Lens Lectures on Aleksandrov-Clark Measures

LENS LECTURES ON ALEKSANDROV-CLARK MEASURES

WILLIAM T. ROSS

1. Introduction In this series of three 90 minute lectures I will give a gentle introduction to the topic of Aleksandrov-Clark measures which turn out to have an uncanny way of appearing in various areas of analysis. In short, for an analytic map ϕ : D → D, where D is the open unit disk, there is a family {µα : |α| = 1} of positive ﬁnite measures on the unit circle T associated with ϕ by the formula 1 − |ϕ(z)|2 Z 1 − |z|2 2 = 2 dµα(ξ), z ∈ D. |α − ϕ(z)| T |ξ − z| These measures, called Aleksandrov-Clark measures, appear as the spectral representing measures for a certain important unitary operator (the rank-one unitary perturbation of the compressed shift) (Clark’s theorem). They disintegrate Lebeague measure on the circle (Aleksandrov’s theorem). The help us compute adjoints and essential norms of composition operators on the Hardy space. The list goes on and on. This course is intended for graduate students with just the basics of real analysis (measure theory, Lebesgue theory), functional analysis (Riesz representation theorem, Hahn-Banach theorem, spectral theorem), complex analysis (Cauchy theory), and of course, the basics of linear algebra. I will build everything from the ground up or provide references for the technical details. Lecture 1. Self maps and measure theory. Lecture 2. Aleksandrov-Clark measures. Lecture 3. Clark theory, Aleksandrov operators, composition operators

2. A motivational example Let H be a complex separable Hilbert space with inner product h·, ·i and A be a bounded cyclic self-adjoint operator on H with cyclic vector ϕ. Here cyclic means that _ n {A ϕ : n ∈ N0} = H. The symbol W will denote the closed linear span in H. The easiest example to think of here is 2 2 Mx : L [0, 1] → L [0, 1],Mxf = xf, ϕ ≡ 1. 1 2 WILLIAM T. ROSS

For the self-adjoint operator A, cyclic vector ϕ, and λ ∈ R let

Aλ := A + λ(ϕ ⊗ ϕ), where ϕ ⊗ ϕ is the rank-one operator on H deﬁned by (ϕ ⊗ ϕ)(v) = hv, ϕiϕ, v ∈ H.

Clearly Aλ is self-adjoint (since λ is real and the rank-one tensor ϕ ⊗ ϕ is self adjoint). Moreover, Aλ is also cyclic with cyclic vector ϕ. Indeed 2 Aλϕ = Aϕ + λkϕk ϕ and so

Aλϕ ∈ span{ϕ, Aϕ}. Continuing in this fashion we see that

_ n _ n {Aλϕ : n ∈ N0} = {A ϕ : n ∈ N0} = H. By the spectral theorem for bounded cyclic self-adjoint operators we know that ∼ 2 Aλ = (Mx,L (µλ)) for some positive ﬁnite Borel measure on R. An interesting result here is the following disintegration theorem. Theorem 2.1. For the situation above Z dµλ dλ = dx in the sense that for all f ∈ Cc(R), Z Z Z f(t)dµλ(t) dλ = f(t)dt. R R R 2 Example 2.2. If A = Mx on L [0, 1] and ϕ ≡ 1, then

Aλ = Mx + λ(1 ⊗ 1). More speciﬁcally, Z 1 2 Aλf = xf + λ f(t)dt, f ∈ L [0, 1]. 0

We won’t give the details here but the spectral measures µλ turns out to be

− 1 χ[0,1] e λ dµλ = dx + δ − 1 . 1−x 2 2 2 2 − 1 (1−e λ )−1 (1 + λ log( x )) + λ π λ (1 − e λ )

Notice that µλ has both absolutely continuous and singular parts (with respect to Lebesgue measures on R). LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 3

2 2 Example 2.3. Let A : C → C be represented by the matrix 1 0 . 0 0 The linear transformation A is certainly cyclic with cyclic (column) vector v = √1 (1, 1)T . The eigenvalues of A = A + λ(v ⊗ v) turn out to be 2 λ 1 p 1 p λ = (1 + λ − 1 + λ2), λ = (1 + λ + 1 + λ2). 1 2 2 2 The spectral measures for Aλ turn out to be √ √ −λ + 1 + λ2 λ + 1 + λ2 µλ = √ δλ + √ δλ . 2 1 + λ2 1 2 1 + λ2 2 So why did we work this example? We wanted to review the spectral theorem as well as showing a student that the material we are about to present in the context of analytic functions on the disk, appear in a much broader context.

3. Self-maps of the disk These notes will focus on a class of measures associated with analytic maps ϕ : D → D. Such maps are often called analytic self-maps of the disk. In the study of such measures is the boundary behavior of the associated map ϕ. On old theorem of Fatou [4] says the following: Theorem 3.1 (Fatou’s theorem). For an analytic self-map ϕ, the radial limit lim ϕ(rζ) r→1− exists for m-almost every ζ ∈ T. We often write ϕ(ζ) for this limit whenever it exists (and is ﬁnite!).

Proposition 3.2. The set {ζ ∈ T : ϕ(ζ) exists} is a Borel set. Actually it turns out that the above limit also exists for m-almost every ζ but the radial limit above is replaced by a non-tangential limit. By this we mean that ϕ(z) → ϕ(ζ) not only as z → ζ along the radius connecting 0 and ζ but as z → ζ inside any Stolz domain |z − ζ| z ∈ : < α , α ∈ (1, ∞). D 1 − |z| This is a triangular shaped region with vertex at ζ. We often write lim ϕ(z) ∠ z→ζ for the non-tangential limit. Deﬁnition 3.3. A self-map ϕ is said to be inner if |ϕ(ζ)| = 1 for m-almost every ζ ∈ T. 4 WILLIAM T. ROSS

Example 3.4. For a ∈ D consider the function z − a ϕ(z) = , z ∈ . 1 − az D Note that eiθ − a e−iθ − a ϕ(eiθ)ϕ(eiθ) = = 1, θ ∈ [0, 2π], 1 − aeiθ 1 − ae−iθ and so, by the maximum modulus theorem, ϕ is an analytic self-map. The above calculation shows that ϕ has unimodular boundary values and so ϕ is inner. Example 3.5. Consider

z+1 ϕ(z) = e z−1 , z ∈ D. Note that 2 z + 1 z + 1 1 − |z| exp = exp < = exp − < 1, z ∈ D. z − 1 z − 1 |1 − z|2 Thus ϕ is an analytic self-map. From the previous identity one can easily check that |ϕ(eiθ)| = 1 if θ ∈ (0, 2π). Thus ϕ is an inner function. If α1, ··· , αn are positive numbers and θ1, ··· , θn ∈ [0, 2π], one can also check that z + eiθ1 z + eiθn ϕ(z) = exp α1 + ··· + αn z − eiθ1 z − eiθn is also an inner function. Due to the limited amount of time, I will not get into all of the details of inner functions here. They are carefully worked out in standard texts [4, 5]. For a sequence of points (an)n≥1 in D \{0} which satisfy the condition ∞ X (1 − |an|) < ∞, n=1 one can show that the inﬁnite product ∞ Y an z − an B(z) := , |a | 1 − a z n=1 n n called a Blaschke product, converges uniformly on compact subsets of D. Work of Blaschke shows the following:

Theorem 3.6 (Blaschke). For a sequence of points (an)n≥1 ⊂ D \{0} sat- P∞ isfying n=1(1 − |an|) < ∞, the Blaschke product B is an inner function. There is another basic type of inner function associated with a singular measure. See the next section for a reminder of the deﬁnition of a singular measure. LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 5

Theorem 3.7. For a positive ﬁnite positive singular Borel measure µ on T the function Z ξ + z s (z) := exp − dµ(ξ) , z ∈ , µ ξ − z D is inner. Proof. As was done previously one can show that Z ξ + z Z 1 − |z|2 |s (z)| = exp < − dµ(ξ) = exp − dµ(ξ) < 1. µ ξ − z |z − ξ|2 2 Thus sµ ∈ H . Now use Proposition 4.14 (below) to show that sµ has unimodular boundary values m-almost everywhere.

Deﬁnition 3.8. The function sµ from the previous theorem is called a singular inner function. Notice how the singular inner function from Example 3.5 is formed from µ = δ1, the point mass at one. Clearly the functions iγ n e z B(z)sµ(z), where γ ∈ R, n ∈ N0, B is a Blaschke product, and sµ is a singular inner function. It turns out that these are all of them. Theorem 3.9 (Nevanlinna). Any inner function ϕ takes the form iγ n ϕ(z) = e z B(z)sµ(z). n The factors z , B, sµ are unique. The deﬁnition of an inner function says that ϕ has unimodular boundary values m-almost everywhere. There is on old theorem of Frostman (and expended by Ahern and Clark) which says when an inner function has unimodular boundary values at a particular point.

Theorem 3.10 (Frostman-Ahern-Clark). An inner function ϕ = Bsµ and all of its inner divisors have limits which are unimodular at ζ ∈ T if and only if X 1 − |λ|2 Z dµ(ξ) + < ∞. |ζ − λ| |ξ − ζ| λ∈B−1({0}) The next concept here is the notion of an angular derivative.

Deﬁnition 3.11. For an analytic self-map ϕ and ζ ∈ T we say that ϕ has an angular derivative at ζ if lim ϕ(z) ∈ T; ∠ z→ζ lim ϕ0(z) exists. ∠ z→ζ 6 WILLIAM T. ROSS

Note that if ϕ has an analytic continuation to a neighborhood of ζ and |ϕ(ζ)| = 1, then the angular derivative exists. For inner functions, there is this following deﬁnitive criterion for the ex- istence of angular derivatives due to Ahern and Clark.

Theorem 3.12 (Frostman-Ahern-Clark). An inner function ϕ = Bsµ has a ﬁnite angular derivative at ζ ∈ T if and only if X 1 − |λ|2 Z dµ(ξ) + < ∞. |ζ − λ|2 |ξ − ζ|2 λ∈B−1({0}) Remark 3.13. Much of the results of this section are discussed in detail in [1, 4, 5].

4. Some measure theory and harmonic analysis reminders Let M denote the space of complex Borel measures on T. We will use dθ m = 2π to denote normalized Lebesgue measure on T. Here is a reminder of some deﬁnitions [8]. Deﬁnition 4.1. (1) For µ ∈ M we say that µ is absolutely continuous with respect to m, written µ m, if µ(A) = 0 whenever A is a Borel subset of T with m(A) =0. (2) We say that µ is singular with respect to m, written µ ⊥ m, if there are disjoint Borel sets A and B with A∪B = T and µ(A) = m(B) = 0. (3) A measure µ is positive, written µ ∈ M+, if µ(E) ≥ 0 for every Borel subset E of T. Theorem 4.2 (Jordan decomposition theorem). Every µ ∈ M can be written uniquely as

µ = (µ1 − µ2) + i(µ3 − µ4), µj ∈ M+. Theorem 4.3 (Radon-Nikodym theorem). A measure µ ∈ M is absolutely continuous if and only if dµ = fdm for some f ∈ L1(m), i.e., Z µ(E) = fdm E for all Borel subset E of T. Theorem 4.4 (Lebesuge decomposition theorem). Every µ ∈ M can be written uniquely as µ = µa + µs, where µa m and µs ⊥ m. Theorem 4.5 (Riesz Representation Theorem). If C is the Banach space of continuous functions on T normed with the usual supremum norm, then the dual C∗ of C is isometrically isomorphic to M via the dual pairing Z hg, µi = fdµ. LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 7

Corollary 4.6. If Z n µb(n) := ξ dµ(ξ), n ∈ Z, the n-th Fourier coeﬃcient of µ ∈ M, is equal to zero for all n ∈ Z, then µ ≡ 0.

Proof. Suppose that µb(n) = 0 for all n ∈ Z, then Z fdµ = 0 for every trigonometric polynomial f. Since such functions are dense in C, we see that the above holds for all f ∈ C. By the Riesz Representation Theorem (Theorem 4.5) µ ≡ 0. Deﬁnition 4.7. 1 − |z|2 P (ζ) := , z ∈ , ζ ∈ , z |ζ − z|2 D T is the standard Poisson kernel. Observe that 1 + ζz Pz(ζ) = < 1 − ζz and so Pz(ζ) is a positive harmonic function of z on D (being the real part of an analytic function). Furthermore, if z = rξ, where ξ ∈ T and r ∈ (0, 1), then writing 1 + ζz ζrξ = 1 + 2 , 1 − ζz 1 − ζrξ using geometric series to expand the last term as a series, and then taking real parts, we see that ∞ X |n| n n (4.8) Prξ(ζ) = r ζ ξ . n=−∞ From here we can us the fact that Z n ζ dm(ζ) = δn,0 T to see that Z Pz(ζ)dm(ζ) = 1, z ∈ D. T Deﬁnition 4.9. For µ ∈ M, let Z (P µ)(z) := Pz(ζ)dµ(ζ) be the Poisson integral of µ. 8 WILLIAM T. ROSS

Using (4.8) we see that ∞ X |n| n (4.10) (P µ)(rξ) = µb(n)r ξ . n=−∞ This following fact will be used several times in these notes.

Proposition 4.11. The set {Pz : z ∈ D} has dense linear span in C.

Proof. Suppose µ ⊥ Pz for all z ∈ D. Then

0 = hPz, µi = (P µ)(z), z ∈ D. But by formula (4.10) we have ∞ X |n| n 0 = µb(n)r ξ , rξ ∈ D. n=−∞

Since |µb(n)| ≤ kµk one can see that for ﬁxed r ∈ (0, 1) the above series converges uniformly in ξ. This means that for each k ∈ Z Z ∞ ∞ Z k X |n| n X |n| n−k |k| 0 = ξ µb(n)r ξ dm(ξ) = r µb(n) ξ dm = r µb(k). T n=−∞ n=−∞ T

Since µb(k) = 0 for all k ∈ Z, one can apply Corollary 4.6 to see that µ ≡ 0. An application of the Hahn-Banach separation theorem completes the proof.

For µ ∈ M+, the Poisson integral P µ is a positive harmonic function on D (diﬀerentiating under the integral). A classical theorem of Herglotz says these are the only ones. Theorem 4.12 (Herglotz). Suppose u is a positive harmonic function on D. Then u = P µ for some unique µ ∈ M+.

Proof. Let ur(ζ) := u(rζ) and note that the family of measures

{urdm : 0 < r < 1} satisfy Z kurdmk = ur(ζ)dm(ζ) = u(0) T and are thus uniformly bounded in total variation norm k · k on M. Notice how we used the positivity of ur as well as the mean-value theorem for harmonic functions. By the Banach-Alaoglu theorem, urn dm → dµ weak-∗ for some sequence rn → 1 and some µ ∈ M+. That is to say Z Z

gurn dm → gdµ, g ∈ C. T T LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 9

Apply this to g = Pz to get Z (P µ)(z) = lim Pz(ζ)ur(ζ)dm(ζ) r →1 n T = lim u(rnz) rn→1 = u(z).

So u = P µ for some measure µ ∈ M+. For uniqueness, suppose u = P µ1 = P µ2. Then P (µ1 − µ2) ≡ 0. By Proposition 4.11, µ1 = µ2. Let’s talk about derivatives of measures.

Deﬁnition 4.13. For µ ∈ M+, deﬁne, for each ζ ∈ T, µ(I(t, ζ)) (Dµ)(ζ) := lim inf ; t→0+ m(I(t, ζ) µ(I(t, ζ)) (Dµ)(ζ) := lim sup , t→0+ m(I(t, ζ) where, for ζ ∈ T and t > 0, I(t, ζ) is the arc of T subtended by the points e−itζ and eitζ. The above derivatives are called the lower and upper sym- metric derivatives of µ. Here are some technical (standard) facts we will not prove since they are carefully worked out in [8].

Proposition 4.14. For µ ∈ M+. we have (1) Dµ = Dµ m-almost everywhere. (2) For every ζ ∈ T, (Dµ)(ζ) ≤ lim inf(P µ)(rζ) ≤ lim sup(P µ)(rζ) ≤ (Dµ)(ζ). r→1 r→1 (3) If Dµ is the m-almost everywhere deﬁned function Dµ = Dµ = Dµ, then if (Dµ)(ζ) exists, then (Dµ)(ζ) = limr→1(P µ)(rζ). (4) If µ = µa + µs, where µa m and µs ⊥ m, is the Lebesgue decomposition then Dµs = 0 and Dµ = Dµa m-almost everywhere. (5) µs is carried by {Dµ = ∞}. (6) µs is carried by {0 < Dµ < ∞}. We used the word carrier in the above theorem. Let us by quite precise about this.

Remark 4.15. For µ ∈ M, consider the union U of all open subsets U ⊂ T for which µ(U) = 0. The set T \U is a closed set called the support of µ. A Borel set H ⊂ T for which µ(H ∩ A) = µ(A) for all Borel sets A ⊂ T is called a carrier for µ. Certainly the support is a carrier but a carrier need not be the support. In fact, it need not even be closed. For example, if f is −1 continuous on T and dµ = fdm, then a carrier of µ is T \ f ({0}) (which is open) while the support of µ is the closure of this set. 10 WILLIAM T. ROSS

5. The basics of Aleksandrov-Clark measures

For an analytic self-map ϕ and |α| = 1 consider the function uα on D deﬁned by 1 − |ϕ(z)|2 u (z) := = P (α). α |α − ϕ(z)|2 ϕ(z)

Since ϕ is analytic and z 7→ Pz(α) is harmonic, then z 7→ Pϕ(z)(α) is harmonic. Clearly uα is also positive on D and so by the Herglotz theorem (Theorem 4.12)

uα(z) = (P µα)(z) for some unique µα ∈ M+. Denote the family

Aϕ := {µα : |α| = 1} as the Aleksandrov-Clark measures (often written as AC measures) associated with ϕ. Why two names? When ϕ is an inner function then this family of measures, along with an associated family of unitary operators (more about this later in the notes), was ﬁrst studied by Clark. General self-maps ϕ were later studied by Aleksandrov. Example 5.1. Let us compute the family of AC measures for ϕ(z) = zn. Here, for each α ∈ T we want to ﬁnd a µα ∈ M+ so that 1 − |ϕ(z)|2 1 − |z|2n Z 1 − |z|2 = = dµ (ξ), z ∈ . |α − ϕ(z)|2 |α − z|2 |ξ − z|2 α D To give a student an idea of the thinking that goes on here (and to fore- shadow later results), let us try to derive the measure rather that just having me give it to you for you to verify. Let z = rζ, where 0 < r < 1 and |ζ| = 1. Then the above identity becomes 1 − r2n Z 1 − r2 (5.2) = dµ (ξ). |α − rnζn|2 |ξ − rζ|2 α When ζn 6= α, then the LHS of the previous identity goes to zero as r → 1. Thus the RHS must do the same. By Proposition 4.14 this says that n µα is placing no mass on T \{ζ : ζ = α}. Let {ζj : 1 ≤ j ≤ n} denote the n n solutions to ζ = α. Note that these are equally spaced points on T. The above analysis says that n X µα = cjδζj . j=1

Thus we are looking to ﬁnd positive constants cj so that n 1 − r2n X 1 − r2 = c , r ∈ (0, 1), ζ ∈ . |α − rnζn|2 j |ζ − rζ|2 T j=1 j LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 11

Fix k so that 1 ≤ k ≤ n and let ζ = ζk in the above formula. This gives us n (since ζk = α) n 1 − r2n X 1 − r2 = c . (1 − rn)2 j |ζ − rζ |2 j=1 j k Write the above expression (after a little simplification) as 1 + rn 1 + r X 1 − r2 n = ck + cj 2 . 1 − r 1 − r |ζj − rζk| j6=k Now multiply the above expression through by 1 − rn and do a little algebra to get 2 n n 2 n−1 X (1 − r )(1 − r ) 1 + r = ck(1 + r)(1 + r + r + ··· + r ) + cj 2 . |ζj − rζk| j6=k Now let r → 1 (the finite sum goes to zero) to get 1 c = . k n We have just computed our first AC measure n 1 X µ = δ , α n ζj j=1 where {ζ, ··· , ζn} are the solutions to ϕ(ζ) = α. Example 5.3. Suppose z + 1 ϕ(z) = exp . z − 1 One can check that 2 z + 1 z + 1 1 − |z| exp = exp < = exp − < 1, z ∈ D, z − 1 z − 1 |1 − z|2 and so ϕ is an analytic self-map of D. Again, for each α ∈ T we want to find a µα ∈ M+ so that 1 − |ϕ(z)|2 Z 1 − |z|2 = dµ (ξ), z ∈ . |α − ϕ(z)|2 |ξ − z|2 α D As in the previous example, we let z = rζ, |ζ| = 1, r ∈ (0, 1) and, as in the previous example, we argue that the LHS approaches zero as r → 1 except for solutions to the equation ϕ(ζ) = α. Solutions to this equations are a discrete set of points i arg α + 2πin + 1 ζ = , n ∈ , n i arg α + 2πin − 1 Z which accumulate at 1. As in the previous example this says that X µα = cnδζn . n∈Z 12 WILLIAM T. ROSS

Proposition 5.5. If µ ∈ M+, then there exists an analytic self-map ϕ of D so that µ ∈ Aϕ. LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 13

Proof. Deﬁne Z ζ + z H (z) := dµ(ζ), z ∈ , µ ζ − z D and note that Z 1 − |z|2 <(H (z)) = dµ(ζ) = (P µ)(z) > 0. µ |ζ − z|2

Thus Hµ maps D onto the right half plane {z : 0}. The map w − 1 w 7→ w + 1 maps

Proposition 5.6. If µα ∈ Aϕ, then 1 − |ϕ(0)|2 µ ( ) = . α T |α − ϕ(0)|2

Proof. Since P0 is the constant function one, we have Z Z 1 − |ϕ(0)|2 µ ( ) = 1dµ = P (ζ)dµ (ζ) = (P µ )(0) = . α T α 0 α α |α − ϕ(0)|2 Corollary 5.7. For ﬁxed ϕ we have

(1) Each µα is a probability measure if and only if ϕ(0) = 0. (2) sup{µα(T): α ∈ T} < ∞. 6. The Aleksandrov disintegration theorem Before getting bogged down in all of the technical details about AC measures, let’s discuss a fascinating result which is the analog of Theorem 2.1 discussed earlier. Theorem 6.1 (Aleksndrov’s disintegration theorem). For a self map ϕ with associated AC measures Aϕ and f ∈ C, Z Z Z f(ζ)dµα(ζ) dm(α) = f(ξ)dm(ξ). 14 WILLIAM T. ROSS

Proof. For a ﬁxed z ∈ D notice that Z Z Z 1 − |ϕ(z)|2 P (ζ)dµ (ζ) dm(α) = dm(α) z α |α − ϕ(z)|2 Z = Pϕ(z)(α)dm(α) = 1 Z = Pz(ζ)dm(ζ).

Thus the disintegration theorem works for finite linear combinations of Pois- son kernels. To get the result for any f ∈ C, let (fn)n≥1 be a sequence of finite linear combinations of Poisson kernels which converge uniformly to f (Proposition 4.11). Let us first note that for each n, the function Z α 7→ fndµα is continuous on T. To see this write N X fn = cjPzj j=1 and, by the definition of an AC measure, observe that

N Z X fndµα = cjPϕ(zj )(α) j=1 which is continuous in the variable α. We can extend this to show that Z α 7→ fdµα is also continuous. Indeed, by Corollary 5.7 we know that

B = sup{µα(T): α ∈ T} < ∞. Then Z Z

fdµα − fndµα ≤ kf − fnk∞µα(T) ≤ kf − fnk∞B which means, via uniform convergence of continuous functions, that Z α 7→ fdµα is continuous LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 15

Finally, Z Z fdm = lim fndm (uniform convergence) n→∞ T T Z Z = lim fn(ζ)dµα(ζ) dm(α) (disintegration formula) n→∞ T T Z Z = f(ζ)dµα(ζ) dmα (dominated convergence theorem). T T This proves the result. Remark 6.2. Aleksandrov showed quite a bit more here in that the continuous functions C can be replaced by L1 in the disintegration theorem. There are quite a few technical issues here. For example, the inner integrals Z f(ζ)dµα(ζ) in the disintegration formula do not seem to be well defined for L1 functions since indeed AC measures can be point masses and L1 functions are defined m-almost everywhere. However, amazingly, the function Z α 7→ f(ζ)dµα(ζ) is defined for m-almost every α and is integrable. An argument with the monotone class theorem is used to prove this more general result. See [1] for the details. A careful reading of the proof of the disintegration theorem will yield the following:

Corollary 6.3. For a self-map ϕ, the function α 7→ µα is continuous from T to the (M, ∗), the space of measures endowed with the weak-∗ topology.

7. Carriers of AC measures

Recall from Proposition 4.14 that for µ = µa + µs ∈ M+, where µa m and µs ⊥ m, a carrier for µa is {0 < Dµ < ∞} and a carrier of µs is {Dµ = ∞}. Suppose µα ∈ Aϕ let us write the Lebesgue decomposition of µα as dµα = hαdm + dσα, 1 where hα ∈ L (m).

Proposition 7.1. The set Eα = {ζ ∈ T : ϕ(ζ) = α} is a Borel set and is a carrier for σα. Proof. Let 1 − |ϕ(z)|2 u (z) := α |α − ϕ(z)|2 16 WILLIAM T. ROSS and note that by Proposition 4.14

{ζ ∈ T :(Dµα)(ζ) = ∞} ⊂ {ζ ∈ T : uα(ζ) = ∞} ⊂ Eα, where uα(ζ) denotes the radial limit. Again by Proposition 4.14 {Dµα = ∞} is a carrier for the singular part of µα, i.e., σα, we see that Eα is a carrier for σα. The proof that Eα is a Borel sets is a bit more technical and can be found in [1]. Corollary 7.2. For an analytic self map ϕ the following are true:

(1) σα ⊥ σβ when α 6= β. (2) ϕ(ζ) = α for σα-almost every ζ ∈ T. (3) µα = σα for all α if and only if ϕ is inner −1 (4) For a Borel subset B ⊂ T, let ϕ (B) be the set of ζ ∈ T such that ϕ(ζ) exists and ϕ(ζ) ∈ B. Then −1 [ ϕ (B) = Eα. α∈B 8. Measure preserving A nice application of the disintegration theorem is this theorem about measure preserving inner functions. Recall for an inner function ϕ and a −1 Borel set A ⊂ T, ϕ (A) is the set of ζ ∈ T such that ϕ(ζ) exists and ϕ(ζ) ∈ B. Theorem 8.1. Suppose ϕ is inner with ϕ(0) = 0. Then for any Borel set −1 A ⊂ T, m(A) = m(ϕ (A)).

Proof. Apply the disintegration theorem to the function f = χϕ−1(A), noting also that µα = σα, since ϕ is inner, to get Z Z Z χϕ−1(A)(ζ)dσα(ζ) dm(α) = χϕ−1(A)(ζ)dm(ζ). T T T The above identity becomes Z −1 −1 σα(ϕ (A))dm(α) = m(ϕ (A)). T Apply Corollary 7.2 to the LHS to get   Z [ −1 σα  Eβ dm(α) = m(ϕ (A)). T β∈A

Apply Corollary 7.2 (Eα is a carrier for σα) to get Z −1 σα(Eα)dm(α) = m(ϕ (A)). A Since we are assuming that ϕ(0) = 0 we know that σα is a probability measure carried by Eα and so σα(Eα) = 1 for all α. Thus the above becomes m(A) = m(ϕ−1(A)) LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 17 and our proof is complete. 9. Point masses and AC measures

We know from Proposition 7.1 that for µα ∈ Aϕ, the measure σα is carried by Eα. When is ζ ∈ Eα a point mass for µα?

Theorem 9.1. An AC measure µα ∈ Aϕ has a point mass at ζ ∈ T if and only if ϕ(ζ) = α and ϕ has a finite angular derivative at ζ. Proof. This proof will be a bit skimpy on the details. By the definition of an AC measure we have 1 − |ϕ(rζ)|2 Z 1 − r2 = dµ (ξ). |α − ϕ(rζ)|2 |ξ − rζ|2 α Multiply both sides of the previous identity by (1 − r)2 1 − r2 to get 2 2 Z 2 1 − r 1 − |ϕ(rζ)| (1 − r) = dµα(ξ). α − ϕ(rζ) 1 − r2 |ξ − rζ|2 Now take limits as r → 1− to get 1 · |ϕ0(ζ)| = µ ({ζ}). |ϕ0(ζ)|2 α Corollary 9.2. For an analytic self-map ϕ and α ∈ T the set 0 Pα := {ζ ∈ T : ϕ(ζ) = α, |ϕ (ζ)| < ∞} is discrete and the pure point part of µα ∈ Aϕ is equal to X 1 δ . |ϕ0(ζ)| ζ ζ∈Pα 10. Clark measures as spectral measures It turns out that the AC measures are the spectral measures for a certain unitary operator, the Clark unitary operator. In fact, this is the real reason they were discovered in the first place. Let us present this in a purely linear algebra way so as to point to the general result without getting tied up in all of the technical definitions. For n ∈ N, let Qn be polynomials of degree at most n−1. We will imagine 2 Qn ⊂ L . As such Qn, inherits an inner product Z Z 2π dθ hp, qi = pqdm = p(eiθ)q(eiθ) . T 0 2π With this inner product note that Z 2π m n imθ −inθ dθ hz , z i = e e = δm,n 0 2π 18 WILLIAM T. ROSS

n−1 and so {1, z, ··· , z } is an orthonormal basis for Qn. Deﬁne the operator 2 Pn : L → Qn; Z 2 2 n−1 n−1 (Pnf)(z) = f(ζ)(1 + ζz + ζ z + ··· ζ z )dm(ζ). T A computation will show that for 0 ≤ k ≤ n − 1 Z k k 2 2 n−1 n−1 k Pnz = ζ (1 + ζz + ζ z + ··· ζ z )dm(ζ) = z T and so PnQn = Qn. With a little be more work, one can check that Pn is 2 actually the orthogonal projection of L onto Qn. For ϕ ∈ L∞, let

Aϕ : Qn → Qn,Aφf = Pn(ϕf). One can check that Z k k 2 2 n−1 n−1 (Aϕz )(z) = ϕ(ζ)ζ (1 + ζz + ζ z + ··· ζ z )dm(ζ) T n−1 X Z = ϕ(ζ)ζk−jdm(ζ) zj j=1 T n−1 X j = ϕb(j − k)z . j=1

This means that the matrix representation of Aϕ with respect to the orthonormal bass {1, z, ··· , zn−1} is a Toeplitz matrix. In fact, every Toeplitz matrix can be thought of in this way. Deﬁne, for |α| = 1, n−1 Uα := Az + α(1 ⊗ z ). This operator is called the Clark unitary operator. One can also see that if |a| = 1, Uα is unitary. Indeed a simple 3 × 3 example is  0 0 α  Ua =  1 0 0  . 0 1 0 The general n ∈ N case follows a similar pattern (1s down the sub-diagonal, α in the right corner, zeros elsewhere). Furthermore if ζn = α one can check that if 2 2 n−1 n−1 kζ (z) = 1 + ζz + ζ z + ··· ζ z , then certainly kζ ∈ Qn and a matrix calculation will show that

Uαkζ = ζkζ and so we have computed the spectral information for Uα, i.e., its eigenvalues and eigenvectors. LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 19

By the spectral theorem from functional analysis, one knows that any cyclic unitary operator is unitarily equivalent to the operator f 7→ ζf (mul- tiplication by the independent variable) on L2(σ) for some finite positive Borel measure σ on T. Computing this measure can be somewhat difficult. For the cyclic (easily checked) unitary operator Uα, one can compute its spectral representation in the following very interesting way: Let σα be the measure on the circle defined by n 1 X dσ = δ , α n ζj j=1 n where {ζ1, ··· , ζn} are the n roots of z = α. Do you recognize σα as the AC measure for ϕ(z) = zn? Define 2 2 Zα : L (σα) → L (σα), (Zαf)(ζ) = ζf(ζ) and note that

Zχζj = ζjχζj , j = 1, ··· , n.

That is to say, χζj are eigenvectors for Zα corresponding to the eigenvalues ζj. Furthermore, note that √ √ { nχζ1 , ··· , nχζj } 2 is an orthonormal basis for L (σα). If we deﬁne √ √ n V nχ = k α ζj n ζj and extend by linearity we see that Vα is an bijective linear transformation 2 √ from L (σα) onto Qn. But since kkζ kL2(m) = n (Parsevel’s theorem) we see that Vα is actually an isometry. Finally, since χζj are the eigenvectors for Zα and kζj are the eigenvectors for Uα we see that

UαVα = VαZα. Thus we have a concrete realization of the spectral measure and spectral representation for Uα. It turns out that this greatly generalizes to model spaces (ΘH2)⊥, where Θ is inner function and H2 is the Hardy space [4, 5]. We will just state the results. The interested reader can find all of the details worked out in [1]. But first we need a crash course in Hardy spaces. The classics texts for this material, together with complete proofs of the main results, are [4, 5, 6]. Consider the set of all power series ∞ X n anz n=0 whose coefficients satisfy ∞ X 2 |an| < ∞. n=0 20 WILLIAM T. ROSS

For any ﬁxed z ∈ D the Cauchy-Schwarz inequality shows that

∞ ∞ !1/2 ∞ !1/2 X n X 2 X 2n |an||z| ≤ |an| |z| n=0 n=0 n=0 ∞ !1/2 X 2 1 = |an| . p 2 n=0 1 − |z| This says that such power series converge uniformly on compact subsets of D and thus form analytic functions on D. This allows us to make the following definition: Definition 10.1. The Hardy space H2 is the set of all analytic functions on D whose power series have square summable coefficients. If we norm H2 by v ∞ u ∞ X n uX 2 anz := t |an| , n=0 H2 n=0 then one can show that H2 is complete vector space – in fact a Hilbert space. There is also this useful alternate definition of H2 involving the integral means Z |f(rζ)|2dm(ζ), 0 < r < 1. T 2 Proposition 10.2. An analytic function f on D belongs to H if and only if Z sup |f(rζ)|2dm(ζ) < ∞. 0

2 Theorem 10.5 (Cauchy integral formula). For f ∈ H and z ∈ D, Z f(ζ) f(z) = dm(ζ). T 1 − zζ LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 21

Note how the above integral makes sense since f(ζ) is well deﬁned for 2 m-almost every ζ ∈ T and ζ 7→ f(ζ) is an L function. The kernel functions 1 kλ(z) := , λ, z ∈ D, 1 − λz are called reproducing kernel functions for H2 since 2 f(λ) = hf, kλi, f ∈ H , that is to say, these functions reproduce the values of f at λ. After that brief interlude on the basics of Hardy spaces, it back to our discussion of Clark theory. Theorem 10.6. For an inner function Θ with Θ(0) = 0, the operator Θ U := S + α 1 ⊗ , α Θ z 2 ⊥ 2 where SΘ = PΘS|(ΘH ) is the compression of the unilateral shift S : H → H2, Sf = zf, is a cyclic unitary operator on (ΘH2)⊥. The eigenvalues of 0 Uα are the ζ ∈ T so that Θ(ζ) = α and |Θ (ζ)| < ∞. The corresponding eigenvector is 1 − Θ(ζ)Θ(z) kΘ := . ζ 1 − ζz

This next theorem says that the spectral measure of Uα (which exists by the spectral theorem for cyclic unitary operators) is indeed the AC measure σα.

Theorem 10.7. For an inner function Θ with Θ(0) = 0, let σα ∈ AΘ. Then the operator Z f(ζ) (Vαf)(z) = (1 − αΘ(z)) dσα(ζ) 1 − ζz 2 2 ⊥ is an isometric operator from L (σα) onto (ΘH ) . Furthermore if 2 2 Zα : L (σα) → L (σα), (Zαf)(ζ) = ζf(ζ), then VαZα = UαVα. 11. A connection to the Paley-Wiener approximation probem

The Paley-Wiener approximation problem: Suppose {xn : n ∈ N} is an orthonormal basis for a Hilbert space H and {yn : n ∈ N} is a sequence in H. If these sequences are ‘close’ to each other (not quite deﬁned yet), does {yn : n ∈ N} span H? As an example of what we mean here, consider the standard orthonormal basis iθ inθ ϕn(e ) = e , n ∈ Z, for L2. When does the sequence

iθ iλnθ ψn(e ) = e , n ∈ Z, 22 WILLIAM T. ROSS

2 where λn ∈ R, span L ? A classical theorem of Paley-Wiener says this is indeed the case when 1 max |λn − n| < 2 . n∈Z π So let Λ ⊂ D be a sequence in D. When does _ Θ 2 ⊥ {kλ : λ ∈ Λ} = (ΘH ) ? An easy exercise using the uniqueness theorem for analytic functions will show that if Λ has accumulation points in either the open unit disk or T \ σb(Θ), then the set of kernel functions does indeed span. Hint: Use annihilators and the Hahn-Banach separation theorem. For other cases, things get a bit more complicated. What we want here is a orthonormal basis for (ΘH2)⊥ consisting of (normalized) kernel functions! Ah ha! But when Uα has discrete spectrum then indeed Uα has an orthoormal basis of eigenvectors ( kΘ ) ζ : |Θ0(ζ)| < ∞, Θ(ζ) = α p|Θ0(ζ)| which can be used formulate Paley-Wiener type approximation theorems for model spaces (ΘH2)⊥.

12. A connection to composition operators For a self-map ϕ of D and an analytic function f on D deﬁne the composition operator

(Cϕf)(z) := f(ϕ(z)). By an application of the Littlewood subordination principle [4], it is known 2 that Cϕ is a bounded operator form H to itself. Two great places to learn about the basics of composition operators are [3, 10]. It also turns out that there are connections to AC measures. The ﬁrst one to mention involves the Aleksandrov operator. For a self map ϕ deﬁne, for a continuous function f on T, Z (Aϕf)(α) = f(ζ)dµα(ζ), α ∈ T. T One can show that since f is continuous on T then Aϕf is also continuous on T. The above operator is called the Aleksandrov operator.

Theorem 12.1 (Aleksandrov (1987)). If ϕ(0) = 0, the operator Aϕ extends to be a bounded operator on H2. Assuming that ϕ(0) = 0 one can work the identity Z ζ + λ α + ϕ(λ) dµα(ζ) = , λ ∈ D, α ∈ T, T ζ − λ α − ϕ(λ) LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 23 to get the formula Z 1 1 dµα(ζ) = . T 1 − λζ 1 − ϕ(λ)α Indeed, Z 1 1 Z ζ + λ dµα(ζ) = + 1 dµα(ζ) 1 − ζλ 2 ζ − λ 1 α + ϕ(λ) = + 1 2 α − ϕ(λ) α = α − ϕ(λ) 1 = . 1 − αϕ(λ) Now take complex conjugates to get the desired identity. If we write 1 kλ(z) = 1 − λz for the reproducing kernel for H2, i.e., 2 hf, kλi = f(λ), f ∈ H , λ ∈ D, we can write the previous identity as 1 (Aϕkλ)(α) = . 1 − ϕ(λ)α We can relate this to the composition operator. Indeed ∗ ∗ (Cϕkλ)(z) = hCϕkλ, kzi = hkλ,Cϕkzi

= hkλ, kz(ϕ(·))i

= hkz(ϕ(·)), kλi

= kz(ϕ(λ)) 1 = . 1 − ϕ(λ)z Thus, ∗ (Aϕkλ)(α) = (Cϕkλ)(α), λ ∈ D, α ∈ T. We are certainly taking a few liberties in the above formula since α ∈ T and there is the issue of boundary limits at a particular point. However, all of this can be made precise by using radial limits m-almost everywhere and in fact we have.

∗ Theorem 12.2. If ϕ(0) = 0 then Cϕ = Aϕ. 24 WILLIAM T. ROSS

There are many other connections between composition operators and AC measures. Let us mention one more. The essential norm of Cϕ, denoted 2 by kCϕke, is the distance from Cϕ to the compact operators K on H , i.e.,

kCϕke := inf{kCϕ − Kk : K ∈ K}.

The essential norm of Cϕ can be computed in terms of something called the Nevanlinna counting function [3, 10] but Cima and Matheson [2] computed it in terms of σα, where µα = hαdm + σα is the Lebesgue decomposition of µα ∈ Aϕ. Theorem 12.3 (Cima-Matheson). For a self map ϕ, r kCϕke = sup σα(T). α∈T I am certainly not doing this subject justice here and there are many other connections between composition operators and AC measures. A more complete account is found in [9].

13. Higher dimensional analogs

Let Θ be a contractive Mn×n(C)-valued analytic function on D. By this we mean that kΘ(z)k, the matrix norm of Θ(z), is bounded by one for all z ∈ D. In the scalar case we know that if Θ is non-constant, then |Θ(z)| < 1 for all z ∈ D. This is no longer the case in the matrix case. What is true is the following:

Proposition 13.1. Every contractive Mn×n(C)-valued function Θ on D can be written in block form as Θ = Θ0 ⊕ Θ1, where Θ0 is a constant unitary matrix and Θ1 is purely contractive, i.e., kΘ1(z)k < 1 for all z ∈ D.

For a purely contractive Θ and a constant unitary matrix A ∈ Mn×n deﬁne the function ∗ ∗ −1 BA(z) := (I + Θ(z)A )(I − Θ(z)A ) , z ∈ D, and note that this function is a purely contractive analytic function on D. Furthermore, assuming that Θ(0) = 0 (the zero matrix!), we have 1

{µA : A ∈ U(n)}, LENS LECTURES ON ALEKSANDROV-CLARK MEASURES 25 where U(n) is the unitary group. As before, these measures are associated with a higher dimensional analog of the Clark unitary operators discussed earlier (see [7] for a reference). We will focus on the following analog of the disintegration theorem: Let mn denote Mn×n-valued measure mI (n copies of normalized Lebesgue measure m along the diagonal) on T. Also let dHn denote Haar measure on U(n). n Theorem 13.2 (Martin (2012)). For any continuous f : T → C , Z Z Z µA(dζ)f(ζ) dH(U) = mn(dζ)f(ζ). U(n) T T We will end here with a remark that much of the theory mentioned earlier for the scalar case (non-tangential limits, Ahern-Clark results, carriers, etc.) have analogs in this matrix-valued case. Details and references are in [7].

References 1. J. A. Cima, A. L. Matheson, and W. T. Ross, The Cauchy transform, Mathematical Surveys and Monographs, vol. 125, American Mathematical Society, Providence, RI, 2006. 2. Joseph A. Cima and Alec L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math. 179 (1997), no. 1, 59–64. 3. Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. 4. P. L. Duren, Theory of Hp spaces, Academic Press, New York, 1970. 5. J. Garnett, Bounded analytic functions, first ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. 6. K. Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Anal- ysis, Prentice-Hall Inc., Englewood Cliffs, N. J., 1962. 7. R.T.W. Martin, Unitary perturbations of compressed n-dimensional shifts, Compl. Anal. Oper. Theory. In press: http://arxiv.org/abs/1107.3439 (2012). 8. Walter Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987. 9. Eero Saksman, An elementary introduction to Clark measures, Topics in complex analysis and operator theory, Univ. Málaga,Málaga,2007, pp. 85–136. 10. Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993.

Department of Mathematics and Computer Science, University of Rich- mond, Richmond, Virginia, 23173, USA E-mail address: [email protected] URL: http://facultystaff.richmond.edu/~wross