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Lens Lectures on Aleksandrov-Clark Measures

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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 ' : D ! D, where D is the open unit disk, there is a family fµα : jαj = 1g of positive finite measures on the unit circle T associated with ' by the formula 1 − j'(z)j2 Z 1 − jzj2 2 = 2 dµα(ξ); z 2 D: jα − '(z)j T jξ − zj These measures, called Aleksandrov-Clark measures, appear as the spec- tral 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 repre- sentation theorem, Hahn-Banach theorem, spectral theorem), complex anal- ysis (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 '. Here cyclic means that _ n fA ' : n 2 N0g = 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; ' ≡ 1: 1 2 WILLIAM T. ROSS For the self-adjoint operator A, cyclic vector ', and λ 2 R let Aλ := A + λ(' ⊗ '); where ' ⊗ ' is the rank-one operator on H defined by (' ⊗ ')(v) = hv; 'i'; v 2 H: Clearly Aλ is self-adjoint (since λ is real and the rank-one tensor ' ⊗ ' is self adjoint). Moreover, Aλ is also cyclic with cyclic vector '. Indeed 2 Aλ' = A' + λk'k ' and so Aλ' 2 spanf'; A'g: Continuing in this fashion we see that _ n _ n fAλ' : n 2 N0g = fA ' : n 2 N0g = 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 2 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 ' ≡ 1, then Aλ = Mx + λ(1 ⊗ 1): More specifically, Z 1 2 Aλf = xf + λ f(t)dt; f 2 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 = p1 (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 p p −λ + 1 + λ2 λ + 1 + λ2 µλ = p δλ + p 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 ' : 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 '. On old theorem of Fatou [4] says the following: Theorem 3.1 (Fatou's theorem). For an analytic self-map ', the radial limit lim '(rζ) r!1− exists for m-almost every ζ 2 T. We often write '(ζ) for this limit whenever it exists (and is finite!). Proposition 3.2. The set fζ 2 T : '(ζ) existsg 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 '(z) ! '(ζ) not only as z ! ζ along the radius connecting 0 and ζ but as z ! ζ inside any Stolz domain jz − ζj z 2 : < α ; α 2 (1; 1): D 1 − jzj This is a triangular shaped region with vertex at ζ. We often write lim '(z) \ z!ζ for the non-tangential limit. Definition 3.3. A self-map ' is said to be inner if j'(ζ)j = 1 for m-almost every ζ 2 T. 4 WILLIAM T. ROSS Example 3.4. For a 2 D consider the function z − a '(z) = ; z 2 : 1 − az D Note that eiθ − a e−iθ − a '(eiθ)'(eiθ) = = 1; θ 2 [0; 2π]; 1 − aeiθ 1 − ae−iθ and so, by the maximum modulus theorem, ' is an analytic self-map. The above calculation shows that ' has unimodular boundary values and so ' is inner. Example 3.5. Consider z+1 '(z) = e z−1 ; z 2 D: Note that 2 z + 1 z + 1 1 − jzj exp = exp < = exp − < 1; z 2 D: z − 1 z − 1 j1 − zj2 Thus ' is an analytic self-map. From the previous identity one can easily check that j'(eiθ)j = 1 if θ 2 (0; 2π). Thus ' is an inner function. If α1; ··· ; αn are positive numbers and θ1; ··· ; θn 2 [0; 2π], one can also check that z + eiθ1 z + eiθn '(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 n f0g which satisfy the condition 1 X (1 − janj) < 1; n=1 one can show that the infinite product 1 Y an z − an B(z) := ; ja j 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 n f0g sat- P1 isfying n=1(1 − janj) < 1, 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 2 ; µ ξ − z D is inner. Proof. As was done previously one can show that Z ξ + z Z 1 − jzj2 js (z)j = exp < − dµ(ξ) = exp − dµ(ξ) < 1: µ ξ − z jz − ξj2 2 Thus sµ 2 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 2 R, n 2 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 ' takes the form iγ n '(z) = e z B(z)sµ(z): n The factors z ; B; sµ are unique. The definition of an inner function says that ' 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 uni- modular boundary values at a particular point. Theorem 3.10 (Frostman-Ahern-Clark). An inner function ' = Bsµ and all of its inner divisors have limits which are unimodular at ζ 2 T if and only if X 1 − jλj2 Z dµ(ξ) + < 1: jζ − λj jξ − ζj λ2B−1(f0g) The next concept here is the notion of an angular derivative. Definition 3.11. For an analytic self-map ' and ζ 2 T we say that ' has an angular derivative at ζ if lim '(z) 2 T; \ z!ζ lim '0(z) exists: \ z!ζ 6 WILLIAM T. ROSS Note that if ' has an analytic continuation to a neighborhood of ζ and j'(ζ)j = 1, then the angular derivative exists.
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