LENS LECTURES ON ALEKSANDROV-CLARK MEASURES

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1 LENS LECURES ON ALEKSANDROV-CLARK MEASURES WILLIAM. ROSS. 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 {µ α : α = } of positive finite measures on the unit circle associated with ϕ by the formula ϕ(z) 2 α ϕ(z) 2 = z 2 ξ z 2 dµ α(ξ), z D. hese 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). hey disintegrate Lebeague measure on the circle (Aleksandrov s theorem). he help us compute adjoints and essential norms of composition operators on the Hardy space. he list goes on and on. his 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. 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, and A be a bounded cyclic self-adjoint operator on H with cyclic vector ϕ. Here cyclic means that {A n ϕ : n N 0 } = H. he symbol will denote the closed linear span in H. he easiest example to think of here is M x : L 2 [0, ] L 2 [0, ], M x f = xf, ϕ.

2 2 WILLIAM. ROSS For the self-adjoint operator A, cyclic vector ϕ, and λ R let A λ := A + λ(ϕ ϕ), where ϕ ϕ is the rank-one operator on H defined by (ϕ ϕ)(v) = v, ϕ ϕ, v H. Clearly A λ is self-adjoint (since λ is real and the rank-one tensor ϕ ϕ is self adjoint). Moreover, A λ is also cyclic with cyclic vector ϕ. Indeed and so A λ ϕ = Aϕ + λ ϕ 2 ϕ A λ ϕ span{ϕ, Aϕ}. Continuing in this fashion we see that {A n λ ϕ : n N 0 } = {A n ϕ : n N 0 } = H. By the spectral theorem for bounded cyclic self-adjoint operators we know that A λ = (Mx, L 2 (µ λ )) for some positive finite Borel measure on R. An interesting result here is the following disintegration theorem. heorem 2.. For the situation above ( dµ λ ) dλ = dx in the sense that for all f C c (R), ( ) f(t)dµ λ (t) dλ = R R R f(t)dt. Example 2.2. If A = M x on L 2 [0, ] and ϕ, then More specifically, A λ f = xf + λ A λ = M x + λ( ). 0 f(t)dt, f L 2 [0, ]. We won t give the details here but the spectral measures µ λ turns out to be dµ λ = χ [0,] e λ ( + λ log( x dx + x ))2 + λ 2 π2 λ 2 ( e λ ) δ ( e λ ). Notice that µ λ has both absolutely continuous and singular parts (with respect to Lebesgue measures on R).

3 LENS LECURES ON ALEKSANDROV-CLARK MEASURES 3 Example 2.3. Let A : C 2 C 2 be represented by the matrix ( ) he linear transformation A is certainly cyclic with cyclic (column) vector v = 2 (, ). he eigenvalues of A λ = A + λ(v v) turn out to be λ = 2 ( + λ + λ 2 ), λ 2 = 2 ( + λ + + λ 2 ). he spectral measures for A λ turn out to be µ λ = λ + + λ 2 2 δ + λ 2 λ + λ + + λ 2 2 δ + λ 2 λ2. 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 hese 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: heorem 3. (Fatou s theorem). For an analytic self-map ϕ, the radial limit lim r ϕ(rζ) exists for m-almost every ζ. We often write ϕ(ζ) for this limit whenever it exists (and is finite!). Proposition 3.2. he set {ζ : 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 D : z < α, α (, ). his is a triangular shaped region with vertex at ζ. We often write for the non-tangential limit. lim z ζ ϕ(z) Definition 3.3. A self-map ϕ is said to be inner if ϕ(ζ) = for m-almost every ζ.

4 4 WILLIAM. ROSS Example 3.4. For a D consider the function Note that ϕ(z) = z a az, z D. ϕ(e iθ )ϕ(e iθ ) = eiθ a e iθ a ae iθ =, θ [0, 2π], ae iθ and so, by the maximum modulus theorem, ϕ is an analytic self-map. he above calculation shows that ϕ has unimodular boundary values and so ϕ is inner. Example 3.5. Consider Note that ( z + exp z ) = exp ( R ϕ(z) = e z+ z, z D. ( )) z + = exp z ( ) z 2 z 2 <, z D. hus ϕ is an analytic self-map. From the previous identity one can easily check that ϕ(e iθ ) = if θ (0, 2π). hus ϕ is an inner function. If α,, α n are positive numbers and θ,, θ n [0, 2π], one can also check that ( z + e iθ ) ϕ(z) = exp α z e iθ + + α z + e iθn n z e iθn 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. hey are carefully worked out in standard texts [4, 5]. For a sequence of points (a n ) n in D \ {0} which satisfy the condition ( a n ) <, n= one can show that the infinite product a n z a n B(z) := a n a n z, n= called a Blaschke product, converges uniformly on compact subsets of D. Work of Blaschke shows the following: heorem 3.6 (Blaschke). For a sequence of points (a n ) n D \ {0} satisfying n= ( a n ) <, the Blaschke product B is an inner function. here 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.

5 LENS LECURES ON ALEKSANDROV-CLARK MEASURES 5 heorem 3.7. For a positive finite positive singular Borel measure µ on the function ( ) ξ + z s µ (z) := exp ξ z dµ(ξ), z D, is inner. Proof. As was done previously one can show that ( ( )) ( ) ξ + z z 2 s µ (z) = exp R ξ z dµ(ξ) = exp z ξ 2 dµ(ξ) <. hus s µ H 2. Now use Proposition 4.4 (below) to show that s µ has unimodular boundary values m-almost everywhere. Definition 3.8. he function s µ from the previous theorem is called a singular inner function. Notice how the singular inner function from Example 3.5 is formed from µ = δ, the point mass at one. Clearly the functions e iγ z n B(z)s µ (z), where γ R, n N 0, B is a Blaschke product, and s µ is a singular inner function. It turns out that these are all of them. heorem 3.9 (Nevanlinna). Any inner function ϕ takes the form he factors z n, B, s µ are unique. ϕ(z) = e iγ z n B(z)s µ (z). he definition of an inner function says that ϕ has unimodular boundary values m-almost everywhere. here 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. heorem 3.0 (Frostman-Ahern-Clark). An inner function ϕ = Bs µ and all of its inner divisors have limits which are unimodular at ζ if and only if λ B ({0}) λ 2 ζ λ + dµ(ξ) ξ ζ <. he next concept here is the notion of an angular derivative. Definition 3.. For an analytic self-map ϕ and ζ we say that ϕ has an angular derivative at ζ if lim z ζ ϕ(z) ; lim z ζ ϕ (z) exists.

6 6 WILLIAM. ROSS Note that if ϕ has an analytic continuation to a neighborhood of ζ and ϕ(ζ) =, then the angular derivative exists. For inner functions, there is this following definitive criterion for the existence of angular derivatives due to Ahern and Clark. heorem 3.2 (Frostman-Ahern-Clark). An inner function ϕ = Bs µ has a finite angular derivative at ζ if and only if λ 2 ζ λ 2 + dµ(ξ) ξ ζ 2 <. λ B ({0}) Remark 3.3. Much of the results of this section are discussed in detail in [, 4, 5]. 4. Some measure theory and harmonic analysis reminders Let M denote the space of complex Borel measures on. We will use m = dθ 2π to denote normalized Lebesgue measure on. Here is a reminder of some definitions [8]. Definition 4.. () 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 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 = 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. heorem 4.2 (Jordan decomposition theorem). Every µ M can be written uniquely as µ = (µ µ 2 ) + i(µ 3 µ 4 ), µ j M +. heorem 4.3 (Radon-Nikodym theorem). A measure µ M is absolutely continuous if and only if dµ = fdm for some f L (m), i.e., µ(e) = fdm for all Borel subset E of. heorem 4.4 (Lebesuge decomposition theorem). Every µ M can be written uniquely as µ = µ a + µ s, where µ a m and µ s m. heorem 4.5 (Riesz Representation heorem). If C is the Banach space of continuous functions on normed with the usual supremum norm, then the dual C of C is isometrically isomorphic to M via the dual pairing g, µ = fdµ. E

7 LENS LECURES ON ALEKSANDROV-CLARK MEASURES 7 Corollary 4.6. If µ(n) := ξ n 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 µ(n) = 0 for all n Z, then 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 heorem (heorem 4.5) µ 0. Definition 4.7. is the standard Poisson kernel. Observe that P z (ζ) := z 2, z D, ζ, ζ z 2 ( ) + ζz P z (ζ) = R ζz and so P z (ζ) is a positive harmonic function of z on D (being the real part of an analytic function). Furthermore, if z = rξ, where ξ and r (0, ), then writing + ζz ζz = + 2 ζrξ ζrξ, using geometric series to expand the last term as a series, and then taking real parts, we see that (4.8) P rξ (ζ) = r n ζ n ξ n. n= From here we can us the fact that ζ n dm(ζ) = δ n,0 to see that Definition 4.9. For µ M, let (P µ)(z) := be the Poisson integral of µ. P z (ζ)dm(ζ) =, z D. P z (ζ)dµ(ζ)

8 8 WILLIAM. ROSS Using (4.8) we see that (4.0) (P µ)(rξ) = n= µ(n)r n ξ n. his following fact will be used several times in these notes. Proposition 4.. he set {P z : z D} has dense linear span in C. Proof. Suppose µ P z for all z D. hen But by formula (4.0) we have 0 = P z, µ = (P µ)(z), z D. 0 = n= µ(n)r n ξ n, rξ D. Since µ(n) µ one can see that for fixed r (0, ) the above series converges uniformly in ξ. his means that for each k Z 0 = ξ k µ(n)r n ξ n dm(ξ) = r n µ(n) ξ n k dm = r k µ(k). n= n= Since µ(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. heorem 4.2 (Herglotz). Suppose u is a positive harmonic function on D. hen u = P µ for some unique µ M +. Proof. Let u r (ζ) := u(rζ) and note that the family of measures satisfy u r dm = {u r dm : 0 < r < } u r (ζ)dm(ζ) = u(0) and are thus uniformly bounded in total variation norm on M. Notice how we used the positivity of u r as well as the mean-value theorem for harmonic functions. By the Banach-Alaoglu theorem, u rn dm dµ weak- for some sequence r n and some µ M +. hat is to say gu rn dm gdµ, g C.

9 LENS LECURES ON ALEKSANDROV-CLARK MEASURES 9 Apply this to g = P z to get (P µ)(z) = lim r n = lim r n u(r nz) = u(z). P z (ζ)u r (ζ)dm(ζ) So u = P µ for some measure µ M +. For uniqueness, suppose u = P µ = P µ 2. hen P (µ µ 2 ) 0. By Proposition 4., µ = µ 2. Let s talk about derivatives of measures. Definition 4.3. For µ M +, define, for each ζ, (Dµ)(ζ) := lim inf t 0 + (Dµ)(ζ) := lim sup t 0 + µ(i(t, ζ)) m(i(t, ζ) ; µ(i(t, ζ)) m(i(t, ζ), where, for ζ and t > 0, I(t, ζ) is the arc of subtended by the points e it ζ and e it ζ. he above derivatives are called the lower and upper symmetric derivatives of µ. Here are some technical (standard) facts we will not prove since they are carefully worked out in [8]. Proposition 4.4. For µ M +. we have () Dµ = Dµ m-almost everywhere. (2) For every ζ, (Dµ)(ζ) lim inf r (P µ)(rζ) lim sup(p µ)(rζ) (Dµ)(ζ). (3) If Dµ is the m-almost everywhere defined function Dµ = Dµ = Dµ, then if (Dµ)(ζ) exists, then (Dµ)(ζ) = lim r (P µ)(rζ). (4) If µ = µ a + µ s, where µ a m and µ s m, is the Lebesgue decomposition then Dµ s = 0 and Dµ = Dµ a m-almost everywhere. (5) µ s is carried by {Dµ = }. (6) µ s is carried by {0 < Dµ < }. We used the word carrier in the above theorem. Let us by quite precise about this. Remark 4.5. For µ M, consider the union U of all open subsets U for which µ(u) = 0. he set \ U is a closed set called the support of µ. A Borel set H for which µ(h A) = µ(a) for all Borel sets A 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 continuous on and dµ = fdm, then a carrier of µ is \ f ({0}) (which is open) while the support of µ is the closure of this set. r

10 0 WILLIAM. ROSS 5. he basics of Aleksandrov-Clark measures For an analytic self-map ϕ and α = consider the function u α on D defined by u α (z) := ϕ(z) 2 α ϕ(z) 2 = P ϕ(z)(α). Since ϕ is analytic and z P z (α) is harmonic, then z P ϕ(z) (α) is harmonic. Clearly u α is also positive on D and so by the Herglotz theorem (heorem 4.2) u α (z) = (P µ α )(z) for some unique µ α M +. Denote the family A ϕ := {µ α : α = } 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.. Let us compute the family of AC measures for ϕ(z) = z n. Here, for each α we want to find a µ α M + so that ϕ(z) 2 α ϕ(z) 2 = z 2n z 2 α z 2 = ξ z 2 dµ α(ξ), z D. o give a student an idea of the thinking that goes on here (and to foreshadow 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 < and ζ =. hen the above identity becomes (5.2) r 2n α r n ζ n 2 = r 2 ξ rζ 2 dµ α(ξ). When ζ n α, then the LHS of the previous identity goes to zero as r. hus the RHS must do the same. By Proposition 4.4 this says that µ α is placing no mass on \ {ζ : ζ n = α}. Let {ζ j : j n} denote the n solutions to ζ n = α. Note that these are equally spaced points on. he above analysis says that n µ α = c j δ ζj. j= hus we are looking to find positive constants c j so that r 2n n α r n ζ n 2 = r 2 c j, r (0, ), ζ. ζ j rζ 2 j=

11 LENS LECURES ON ALEKSANDROV-CLARK MEASURES Fix k so that k n and let ζ = ζ k in the above formula. his gives us (since ζk n = α) r 2n n ( r n ) 2 = r 2 c j ζ j rζ k 2. j= Write the above expression (after a little simplification) as + r n r n = c + r k r + j k c j r 2 ζ j rζ k 2. Now multiply the above expression through by r n and do a little algebra to get + r n = c k ( + r)( + r + r r n ) + j k Now let r (the finite sum goes to zero) to get c k = n. We have just computed our first AC measure µ α = n δ ζj, n j= where {ζ,, ζ n } are the solutions to ϕ(ζ) = α. Example 5.3. Suppose One can check that ( z + exp z ) = exp ( R ϕ(z) = exp ( ) z +. z ( )) z + = exp z c j ( r 2 )( r n ) ζ j rζ k 2. ( ) z 2 z 2 <, z D, and so ϕ is an analytic self-map of D. Again, for each α we want to find a µ α M + so that ϕ(z) 2 z 2 α ϕ(z) 2 = ξ z 2 dµ α(ξ), z D. As in the previous example, we let z = rζ, ζ =, r (0, ) and, as in the previous example, we argue that the LHS approaches zero as r except for solutions to the equation ϕ(ζ) = α. Solutions to this equations are a discrete set of points i arg α + 2πin + ζ n = i arg α + 2πin, n Z, which accumulate at. As in the previous example this says that µ α = n Z c n δ ζn.

12 2 WILLIAM. ROSS Now we need to compute positive constants c n which satisfy ϕ(rζ) 2 α ϕ(rζ) 2 = r 2 c n, ζ, r (0, ). ζ n rζ 2 n Z Let ζ = ζ k and observe that the previous equation becomes Multiply both sides by to get ϕ(rζ k ) 2 α ϕ(rζ k ) 2 = c r 2 k ( r) 2 + r 2 ζ n rζ k 2. n k α ϕ(rζ k ) 2 r 2 ϕ(rζ k ) 2 r 2 = c k α ϕ(rζ k ) r 2 + n k α ϕ(rζ k ) 2 ζ n rζ k 2. Now take limits as r and argue (since ϕ is analytic in a neighborhood of each ζ k ) that ϕ (ζ k ) = c k ϕ (ζ k ) and hence c k = ϕ (ζ k ). In summary, dµ α = n Z ϕ (ζ n ) δ ζ n. Example 5.4. Suppose that ϕ is an analytic self-map of D which maps D to a compact subset of D (something like ϕ(z) = z/2). hen we have ϕ(rζ) 2 α ϕ(rζ) 2 = P rζ (ξ)dµ α (ξ). aking limits as r we use our earlier discussion of Poisson integrals (Proposition 4.4) to see that dµ α ϕ(ζ) 2 (ζ) = dm α ϕ(ζ) 2 for m-almost every ζ. Since the limits never vanish, we see that there is no singular part to µ α and so indeed dµ α = ϕ(ζ) 2 α ϕ(ζ) 2 dm. At this point, the reader might think that AC measures have to be very special and form an exclusive club. hey don t. Proposition 5.5. If µ M +, then there exists an analytic self-map ϕ of D so that µ A ϕ.

13 LENS LECURES ON ALEKSANDROV-CLARK MEASURES 3 Proof. Define and note that H µ (z) := ζ + z dµ(ζ), z D, ζ z z 2 R(H µ (z)) = dµ(ζ) = (P µ)(z) > 0. ζ z 2 hus H µ maps D onto the right half plane {z : Rz > 0}. he map w w w + maps Rz onto D and so is an analytic self-map of D. A little algebra yields and so ϕ(z) := H µ(z) H µ (z) +. H µ (z) = + ϕ(z) ϕ(z) (P µ)(z) = R(H µ (z)) = ϕ(z) 2 ϕ(z) 2 which means µ = µ A ϕ. One last basic fact about AC measures is the total mass. Proposition 5.6. If µ α A ϕ, then µ α () = ϕ(0) 2 α ϕ(0) 2. Proof. Since P 0 is the constant function one, we have µ α () = dµ α = P 0 (ζ)dµ α (ζ) = (P µ α )(0) = ϕ(0) 2 α ϕ(0) 2. Corollary 5.7. For fixed ϕ we have () Each µ α is a probability measure if and only if ϕ(0) = 0. (2) sup{µ α () : α } <. 6. he 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 heorem 2. discussed earlier. heorem 6. (Aleksndrov s disintegration theorem). For a self map ϕ with associated AC measures A ϕ and f C, ( ) f(ζ)dµ α (ζ) dm(α) = f(ξ)dm(ξ).

14 4 WILLIAM. ROSS Proof. For a fixed z D notice that ( ) ( ) ϕ(z) 2 P z (ζ)dµ α (ζ) dm(α) = α ϕ(z) 2 dm(α) = P ϕ(z) (α)dm(α) = = P z (ζ)dm(ζ). hus the disintegration theorem works for finite linear combinations of Poisson kernels. o get the result for any f C, let (f n ) n be a sequence of finite linear combinations of Poisson kernels which converge uniformly to f (Proposition 4.). Let us first note that for each n, the function α f n dµ α is continuous on. o see this write f n = N c j P zj j= and, by the definition of an AC measure, observe that f n dµ α = N c j P ϕ(zj )(α) which is continuous in the variable α. We can extend this to show that α fdµ α j= is also continuous. Indeed, by Corollary 5.7 we know that B = sup{µ α () : α } <. hen fdµ α f n dµ α f f n µ α () f f n B which means, via uniform convergence of continuous functions, that α fdµ α is continuous

15 LENS LECURES ON ALEKSANDROV-CLARK MEASURES 5 Finally, fdm = lim f n dm (uniform convergence) n ( ) = lim f n (ζ)dµ α (ζ) dm(α) (disintegration formula) n ( ) = f(ζ)dµ α (ζ) dm α (dominated convergence theorem). his proves the result. Remark 6.2. Aleksandrov showed quite a bit more here in that the continuous functions C can be replaced by L in the disintegration theorem. here are quite a few technical issues here. For example, the inner integrals f(ζ)dµ α (ζ) in the disintegration formula do not seem to be well defined for L functions since indeed AC measures can be point masses and L functions are defined m-almost everywhere. However, amazingly, the function α f(ζ)dµ α (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 [] 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 µ α is continuous from to the (M, ), the space of measures endowed with the weak- topology. 7. Carriers of AC measures Recall from Proposition 4.4 that for µ = µ a + µ s M +, where µ a m and µ s m, a carrier for µ a is {0 < Dµ < } and a carrier of µ s is {Dµ = }. Suppose µ α A ϕ let us write the Lebesgue decomposition of µ α as dµ α = h α dm + dσ α, where h α L (m). Proposition 7.. he set E α = {ζ : ϕ(ζ) = α} is a Borel set and is a carrier for σ α. Proof. Let u α (z) := ϕ(z) 2 α ϕ(z) 2

16 6 WILLIAM. ROSS and note that by Proposition 4.4 {ζ : (Dµ α )(ζ) = } {ζ : u α (ζ) = } E α, where u α (ζ) denotes the radial limit. Again by Proposition 4.4 {Dµ α = } is a carrier for the singular part of µ α, i.e., σ α, we see that E α is a carrier for σ α. he proof that E α is a Borel sets is a bit more technical and can be found in []. Corollary 7.2. For an analytic self map ϕ the following are true: () σ α σ β when α β. (2) ϕ(ζ) = α for σ α -almost every ζ. (3) µ α = σ α for all α if and only if ϕ is inner (4) For a Borel subset B, let ϕ (B) be the set of ζ such that ϕ(ζ) exists and ϕ(ζ) B. hen ϕ (B) = E α. 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 Borel set A, ϕ (A) is the set of ζ such that ϕ(ζ) exists and ϕ(ζ) B. heorem 8.. Suppose ϕ is inner with ϕ(0) = 0. hen for any Borel set A, m(a) = m(ϕ (A)). Proof. Apply the disintegration theorem to the function f = χ ϕ (A), noting also that µ α = σ α, since ϕ is inner, to get ( ) χ ϕ (A)(ζ)dσ α (ζ) dm(α) = χ ϕ (A)(ζ)dm(ζ). he above identity becomes σ α (ϕ (A))dm(α) = m(ϕ (A)). Apply Corollary 7.2 to the LHS to get dm(α) = m(ϕ (A)). σ α β A Apply Corollary 7.2 (E α is a carrier for σ α ) to get σ α (E α )dm(α) = m(ϕ (A)). A E β Since we are assuming that ϕ(0) = 0 we know that σ α is a probability measure carried by E α and so σ α (E α ) = for all α. hus the above becomes m(a) = m(ϕ (A))

17 LENS LECURES ON ALEKSANDROV-CLARK MEASURES 7 and our proof is complete. 9. Point masses and AC measures We know from Proposition 7. that for µ α A ϕ, the measure σ α is carried by E α. When is ζ E α a point mass for µ α? heorem 9.. An AC measure µ α A ϕ has a point mass at ζ if and only if ϕ(ζ) = α and ϕ has a finite angular derivative at ζ. Proof. his proof will be a bit skimpy on the details. By the definition of an AC measure we have ϕ(rζ) 2 α ϕ(rζ) 2 = r 2 ξ rζ 2 dµ α(ξ). Multiply both sides of the previous identity by to get ( r) 2 r 2 r 2 ϕ(rζ) 2 ( r) 2 α ϕ(rζ) r 2 = ξ rζ 2 dµ α(ξ). Now take limits as r to get ϕ (ζ) 2 ϕ (ζ) = µ α ({ζ}). Corollary 9.2. For an analytic self-map ϕ and α the set P α := {ζ : ϕ(ζ) = α, ϕ (ζ) < } is discrete and the pure point part of µ α A ϕ is equal to ϕ (ζ) δ ζ. ζ P α 0. 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 Q n be polynomials of degree at most n. We will imagine Q n L 2. As such Q n, inherits an inner product p, q = pqdm = With this inner product note that z m, z n = 2π 0 2π 0 p(e iθ )q(e iθ ) dθ 2π. e imθ e inθ dθ 2π = δ m,n

18 8 WILLIAM. ROSS and so {, z,, z n } is an orthonormal basis for Q n. Define the operator P n : L 2 Q n ; (P n f)(z) = f(ζ)( + ζz + ζ 2 z 2 + ζ n z n )dm(ζ). A computation will show that for 0 k n P n z k = ζ k ( + ζz + ζ 2 z 2 + ζ n z n )dm(ζ) = z k and so P n Q n = Q n. With a little be more work, one can check that P n is actually the orthogonal projection of L 2 onto Q n. For ϕ L, let One can check that (A ϕ z k )(z) = A ϕ : Q n Q n, n ( = = j= n j= A φ f = P n (ϕf). ϕ(ζ)ζ k ( + ζz + ζ 2 z 2 + ζ n z n )dm(ζ) ϕ(ζ)ζ k j dm(ζ) ϕ(j k)z j. his means that the matrix representation of A ϕ with respect to the orthonormal bass {, z,, z n } is a oeplitz matrix. In fact, every oeplitz matrix can be thought of in this way. Define, for α =, ) z j U α := A z + α( z n ). his operator is called the Clark unitary operator. One can also see that if a =, U α is unitary. Indeed a simple 3 3 example is U a = 0 0 α he general n N case follows a similar pattern (s down the sub-diagonal, α in the right corner, zeros elsewhere). Furthermore if ζ n = α one can check that if k ζ (z) = + ζz + ζ 2 z 2 + ζ n z n, then certainly k ζ Q n and a matrix calculation will show that U α k ζ = ζk ζ and so we have computed the spectral information for U α, i.e., its eigenvalues and eigenvectors.

19 LENS LECURES ON ALEKSANDROV-CLARK MEASURES 9 By the spectral theorem from functional analysis, one knows that any cyclic unitary operator is unitarily equivalent to the operator f ζf (multiplication by the independent variable) on L 2 (σ) for some finite positive Borel measure σ on. Computing this measure can be somewhat difficult. For the cyclic (easily checked) unitary operator U α, one can compute its spectral representation in the following very interesting way: Let σ α be the measure on the circle defined by dσ α = n δ ζj, n where {ζ,, ζ n } are the n roots of z n = α. Do you recognize σ α as the AC measure for ϕ(z) = z n? Define Z α : L 2 (σ α ) L 2 (σ α ), (Z α f)(ζ) = ζf(ζ) and note that j= Zχ ζj = ζ j χ ζj, j =,, n. hat is to say, χ ζj are eigenvectors for Z α corresponding to the eigenvalues ζ j. Furthermore, note that { nχ ζ,, nχ ζj } is an orthonormal basis for L 2 (σ α ). If we define n V α nχζj = n k ζ j and extend by linearity we see that V α is an bijective linear transformation from L 2 (σ α ) onto Q n. But since k ζ L 2 (m) = n (Parsevel s theorem) we see that V α is actually an isometry. Finally, since χ ζj are the eigenvectors for Z α and k ζj are the eigenvectors for U α we see that U α V α = V α Z α. hus we have a concrete realization of the spectral measure and spectral representation for U α. It turns out that this greatly generalizes to model spaces (ΘH 2 ), where Θ is inner function and H 2 is the Hardy space [4, 5]. We will just state the results. he interested reader can find all of the details worked out in []. But first we need a crash course in Hardy spaces. he classics texts for this material, together with complete proofs of the main results, are [4, 5, 6]. Consider the set of all power series a n z n whose coefficients satisfy n=0 a n 2 <. n=0

20 20 WILLIAM. ROSS For any fixed z D the Cauchy-Schwarz inequality shows that ( ) /2 ( ) /2 a n z n a n 2 z 2n n=0 = n=0 ( n=0 n=0 a n 2 ) /2 z 2. his says that such power series converge uniformly on compact subsets of D and thus form analytic functions on D. his allows us to make the following definition: Definition 0.. he Hardy space H 2 is the set of all analytic functions on D whose power series have square summable coefficients. If we norm H 2 by a n z n H n=0 2 := a n 2, n=0 then one can show that H 2 is complete vector space in fact a Hilbert space. here is also this useful alternate definition of H 2 involving the integral means f(rζ) 2 dm(ζ), 0 < r <. Proposition 0.2. An analytic function f on D belongs to H 2 if and only if sup f(rζ) 2 dm(ζ) <. 0<r< heorem 0.3. For f H 2 we have the following: () lim f(rζ) := f(ζ) exists for m-almost every ζ. r (2) sup f(rζ) 2 dm(ζ) = f(ζ) 2 dm(ζ) = a n 2. 0<r< Remark 0.4. Notice how the norm of an H 2 function is equal to the L 2 norm of its boundary function. It is worth pointing out a Hardy space version of the classical Cauchy integral formula. heorem 0.5 (Cauchy integral formula). For f H 2 and z D, f(ζ) f(z) = zζ dm(ζ). n=0

21 LENS LECURES ON ALEKSANDROV-CLARK MEASURES 2 Note how the above integral makes sense since f(ζ) is well defined for m-almost every ζ and ζ f(ζ) is an L 2 function. he kernel functions k λ (z) :=, λ, z D, λz are called reproducing kernel functions for H 2 since f(λ) = f, k λ, f H 2, 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. heorem 0.6. For an inner function Θ with Θ(0) = 0, the operator ( U α := S Θ + α Θ ), z where S Θ = P Θ S (ΘH 2 ) is the compression of the unilateral shift S : H 2 H 2, Sf = zf, is a cyclic unitary operator on (ΘH 2 ). he eigenvalues of U α are the ζ so that Θ(ζ) = α and Θ (ζ) <. he corresponding eigenvector is kζ Θ Θ(ζ)Θ(z) :=. ζz his next theorem says that the spectral measure of U α (which exists by the spectral theorem for cyclic unitary operators) is indeed the AC measure σ α. heorem 0.7. For an inner function Θ with Θ(0) = 0, let σ α A Θ. hen the operator f(ζ) (V α f)(z) = ( αθ(z)) ζz dσ α(ζ) is an isometric operator from L 2 (σ α ) onto (ΘH 2 ). Furthermore if then V α Z α = U α V α. Z α : L 2 (σ α ) L 2 (σ α ), (Z α f)(ζ) = ζf(ζ),. A connection to the Paley-Wiener approximation probem he Paley-Wiener approximation problem: Suppose {x n : n N} is an orthonormal basis for a Hilbert space H and {y n : n N} is a sequence in H. If these sequences are close to each other (not quite defined yet), does {y n : n N} span H? As an example of what we mean here, consider the standard orthonormal basis ϕ n (e iθ ) = e inθ, n Z, for L 2. When does the sequence ψ n (e iθ ) = e iλnθ, n Z,

22 22 WILLIAM. ROSS where λ n R, span L 2? A classical theorem of Paley-Wiener says this is indeed the case when max n Z λ n n < π 2. So let Λ D be a sequence in D. When does {k Θ λ : λ Λ} = (ΘH 2 )? 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 \ σ b (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 (ΘH 2 ) consisting of (normalized) kernel functions! Ah ha! But when U α has discrete spectrum then indeed U α has an orthoormal basis of eigenvectors { } k Θ ζ Θ (ζ) : Θ (ζ) <, Θ(ζ) = α which can be used formulate Paley-Wiener type approximation theorems for model spaces (ΘH 2 ). 2. A connection to composition operators For a self-map ϕ of D and an analytic function f on D define the composition operator (C ϕ f)(z) := f(ϕ(z)). By an application of the Littlewood subordination principle [4], it is known that C ϕ is a bounded operator form H 2 to itself. wo great places to learn about the basics of composition operators are [3, 0]. It also turns out that there are connections to AC measures. he first one to mention involves the Aleksandrov operator. For a self map ϕ define, for a continuous function f on, (A ϕ f)(α) = f(ζ)dµ α (ζ), α. One can show that since f is continuous on then A ϕ f is also continuous on. he above operator is called the Aleksandrov operator. heorem 2. (Aleksandrov (987)). If ϕ(0) = 0, the operator A ϕ extends to be a bounded operator on H 2. Assuming that ϕ(0) = 0 one can work the identity ζ + λ ζ λ dµ α(ζ) = α + ϕ(λ), λ D, α, α ϕ(λ)

23 to get the formula Indeed, LENS LECURES ON ALEKSANDROV-CLARK MEASURES 23 λζ dµ α(ζ) = ϕ(λ)α. ζλ dµ α(ζ) = ( ) ζ + λ 2 ζ λ + dµ α (ζ) = ( ) α + ϕ(λ) 2 α ϕ(λ) + α = α ϕ(λ) = αϕ(λ). Now take complex conjugates to get the desired identity. If we write k λ (z) = λz for the reproducing kernel for H 2, i.e., we can write the previous identity as f, k λ = f(λ), f H 2, λ D, (A ϕ k λ )(α) = ϕ(λ)α. We can relate this to the composition operator. Indeed hus, (C ϕk λ )(z) = C ϕk λ, k z = k λ, C ϕ k z = k λ, k z (ϕ( )) = k z (ϕ( )), k λ = k z (ϕ(λ)) = ϕ(λ)z. (A ϕ k λ )(α) = (C ϕk λ )(α), λ D, α. We are certainly taking a few liberties in the above formula since α 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. heorem 2.2. If ϕ(0) = 0 then C ϕ = A ϕ.

24 24 WILLIAM. ROSS here are many other connections between composition operators and AC measures. Let us mention one more. he essential norm of C ϕ, denoted by C ϕ e, is the distance from C ϕ to the compact operators K on H 2, i.e., C ϕ e := inf{ C ϕ K : K K}. he essential norm of C ϕ can be computed in terms of something called the Nevanlinna counting function [3, 0] but Cima and Matheson [2] computed it in terms of σ α, where µ α = h α dm + σ α is the Lebesgue decomposition of µ α A ϕ. heorem 2.3 (Cima-Matheson). For a self map ϕ, C ϕ e = σ α (). sup α 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]. 3. Higher dimensional analogs Let Θ be a contractive M n n (C)-valued analytic function on D. By this we mean that Θ(z), 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) < for all z D. his is no longer the case in the matrix case. What is true is the following: Proposition 3.. Every contractive M n n (C)-valued function Θ on D can be written in block form as Θ = Θ 0 Θ, where Θ 0 is a constant unitary matrix and Θ is purely contractive, i.e., Θ (z) < for all z D. For a purely contractive Θ and a constant unitary matrix A M n n define the function B A (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 RB A (z) := 2 (B A(z) + B A (z) ) is a positive definite and, by a matrix version of the Herglotz formula, we can write ζ + z B A (z) = ζ z µ A(dζ), where µ A is a positive matrix-valued measure on. hat is to say, for every Borel set E, µ A (E) is a positive definite matrix. his yields the family of matrix-valued measures on, the Aleksandrov-Clark measures {µ A : A U(n)},

25 LENS LECURES 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 m n denote M n n -valued measure mi (n copies of normalized Lebesgue measure m along the diagonal) on. Also let dh n denote Haar measure on U(n). heorem 3.2 (Martin (202)). For any continuous f : C n, ( ) µ A (dζ)f(ζ) dh(u) = m n (dζ)f(ζ). U(n) 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. J. A. Cima, A. L. Matheson, and W.. Ross, he Cauchy transform, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, Joseph A. Cima and Alec L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math. 79 (997), no., Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, P. L. Duren, heory of H p spaces, Academic Press, New York, J. Garnett, Bounded analytic functions, first ed., Graduate exts in Mathematics, vol. 236, Springer, New York, K. Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall Inc., Englewood Cliffs, N. J., R..W. Martin, Unitary perturbations of compressed n-dimensional shifts, Compl. Anal. Oper. heory. In press: (202). 8. Walter Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, Eero Saksman, An elementary introduction to Clark measures, opics in complex analysis and operator theory, Univ. Málaga, Málaga, 2007, pp Joel H. Shapiro, Composition operators and classical function theory, Universitext: racts in Mathematics, Springer-Verlag, New York, 993. Department of Mathematics and Computer Science, University of Richmond, Richmond, Virginia, 2373, USA address: wross@richmond.edu URL: