surfaces in maximal ideal spaces,” Jour. Funct. Anal. 2 [1968] 123-146 9 J.B. Garnett “Applications of Harmonic Measure,” University of Arkansas Lecture Notes in the Mathematical Sciences, John Wiley, New York, 1986 10 W.W. Hastings “Subnormal operators quasisimilar to an isometry,” Trans. Amer. Math. Soc. 256 [1979] 145-161 11 M. Raphael “Quasisimilarity and essential spectra for subnormal operators,” Indiana Univ. Math. Jour 31 [1982] 243-246 12 M. Raphael “Commutants of quasisimilar subnormal operators,” Pac. Jour. Math. 122 [1986] 449-454 13 T.T. Trent “H^{2}(μ) spaces and bounded point evaluations,” Pacific Jour. Math. 80 [1979] 279-292 14 Yang Liming “Equality of essential spectra of quasisimilar subnormal operators,” preprint

15 T. Yoshino “Subnormal operators with a cyclic vector,” Tohoku Math. 21 [1969] 47-55

Department of Mathematics Indiana University Bloomington IN 47405