LECTURES ON COMPOSITION OPERATORS AND ANALYTIC FUNCTION THEORY

JOEL H. SHAPIRO

1. Invertibility and the Schwarz Lemma

1.1. Introduction. At first we will work in H(U), the collection of all complexvalued functions that are holomorphic (i.e., analytic) on on the unit disc U = {z ∈ C : |z| < 1} of the complex plane. Because pointwise sums and products of analytic functions are again analytic, these pointwise operations make H(U) into an algebra, and in particular a vector space, over the field of complex numbers.

Suppose φ is a holomorphic self-map of U, i.e. φ is holomorphic on U and φ(U) ⊂ U. From now on we always use the symbol φ to represent—usually without further comment—a holomorphic self-map of U. For f ∈ H(U) and z ∈ U define C_{φ}f : U → C by:

(C_{φ}f)(z) = f(φ(z)).

Since compositions of analytic functions are analytic when the domains and ranges match up correctly (which they do here), we see that C_{φ}f ∈ H(U), so we have defined a mapping C_{φ} : H(U) → H(U) that’s called the composition operator induced by φ.

It is easy to check that C_{φ} is a linear transformation on H(U). The point of studying composition operators is to understand how the properties of the analytic function φ influence those of the linear map C_{φ}, and vice versa.

We start with what is perhaps the most basic question you can ask about any linear transformation: when is it invertible? In other words, when is it one-to-one and onto? For composition operators, one-to-one is never an issue:

1.2. Theorem. If φ is not constant than C_{φ} is one-to-one on H(U).

Proof. Suppose that for some f and g in H(U) we have C_{φ}f = C_{φ}g, i.e. that f(φ(z)) = g(φ(z)) for all z ∈ U. Then f ≡ g on φ(U). But nonconstant holomorphic functions are open mappings, so f ≡ g on a nonempty open subset of U. Thus f and g agree on a subset of U having a limit point in U, so by the uniqueness theorem f ≡ g on U. Thus C_{φ} is one-to-one.

That was easy, but note that it required two important results about analytic functions—the open mapping theorem and the uniqueness theorem.

Date: February 16, 1999.

2 JOEL H. SHAPIRO

1.3. Exercise. Let C(U) denote the collection of complex valued continuous functions on U. For ψ : U → U a continuous self-map of U, define C_{ψ} on C(U) in the obvious way. Determine those ψ’s for which C_{ψ} is one-to-one.

In order to characterize the composition operators that are invertible on H(U), we have to work a little harder.

1.4. Theorem. C_{φ} is invertible on H(U) if and only if φ maps U one-to-one onto itself.

Proof. It’s clear that if φ maps U one-to-one onto itself then C_{φ} is invertible, since then the inverse is C_{φ^{−1}}.

The converse takes more work. We will show that if φ is not an automorphism, then the range C_{φ} is not H(U). If φ is not one-to-one this is easy: there exist distinct points a and b in U with φ(a) = φ(b), hence for every f ∈ H(U) the image function C_{φ}f = f ∘ φ takes the same value at a that it takes at b. In particular, the identity map f(z) ≡ z cannot belong to ran C_{φ}, so C_{φ} is not invertible.

One case remains: suppose φ(U) ≠ U. Then we can choose w ∈ U\φ(U) and consider the functions

f(z) = \frac{1}{z − w}, and g(z) = \frac{1}{φ(z) − w} (z ∈ U).

Because w ∉ φ(U), we see that f is holomorphic on φ(U), and g is holomorphic on U.

I claim that g ∉ ran C_{φ}. Suppose otherwise. Then there exists h ∈ H(U) such that h(φ(z)) = g(z) = f(φ(z)) for every z ∈ U\{w}. Thus h = f on a nonempty open subset of U\{w}, and so h = f on all of U\{w}, again by the identity theorem. Thus f must be bounded in a neighborhood of w, which it obviously is not. This shows that ran C_{φ} ≠ H(U), as desired.

1.5. Exercise. Show that if ψ is a continuous self-map of the unit disc, then C_{ψ} is an isomorphism of C(U) if and only if ψ is a homeomorphism of U onto itself.

Conformal automorphisms of U. One-to-one holomorphic mappings are usually called univalent. If a holomorphic self-map of the unit disc is univalent and maps U onto itself, we call it a conformal automorphism (or just an automorphism of U. Theorem 1.4 can now be rephrased:

C_{φ} is invertible on H(U) if and only if φ is a conformal automorphism of U.

To make this result meaningful, however, we have to characterize conformal automorphisms in some concrete way. A good beginning involves finding a large class of examples.

For p ∈ U define

α_{p}(z) =^{def} \frac{p − z}{1 − p̄z} (z ∈ U). (1.1)

COMPOSITION OPERATORS 3

Because the denominator is zero only at z = 1/p̄, the function α_{p} is holomorphic on the disc {|z| < 1/|p|}, which contains the closed unit disc. Note that α_{p} interchanges the origin and the point p.

1.6. Theorem. α_{p} is a conformal automorphism of U.

Proof. The proof hinges on the formula below, which I leave to you as an exercise.

1 − |α_{p}(z)|^{2} = \frac{(1 − |p|^{2})(1 − |z|^{2})}{|1 − p̄z|^{2}} (z ∈ C\\{1/p̄\}). (1.2)

The right-hand side of this equation is > 0 for every z ∈ U, and = 0 for every z ∈ ∂U. Thus α_{p} maps U into itself, and ∂U into itself.

The fact that α_{p} is a conformal automorphism follows from this simple exercise: α_{p} is its own compositional inverse, in other words, α_{p}(α_{p}(z)) = z . This is most easily seen by reducing the equation w = α_{p}(z) to its equivalent form w − wp̄z + z = p, in which w and z play symmetric roles.

1.7. Exercise. If p and q are points of U then there is a conformal automorphism of U that maps p onto q.

Clearly the conformal automorphisms of U form a group under composition. The exercise above says that this group acts transitively on U.

We still haven’t classified all the conformal automorphisms of U. The largest class that comes to mind, given what we’ve already done, is the one you get by composing maps α_{p} with rotations, i.e., maps of the form α_{p}(ωz) or ωα_{p}(z) for p ∈ U, and then composing these maps with each other …. It is a remarkable fact that nothing new arises from this process.

1.8. Theorem. If φ is a conformal automorphism of U then there exists p ∈ U and ω, η ∈ ∂U such that φ(z) = ωα_{p}(z) = α_{p}(ηz) for all z ∈ U.

Proof. I’ll prove the first equality and leave the second one as an exercise. Since φ maps U onto itself there is a point p ∈ U such that φ(p) = 0. Thus φ(α_{p}(0)) = 0, so if we define ψ = φ ∘ α_{p} then ψ is a conformal automorphism of U that fixes the origin. Now the only such maps that come quickly to mind are the rotations. In fact we’ll see shortly that there are no others. Granting this, there must be a complex number ω of modulus one such that

ωz = ψ(z) = φ(α_{p}(z)) (z ∈ U).

Upon replacing z by α_{p}(z) in this equation and using the self-inverse property of α_{p}, we see that ωα_{p}(z) = φ(z) for each z ∈ U, as desired.

It remains to prove that the only conformal automorphisms of U that fix the origin are the rotations. This is not a triviality—it requires one of the most important theorems in complex analysis:

4 JOEL H. SHAPIRO

1.9. The Schwarz Lemma. If φ is a holomorphic self-map of U with φ(0) = 0, then

(a) |φ(z)| ≤ |z| for every z in U, with equality for some z ∈ U if and only if φ is a rotation (about the origin).

(b) |φ′(0)| ≤ 1, with equality if and only if φ is a rotation.

Proof. Everything follows from an analysis of the function g defined on U by: g(z) = φ(z)/z if z ∈ U\{0}, and φ′(0). Then g is holomorphic on U. Fix z_{0} ∈ U and consider a number r with |z_{0}| < r < 1. Note that since |g(z)| ≤ 1/r on the circle {|z| = r}, the maximum modulus principle shows that |g(z_{0})| ≤ 1/r. Upon letting r → 1− we see that in fact |g(z_{0})| ≤ 1, hence |φ(z_{0})| ≤ |z_{0}|.

As for equality in part (a), suppose that for some z_{0} ∈ U we had |φ(z_{0})| = |z_{0}|. Then |g(z_{0})| = 1 so if g were not constant, then by the maximum principle we would have |g| > 1 somewhere on U. Since this does not happen, g is constant, necessarily a constant of modulus one, and therefore φ is a rotation. This proves part (a) of the Schwarz Lemma.

The “inequality” part of part (b) follows immediately from the corresponding part of part (a). As for the rest, if |φ′(0)| = 1 then g attains its maximum at the origin, so as before, g has to be constant, and therefore φ is a rotation.

1.10. Corollary. If φ is a conformal automorphism of U that fixes the origin, then there exists ω ∈ ∂U such that φ(z) = ωz for every z ∈ U.

Proof. Since φ is a holomorphic self-map of U that fixes the origin, part (b) of the Schwarz Lemma guarantees that |φ′(0)| ≤ 1. But since φ is an automorphism, it has a compositional inverse ψ that also obeys the hypotheses of the Schwarz Lemma, hence |ψ′(0)| < 1. By the chain rule, φ′(0)ψ′(0) = 1, hence |φ′(0)| = 1, and so by the “equality part” of part (b) of the Schwarz Lemma, φ is a rotation.

With this, the proof of Theorem 1.8 is complete.

The Schwarz Lemma will prove to be the single most valuable result in this course. Here is another application, seemingly not related to invertibility, which we will need in the next section. For this one we use the notation

φ_{n} = φ ∘ φ ∘ … ∘ φ (n times).

to denote the n-th iterate of φ for n a positive integer.

1.11. Theorem. If φ is a holomorphic self-map of U with φ(0) = 0, and φ is not a rotation, then φ_{n} → 0 uniformly on compact subsets of U.

Proof. Fix a compact subset K of U, and fix 0 < r < 1 so that K ⊂ {|z| < r}. By the Schwarz Lemma we know that |φ(z)| < r on the circle γ_{r} = {|z| = r}, the strict inequality coming from the fact that φ is not a rotation. Since |φ| is continuous and γ_{r} compact, we know that |φ| attains its maximum M(r) on γ_{r}, so also M(r) < r. This means that the function ψ, defined on U by

ψ(w) = \frac{1}{M(r)}φ(rw) (w ∈ U),

is a holomorphic self-map of U that fixes the origin. So by the Schwarz Lemma, ψ(w) ≤ |w| for all w ∈ U (in fact, there is strict inequality since ψ is not a rotation either). Translating this estimate into the language of φ we obtain

|φ(z)| ≤ δ|z| (|z| < r) (1.3)

where δ = M(r)/r < 1. Since φ takes the disc {|z| < r} into itself, the above estimate can be iterated, resulting in

|φ_{n}(z)| ≤ δ^{n}|z| (|z| < r, n = 1, 2, …). (1.4)

In particular, max_{z∈K} |φ_{n}(z)| ≤ δ^{n} max_{z∈K} |z|, hence φ_{n} → 0 uniformly on K.

1.12. Exercise. Show that if φ is a holomorphic self-map of U that fixes a point p ∈ U, and is not a conformal automorphism, then φ_{n} → p uniformly on compact subsets of U (the above theorem is the case p = 0).

Let’s close this section with a few more exercises that illustrate further uses of the Schwarz lemma.

1.13. Exercise. Show that if φ is a holomorphic self-map of U then for every p ∈ U,

|φ′(p)| ≤ \frac{1 − |φ(p)|}{1 − |p|}

with equality if and only if there is a complex number of modulus one such that φ(z) = ωα_{p}(z) for all z ∈ U. (Hint: Let q = φ(p) and consider α_{q} ∘ φ ∘ α_{p}).

1.14. Exercise. Show that if a holomorphic self-map φ of U fixes two distinct points of U, then φ is the identity map. (Hint: Consider first the case where one of the fixed points is the origin.)

1.15. Exercise. Show that if φ is a holomorphic self-map of U that fixes a point p ∈ U, then |φ′(p)| ≤ 1, with equality if and only if φ is an automorphism.

6 JOEL H. SHAPIRO

2. Eigenvalues and Schröder’s equation

Introduction. For linear transformations the next most fundamental topic after invertibility is eigenvalues. The eigenvalue equation for a composition operator C_{φ} on H(U) is C_{φ}f = λf, or more classically:

f ∘ φ = λf, (2.1)

a functional equation which, given φ, is to be solved for f and λ. Equation (2.1) is called Schröder’s equation; it was first studied by Ernst Schröder during the 1860’s in connection with iteration of holomorphic functions.

To understand this connection, suppose we are lucky enough to have solved Schröder’s equation with a univalent eigenfunction f. Then φ = f^{−1} ∘ M_{λ} ∘ f, where M_{λ} is the map of multiplication by the complex number λ, acting on f(U) (Schröder’s equation insures that M_{λ} takes f(U) into itself). In other words, the action of φ on U is “modeled” by the action of the much simpler mapping M_{λ} on the more complicated domain f(U). For obvious reasons, then, Schröder’s equation is said to “linearize” the action of φ.

The connection with iteration comes from the fact that

φ_{n} = f^{−1} ∘ M_{λ^{n}} ∘ f (n = 1, 2, …). (2.2)

For example, if you wish to study the φ-orbit {φ_{n}(z_{0})} of a point z_{0} ∈ U, all you have to do is pull back the easily visualized M_{λ}-orbit {λ^{n}f(z_{0})} via the map f.

It is rare that Schröder’s equation can be solved explicitly. Here are two examples where you can do it. The first one has a fixed point at the origin, and the second has no fixed point in U. Note that the properties of the solutions are very different.

2.1. Exercise.

(a) Suppose ω is a complex number of modulus 1, and set φ(z) = ωz. Show that the numbers 1, ω, ω^{2}, … are all eigenvalues of C_{φ}. Show eigenvalues have multiplicity one if ω is not a root of unity, and infinite multiplicity otherwise.

(b) Let φ(z) = \frac{1+z}{2}. Show that every nonzero complex number is an eigenvalue of C_{φ}. Do these eigenvalues have finite multiplicity? (Hint: Consider the functions f_{α}(z) = (1 − z)^{α} for α ∈ C.)

Schröder never succeeded in getting a solution to his equation for any significant classes of maps φ, and was forced to settle for what he called the “opposite approach” wherein one constructs examples by choosing a mapping f that takes the unit disc univalently onto a domain that is invariant under multiplication by a complex number λ of modulus < 1, and defining φ on U by equation (2.2). Here are some examples for you to work out.

2.2. Exercise. Let G denote the half-plane {w ∈ C : Re w > −1}, and let f(z) = \frac{2z}{1−z} = \frac{1+z}{1−z} − 1. Prove that f maps U onto G, taking the unit circle to the line Re z = −1 (with 1 going to ∞). Show that the multiplication map M_{λ} takes Π into itself if and only if λ > 0. Thus for each positive λ the right-hand side of equation

COMPOSITION OPERATORS 7

(2.2) defines a holomorphic self-map φ_{λ} of U. Show that φ_{λ}(0) = 0, and that φ is not an automorphism. Determine the orbit of an arbitrary point z ∈ U under the action of φ. Show that the numbers λ^{n} are eigenvalues of C_{φ_{λ}} (n = 0, 1, 2, …). Find the corresponding eigenfunctions.

2.3. Exercise. Same setup as in the previous exercise, but replace G by the right half-plane Π = \{w ∈ C : Re w > 0\}, and set f(z) = \frac{1+z}{1−z}. This time show that f maps U onto Π, and that for each λ > 0: φ_{λ} is an automorphism of U that fixes the points ±1 and no others. In particular, φ_{λ} has no fixed point in U. Show that the numbers λ^{n} are eigenvalues of C_{φ_{λ}} (n = 0, 1, 2, …). Find the corresponding eigenfunctions.

Real progress on the theory of Schröder’s equation had to wait until 1884. In that year Gabriel Koenigs published his work on solutions of Schröder’s equation for the class of holomorphic self-maps φ that are not conformal automorphisms, that fix a point of p ∈ U and for which φ′(p) ≠ 0 (so that φ is, in particular, nonconstant). Let us call these mappings Koenigs maps. Note that by Exercise 1.15 we have 0 < |φ′(p)| < 1 for each Koenigs map φ with fixed point at p ∈ U.

Here is the formal statement of the result that Koenigs proved.

2.4. Theorem of Koenigs. If φ is a Koenigs map, then the eigenvalues of C_{φ} are precisely the numbers φ′(p)^{n} for n = 0, 1, 2, . . .. Moreover these are the only eigenvalues, and they all have multiplicity one. If φ is univalent, then so is the eigenfunction σ corresponding to the eigenvalue φ′(p).

The “multiplicity one” property of eigenvalues justifies our reference to “the” eigenfunction corresponding to the eigenvalue φ′(p). Note that it is clear that once you have an eigenvector σ for φ′(p), then for any positive integer n, φ′(p)^{n} is an eigenvalue of C_{φ} with eigenvector σ^{n}. Multiplicity 1 insures that, up to constant multiples, σ^{n} is the only such eigenvector. The statement about univalence is important since it implies that if φ is univalent, then the eigenfunction σ “linearizes” φ to M_{φ′(p)} in accordance with our discussion at the beginning of this section.

To prove Theorem 2.4 it is enough to consider the case where the fixed point p is the origin. Once this is finished, given a Koenigs map whose fixed point p is not 0 we apply the “p = 0” result to ψ = α_{p} ∘ φ ∘ α_{p}, which fixes the origin, and use the automorphism α_{p} to translate the results for ψ into results for φ. I leave the details as an exercise.

Toward the proof of Theorem 2.4. Before jumping into the proof of this result, it is important to understand the roles played by its various hypotheses. For example, it is desirable that φ have an interior fixed point because of part (b) of Exercise 2.1, which shows that if φ has no such fixed point then C_{φ} can have eigenvalues of infinite multiplicity. Part (a) of that same exercise shows that it is important for φ not to be an automorphism of U, since otherwise there may again be eigenvalues of infinite multiplicity. Here is a preliminary result that gets us started toward further understanding.

8 JOEL H. SHAPIRO

2.5. Lemma. Suppose φ is a nonconstant holomorphic self-map of U, and that and C_{φ}f = λf for some f ∈ H(U) and λ ∈ C. Then:

(a) λ ≠ 0.

(b) If f is not constant, and φ fixes p ∈ U, then λ ≠ 1, and f(p) = 0.

Proof. Part (a) is just the statement that C_{φ} is one-to-one, which we proved in the last section.

For part (b), since φ is not an automorphism we know from Theorem 1.11 and the Exercise that follow it that the iterates φ_{n} converge to p uniformly on compact subsets of U. Now if λ were equal to 1, then we would have C_{φ}f = f, and therefore for each z ∈ U:

f(z) = C_{φ}^{n}f(z) = f(φ_{n}(z)) → f(p),

so f ≡ f(p), which contradicts our hypothesis that f is not constant.

To see that f(p) = 0 just set z = p in the eigenvalue equation f(φ(z)) = λf(z) to get f(p) = λf(p) which, since λ ≠ 1, gives the desired result.

We next turn to the question of why the eigenvalues in Theorem 2.4 are what they are.

2.6. Proposition. Suppose φ is a holomorphic self-map of U that fixes a point p ∈ U. Suppose that C_{φ}f = λf for some nonconstant f ∈ H(U) and some λ ∈ C. Then λ = φ′(p)^{n} for some n = 0, 1, 2, ….

Proof. Here is the proof for the special case p = 0. I leave it to you to employ the conformal automorphism α_{p} to translate the result into one for arbitrary p ∈ U.

The eigenfunction f is not constant, so by the previous result it vanishes at the origin, hence there exists a positive integer n such that

f(z) = a_{n}z^{n} + a_{n+1}z^{n+1} + ⋯,

where a_{n} ≠ 0. Upon solving the eigenvalue equation f ∘ φ = λf for λ we obtain for each z ∈ U:

λ = \frac{f(φ(z))}{f(z)} = (\frac{φ(z)}{z})^{n} \frac{a_{n} + a_{n+1}φ(z) + a_{n+2}φ(z)^{2} + ⋯}{a_{n} + a_{n+1}z + a_{n+2}z^{2} + ⋯}.

Clearly the right-hand side of this equation converges to φ′(0)^{n} as z → 0, which establishes the desired result.

Note that in the last result we did not rule out the possibility that φ′(p) = 0. Since 0 cannot be an eigenvalue of C_{φ} (for φ nonconstant), the Proposition tells us that if φ′(p) = 0 then 1 is the only eigenvalue of C_{φ}. Or, to put it differently, if C_{φ} has a nonconstant eigenfunction, then φ′(p) ≠ 0.

So far we have justified all the elements that go into the definition of “Koenigs map”. Now let’s tackle the question of eigenvalue multiplicity.

COMPOSITION OPERATORS 9

2.7. Proposition. If φ is a Koenigs map and λ an eigenvalue of C_{φ}, then λ has multiplicity one, in other words: if g ∈ H(U) and C_{φ}g = λg, then g is a constant multiple of f.

Proof. Note that we have already established a special case of this: in the proof of Lemma 2.5 we saw that the only eigenvectors for the eigenvalue 1 are the constant functions.

To make further progress, successively differentiate both sides of Schröder’s equation, and evaluate each result at z = 0. The calculation shows that for every n ≥ 2, the quantity (λ − λ^{n})f^{(n)}(0) is given by an expression that involves the derivatives φ^{(k)}(0) for 1 ≤ k ≤ n, and f^{(j)}(0) for 1 ≤ j ≤ n − 1. Since λ is neither 0 nor 1, an induction argument shows that for n ≥ 2 the derivative f^{(n)}(0) is determined solely by φ and f′(0). So given φ, the coefficients of f in its Taylor expansion about the origin are determined solely by f′(0) (recall that, because f(0) = 0, the constant coefficient is zero). This completes the proof.

2.8. Exercise. Go back to Exercises 2.2 and 2.3. For the first of these, the work we have done so far shows that the eigenvalues {λ^{n}}_{0}^{∞} found there are the only eigenvalues for C_{φ_{λ}}, and they have multiplicity one. Show that by contrast, the mappings created in Exercise 2.3 induce composition operators with many more eigenvalues, all of infinite multiplicity! (Hint: go over to the right half-plane Π and consider the composition operator induced by M_{λ} on H(Π).)

Now we understand everything about what eigenvalues for C_{φ} have to look like when φ is a Koenigs function. What we have not yet proved is that there are any nontrivial (i.e. not = 1) eigenvalues, or, equivalently, nontrivial (not ≡ constant) eigenvectors. We remedy this deficiency with the next result, which almost completes our proof of the Theorem of Koenigs.

2.9. Theorem. Suppose φ is a Koenigs map with fixed point p ∈ U. Then there exists σ ∈ H(U) such that σ ∘ φ = φ′(p)σ.

Proof. As usual, I’ll do the proof only for the case p = 0. The function σ will be obtained as the limit of a sequence of “normalized iterates” of φ. As previously discussed, we may take p = 0, so that φ(0) = 0. By the Schwarz Lemma, 0 < |φ′(0)| < 1. Let λ = φ′(0), and for each positive integer n set σ_{n} = λ^{−n}φ_{n} (noting that while the subscript n on the right-hand side of the equation denotes an iterate, the one on the left does not). Since σ_{n} ∘ φ = λσ_{n+1} for each n, the solution σ we seek will be the limit of the sequence of functions {σ_{n}}, if only we can prove that this limit exists uniformly on compact subsets of U.

To do so, write

σ_{n}(z) = z · \frac{φ(z)}{λz} · \frac{φ_{2}(z)}{λφ(z)} · · · · · \frac{φ_{n}(z)}{λφ_{n−1}(z)} = z ∏_{j=0}^{n−1} F (φ_{j}(z)),

where in the product on the right, φ_{0}(z) ≡ z, and F (z) = φ(z)/λz. Thus our task is to prove that the infinite product

∏_{j=0}^{∞} F (φ_{j}(z))

converges uniformly on compact subsets of U. For this it is enough to show that the infinite series

∑_{j=0}^{∞} |1 − F (φ_{j}(z))| (2.3)

converges likewise ^{1}. We estimate the size of each term. Since φ(0) = 0, the function F is holomorphic on U. Let ∥F ∥_{∞} denote the supremum of the values of |F | over the unit disc. By the Maximum Principle, ∥F ∥_{∞} ≤ |λ^{−1}|, hence

∥1 − F ∥_{∞} ≤ 1 + ∥F ∥_{∞} ≤ 1 + \frac{1}{|λ|} =^{def} A.

Finally, the definition λ = φ′(0) forces F (0) = 1, so the Schwarz Lemma, applied to the function (1 − F)/A, yields

|1 − F (z)| ≤ A|z| (z ∈ U). (2.4)

Now fix 0 < r < 1. In the proof of Theorem 1.11 we observed that a slight refinement of the Schwarz Lemma produces a constant δ < 1 such that for each non-negative integer j,

|φ_{j}(z)| ≤ δ^{j}|z|

whenever |z| ≤ r. Upon substituting this inequality in (2.4) we see that

|1 − F (φ_{j}(z))| ≤ A|φ_{j}(z)| ≤ Aδ^{j}|z| (z ∈ r\overline{U}).

So on the closed disc r\overline{U}, each term of the series (2.3) is bounded uniformly by the corresponding term of a convergent geometric series, hence the original series converges uniformly on that disc. This establishes the desired convergence for the sequence {σ_{n}}.

We refer to the function σ constructed above as the principal eigenfunction of C_{φ}. It is the (essentially unique) eigenvector corresponding to the largest nontrivial eigenvalue, namely φ′(0).

All that’s left in our quest to prove the Theorem of Koenigs is the result about univalence.

1See the Appendix at the end of this section

COMPOSITION OPERATORS 11

2.10. Corollary. If φ is a univalent Koenigs map, then the principle eigenfunction of C_{φ} is also univalent

Proof. The univalence of φ gets inherited by each iterate φ_{n}, and therefore by each normalized iterate σ_{n} = φ_{n}/λ^{n}. Now suppose the limit function σ were not univalent. Then there would be points a, b ∈ U with σ(a) = σ(b). Call this common value w. Fix a number r with max{|a|, |b|} < r < 1, so that φ does not take the value w on the circle γ = {|z| = r}. Give γ the positive orientation. By the Argument Principle, the integral

I =^{def} \frac{1}{2πi} ∫_{γ} \frac{σ′(z)}{σ(z) − w} dz

counts the number of points in the disc {|z| < r} that φ maps onto w, so this integral is at least 2. On the other hand, the same reasoning applied to the univalent maps σ_{n} shows that for each n,

I_{n} =^{def} \frac{1}{2πi} ∫_{γ} \frac{σ′_{n}(z)}{σ_{n}(z) − w} dz ≤ 1.

Since σ_{n} → σ uniformly on compact subsets of U, the integrand of I_{n} converges uniformly on γ to the integrand of I, and hence I_{n} → I. But preceding estimates show that this cannot happen, so the assumption that σ is not univalent has led to a contradiction.

The Theorem of Koenigs is now proved.

2.11. Appendix. Suppose (F_{n})_{1}^{∞} is a sequence of functions in H(U), and that for every compact subset K of U we have

sup_{z∈K} ∑_{n=1}^{∞} |1 − F_{n}(z)| < ∞. (2.5)

Then the sequence (∏_{n=1}^{∞} F_{n})_{1}^{∞} converges uniformly on compact subsets of U.

Proof. Write u_{n} = F_{n} − 1, so (2.5) becomes

∑_{n=1}^{∞} |u_{n}| converges uniformly on compact subsets of U. Let P_{n} = ∏_{N=1}^{∞}(1 + u_{n}). Our goal is to show that the sequence (P_{N})_{1}^{∞} is Cauchy

uniformly on compact subsets of U. For this we need an elementary numerical estimate.

Suppose (a_{n})_{1}^{N} is an N-tuple of non-negative numbers. Then an induction shows that

∏_{1}^{N}(1 + a_{n}) = 1 + S_{1} + S_{2} + … + S_{N} (2.6)

where S_{1} is the sum of the terms a_{1}, … a_{N}, S_{2} is the sum of the products a_{i}a_{j}, where 1 ≤ i < j ≤ N, S_{3} is the sum of the products a_{i}a_{j}a_{k} where 1 ≤ i < j < k ≤ N, etc.

12 JOEL H. SHAPIRO

(the last sum, S_{N} is just the product of all the terms a_{1}, …, a_{N}). Now the binomial theorem shows that

S_{k} ≤ S_{1}^{k} (1 ≤ k ≤ N), (2.7)

which is the numerical estimate we need.

Returning to the main part of the proof, fix a compact subset K of U. Our hypothesis that ∑_{1}^{∞} |u_{n}| converges uniformly on K asserts that

lim_{N→∞} ∑_{n=N}^{∞} |u_{n}| = 0

uniformly on K, so if we throw away a finite collection of u_{n}’s, which entails no loss of generality in our problem, we may assume that

δ =^{def} sup_{z∈K} ∑_{n=1}^{∞} |u_{n}(z)| < 1. (2.8)

Now apply (2.6), (2.8), and the numerical estimate (2.7) with a_{n} = |u_{n}(z)| in the third line of the estimate below:

max_{z∈K} |P_{N}(z)| = max_{z∈K} ∏_{n−1}^{N} |1 + u_{n}(z)|

≤ max_{z∈K} ∏_{n−1}^{N}(1 + |u_{n}(z)||
≤ 1 + δ + δ^{2} + … δ^{N}
< \frac{1}{1 − δ} ,

where the last line uses the fact that δ < 1. This shows that the sequence (P_{N})_{1}^{∞} is unformly bounded on K.

We finish the proof by showing that (P_{N})_{1}^{∞} is uniformly Cauchy on K. For 1 ≤ M < N set

δ_{M,N} = δ_{M,N}(K) = max_{z∈K} ∑_{n=M+1}^{N} |u_{n}(z)|.

Our hypothesis says that δ_{M,N} → 0 as M, N → ∞.

Now for fixed z ∈ K we have:

|P_{N}(z) − P_{M}(z)| = |P_{M}(z)| |\frac{P_{N}(z)}{P_{M}(z)} − 1|
≤ \frac{1}{1 − δ} | ∏_{n=M+1}^{N}(1 + u_{n}(z)) − 1|
= \frac{1}{1 − δ} |S_{1} + S_{2} + … S_{N−(M+1)}|,

COMPOSITION OPERATORS 13

where S_{k} in the last line is defined by (2.6), but now for the product of the terms u_{M+1}(z), … u_{N}(z). From the last inequality, (2.7) and the definition of δ_{M,N} we therefore have

|P_{N}(z) − P_{M}(z)| ≤ \frac{1}{1 − δ} (δ_{M,N} + δ_{M,N}^{2} + … + δ_{M,N}^{N−(M+1)})
< \frac{δ_{M,N}}{(1 − δ)(1 − δ_{M,N})}.

Since the expression in the last line goes to zero as M, N → ∞, the sequence (P_{N}) is Cauchy uniformly on K, as we wished to show.

14 JOEL H. SHAPIRO

3. Composition operators on H^{2}

The scene now shifts to the Hardy space H^{2}, a subspace of H(U) that is a Hilbert space. This is arguably the best place to study the interaction between the theory of linear operators and analytic function theory. The main result of this section is that every composition operator restricts to a continuous mapping of H^{2} into itself. At the core of this result lies the famous Littlewood Subordination Principle.

3.1. Taylor series. For f ∈ H(U) and every non-negative integer n, let f̂(n) = f^{(n)}(0)/n!. Then the series ∑_{n=0}^{∞} f̂(n)z^{n} is the Taylor series of f with center at the origin: it converges uniformly on compact subsets of U to f.

3.2. The Hardy space. The Hardy space H^{2} is the collection of functions f ∈ H(U) with ∑_{n=0}^{∞} |f̂(n)|^{2} < ∞.

3.3. Exercise. For α real let f_{α}(z) = (1 − z)^{−α}. Show that f_{α} ∈ H^{2} if and only if α < 1/2. (Hint: Use the Binomial theorem and Stirling’s formula to show that f̂_{α}(n) ≈ n^{α−1}.

We equip H^{2} with the norm that is naturally associated with its definition:

∥f∥ = (∑_{n=0}^{∞} |f̂(n)|^{2})^{1/2} , (3.1)

and note that this norm arises from the natural inner product

⟨f, g⟩ =^{def} ∑_{n=0}^{∞} f̂(n)\overline{ĝ(n)} (f, g ∈ H^{2}). (3.2)

Let T be the “Taylor transformation” from H^{2} into the sequence space ℓ^{2} defined by T f = f̂. The mapping T is clearly linear, and from the definition of the H^{2} norm, it is an isometry: ∥T f∥ = ∥f∥ for every f ∈ H^{2}.

3.4. Proposition. T maps H^{2} onto ℓ^{2}. In particular, H^{2} is a Hilbert space in the inner product (3.2).

Proof. Because square-summable sequences are bounded, a simple geometric series estimate shows that if the complex sequence a⃗ = {a_{n}}_{0}^{∞} lies in ℓ^{2}, then the associated power series ∑_{n=0}^{∞} a_{n}z^{n} converges uniformly on compact subsets of U to an analytic function f. By the uniqueness of power series representations, a_{n} = f̂(n) for every n, hence T f = a⃗, so T (H^{2}) = ℓ^{2}.

Thus H^{2} is the sequence space ℓ^{2}, disguised as a space of analytic functions. Note in particular that:

3.5. Proposition. The sequence of monomials {z^{n} : n = 0, 1, 2, … } is an orthonormal basis for H^{2}.

Some properties of the functions in H^{2} can be easily discerned from the definition of the space. Here is one.

3.6. Growth Estimate. For every f ∈ H^{2} and z ∈ U,

|f(z)| ≤ \frac{∥f∥}{(1 − |z|^{2})^{1/2}} .

Proof. Use successively the triangle inequality and the Cauchy-Schwarz Inequality on the power series representation for f:

|f(z)| = | ∑_{n=0}^{∞} f̂(n)z^{n}|

≤ ∑_{n=0}^{∞} |f̂(n)∥z|^{n}

≤ (∑_{n=0}^{∞} |f̂(n)|^{2})^{1/2} (∑_{n=0}^{∞} |z|^{2n})^{1/2}
= ∥f∥\frac{1}{(1 − |z|^{2})^{1/2}} .

3.7. Corollary. Convergence in H^{2} implies uniform convergence on compact subsets of U.

Proof. Suppose {f_{n}} is a sequence of functions in H^{2}, f is a function in H^{2}, and ∥f_{n} − f∥ → 0. Our goal is to show that f_{n} → f uniformly on compact subsets of U.

For this, suppose K is a compact subset of U. Let r = max{|z| : z ∈ K}. Then for z ∈ K, the Growth Estimate yields:

|f_{n}(z) − f(z)| ≤ \frac{∥f_{n} − f∥}{(1 − |z|^{2})^{1/2}} ≤ \frac{∥f_{n} − f∥}{(1 − |r|^{2})^{1/2}} ,

which shows that as n → ∞,

max_{z∈K} |f_{n}(z) − f(z)| ≤ \frac{∥f_{n} − f∥}{(1 − |r|^{2})^{1/2}} → 0,

i.e. that f_{n} → f uniformly on K.

3.8. Exercise. For α ∈ C, let f_{α}(z) = (1 − z)^{α}. Show that f_{α} ∉ H^{2} whenever Re α ≤ −1/2.

However some properties of H^{2} do not follow easily from the definition. For example, is every bounded analytic function in H^{2}? In order to answer this question reasonably, we need a different description of the norm.

3.9. Proposition. A function f ∈ H(U) belongs to H^{2} if and only if

lim_{r→1−} \frac{1}{2π} ∫_{0}^{2π} |f(re^{iθ})|^{2} dθ < ∞.

When this happens, the limit of integrals on the left is ∥f∥^{2}.

Proof. The functions e^{inθ} form an orthonormal set in the space L^{2}([0, 2π]), hence for each 0 ≤ r < 1 the integral on the right is ∑_{n=0}^{∞} |f̂(n)|r^{2n}. The result now follows from the monotone convergence theorem.

3.10. Exercise. Show that the function f_{α} of Exercise 3.8 is in H^{2} whenever Re α > −1/2.

It is now an easy matter to show that every bounded function in H(U) belongs to H^{2}. In fact, we can do better. Let H^{∞} denote the collection of bounded analytic functions on U, and for b ∈ H^{∞} let ∥b∥_{∞} = sup_{z∈U} |b(z)|. The integral representation given above for the H^{2} norm shows immediately:

3.11. Proposition. If b ∈ H^{∞} and f ∈ H^{2} then bf ∈ H^{2} and ∥bf∥ ≤ ∥b∥_{∞}∥f∥.

In particular, upon taking f ≡ 1 we obtain:

3.12. Corollary. If b ∈ H^{∞} then b ∈ H^{2} with ∥b∥ ≤ ∥b∥_{∞}.

3.13. Multiplication operators act on H^{2}. Proposition 3.11 reveals an interesting class of linear transformations on H^{2}. For b ∈ H^{∞} let M_{b} denote the operator of (pointwise) multiplication by b. That is, M_{b}f = bf. Clearly M_{b}, when viewed as a mapping on all of H(U), is linear (note that for this we don’t need b to be bounded). According to Proposition 3.11 M_{b} maps H^{2} into itself, with ∥M_{b}f∥ ≤ ∥b∥_{∞}∥f∥ for each f ∈ H^{2}. It is not difficult to show from this inequality that that M_{b} is continuous on H^{2} (see §3.18 and Proposition 3.24 below). We call M_{b} the multiplication operator induced by b. The most famous of these is the one induced by the identity map b(z) ≡ z. Because it shifts the Taylor coefficients of any function on which it acts one unit to the right, this operator of “multiplication by z” is often called the “Forward Shift.”

3.14. Do Composition operators act on H^{2}? This is not a trivial question. Suppose you have f ∈ H^{2} and want to determine if C_{φ}f ∈ H^{2}. Using the definition of H^{2} we would substitute φ(z) for z in the power series expansion of f, expand the various powers of the power series of φ by the binomial theorem, and regroup the resulting double series to identify the new powers of z, which are now complicated numerical series involving the coefficients of f and those of the powers of φ. Done this way, there seems to be no reason why C_{φ}f should be in H^{2}. A calculation using the alternate characterization of H^{2} provided by Proposition 3.9 fares just as badly, since it raises the specter of an unpleasant, and possibly non-univalent, change of variable in an integral.

COMPOSITION OPERATORS 17

After these pessimistic observations, it is remarkable that composition operators do preserve the space H^{2}, and do so continuously. The key to this is the following result, proved by Littlewood and published in 1925.

3.15. Littlewood’s Subordination Theorem. Suppose φ is a holomorphic selfmap of U and φ(0) = 0. Then C_{φ}f ∈ H^{2}, and ∥C_{φ}f∥ ≤ ∥f∥ for every f ∈ H^{2}.

Proof. The proof is helped significantly by the backward shift operator B, defined on H^{2} by

Bf(z) = ∑_{n=0}^{∞} f̂(n + 1)z^{n} (f ∈ H^{2}).

The name comes from the fact that B shifts the power series coefficients of f one unit to the left, and drops off the constant term. Clearly, ∥Bf∥ ≤ ∥f∥ for each f ∈ H^{2}, and one might expect this fact to play an important role in the proof, but surprisingly it does not! Only the following two identities are needed, and they hold for any f ∈ H(U):

f(z) = f(0) + zBf(z) (z ∈ U), (3.3)

B^{n}f(0) = f̂(n) (n = 0, 1, 2, …). (3.4)

To begin the proof, suppose first that f is a (holomorphic) polynomial. Then f ∘ φ is bounded on U, so by the work of the last section there is no doubt that it lies in H^{2}; the real issue is its norm.

We begin the norm estimate by substituting φ(z) for z in (3.3) to obtain

f(φ(z)) = f(0) + φ(z)(Bf)(φ(z)) (z ∈ U).

Let us rewrite this equation in the language of composition and multiplication operators:

C_{φ}f = f(0) + M_{φ}C_{φ}Bf . (3.5)

At this point, the assumption φ(0) = 0 makes its first (and only) appearance. It asserts that all the terms of the power series for φ have a common factor of z, hence the same is true for the second term on the right side of equation (3.5), rendering it orthogonal in H^{2} to the constant function f(0). Thus,

∥C_{φ}f∥^{2} = |f(0)|^{2} + ∥M_{φ}C_{φ}Bf∥^{2} ≤ |f(0)|^{2} + ∥C_{φ}Bf)∥^{2}, (3.6)

where the last inequality follows from Proposition 3.11 above (since ∥φ∥_{∞} ≤ 1). Now successively substitute Bf, B^{2}f, ⋯ for f in (3.6) to obtain:

∥C_{φ}Bf∥^{2} ≤ |Bf(0)|^{2} + ∥C_{φ}B^{2}f∥^{2}
∥C_{φ}B^{2}f∥^{2} ≤ |B^{2}f(0)|^{2} + ∥C_{φ}B^{3}f∥^{2}
⋮  ⋮
∥C_{φ}B^{n}f∥^{2} ≤ |B^{n}f(0)|^{2} + ∥C_{φ}B^{n+1}f∥^{2}.

Putting all these inequalities together, we get

∥C_{φ}f∥^{2} ≤ ∑_{k=0}^{n} |(B^{k}f)(0)|^{2} + ∥C_{φ}B^{n+1}f∥^{2}

for each non-negative integer n.

Now recall that f is a polynomial. If we choose n be the degree of f, then B^{n+1}f = 0, and this reduces the last inequality to

∥C_{φ}f∥^{2} ≤ ∑_{k=0}^{n} |(B^{k}f)(0)|^{2}

= ∑_{k=0}^{n} |f̂(k)|^{2}
= ∥f∥^{2},

where the middle line comes from property (3.4) of the backward shift. This shows that C_{φ} is an H^{2}-norm contraction, at least on the vector space of holomorphic polynomials.

To finish the proof, suppose f ∈ H^{2} is not a polynomial. Let f_{n}(z) = ∑_{k=0}^{n} f̂(k)z^{k}, the n-th partial sum of the Taylor series of f. Then f_{n} → f in the norm of H^{2}, so by Corollary 3.7 f_{n} → f uniformly on compact subsets of U, hence f_{n} ∘ φ → f ∘ φ in the same manner. It is clear that ∥f_{n}∥ ≤ ∥f∥, and we have just shown that ∥f_{n} ∘ φ∥ ≤ ∥f_{n}∥. Thus for each fixed 0 < r < 1 we have

\frac{1}{2π} ∫_{0}^{2π} |f_{n}(φ(re^{iθ}))|^{2} dθ = lim_{n→∞} \frac{1}{2π} ∫_{0}^{2π} |f_{n}(φ(re^{iθ}))|^{2} dθ
≤ lim sup_{n→∞} ∥f_{n} ∘ φ∥
≤ lim sup_{n→∞} ∥f_{n}∥
≤ ∥f∥.

To complete the proof, let r tend to 1, and appeal one last time to Proposition 3.9.

To prove that C_{φ} is bounded even when φ does not fix the origin, we utilize the conformal automorphisms α_{p} introduced in Definition 1.1 to move points of U from where they are to where we want them. For each point p ∈ U, recall that α_{p} takes U onto itself, interchanges p with the origin, and is its own inverse. Write p = φ(0). Then the holomorphic function ψ = α_{p} ∘ φ takes U into itself and fixes the origin. By the self-inverse property of α_{p} we have φ = α_{p} ∘ ψ, and this translates into the operator equation C_{φ} = C_{ψ}C_{α_{p}}. We have just seen that C_{ψ} maps H^{2} into itself. Thus, the fact that C_{φ} does the same will follow from the first sentence of the next result.

3.16. Lemma. For each p ∈ U the operator C_{α_{p}} maps H^{2} into itself. Moreover,

∥C_{α_{p}}∥ ≤ (\frac{1 + |p|}{1 − |p|})^{\frac{1}{2}} .

Proof. Suppose first that f is holomorphic in a neighborhood of the closed unit disc, say in RU = {|z| < R} for some R > 1. Then the limit in formula (3.9) can be passed inside the integral sign, with the result that

∥f∥^{2} = \frac{1}{2π} ∫_{0}^{2π} |f(e^{iθ})|^{2}dθ.

This opens the door to a simple change of variable in which the self-inverse property of α_{p} figures prominently:

∥f ∘ α_{p}∥^{2} = \frac{1}{2π} ∫_{0}^{2π} |f(α_{p}(e^{iθ}))|^{2}dθ
= \frac{1}{2π} ∫_{0}^{2π} |f(e^{it})|^{2}|α′_{p}(e^{it})|dt
= \frac{1}{2π} ∫_{0}^{2π} |f(e^{it})|^{2} \frac{1 − |p|^{2}}{|1 − p̄e^{it}|^{2}} dt
≤ \frac{1 − |p|^{2}}{(1 − |p|)^{2}} · (\frac{1}{2π} ∫_{0}^{2π} |f(e^{it})|^{2}dt)
= \frac{1 + |p|}{1 − |p|} · ∥f∥^{2}.

Thus the desired inequality holds for all functions holomorphic in RU; in particular it holds for polynomials. It remains only to transfer the result to the rest of H^{2}, and for this we simply repeat the argument used to finish the proof of Littlewood’s Subordination Theorem.

At this point we have assembled everything we need to show that composition operators map H^{2} into itself.

3.17. Theorem. Suppose φ is a holomorphic self-map of U. Then C_{φ} is a bounded operator on H^{2}, and

∥C_{φ}f∥ ≤ \sqrt{\frac{1 + |φ(0)|}{1 − |φ(0)|}}∥f∥.

for every f ∈ H^{2}.

Proof. As outlined earlier, we have C_{φ} = C_{ψ}C_{α_{p}}, where p = φ(0), and ψ fixes the origin. Since each of the operators on the right-hand side of this equation sends H^{2} into itself, the same is true of C_{φ}.

As for the inequality, this follows from Lemmas 3.15 and 3.16. I leave the details to you.

20 JOEL H. SHAPIRO

3.18. Normed spaces and bounded linear transformations. Suppose that X is a normed space, that is, a vector space over the real or complex field on which there is defined a functional ∥ · ∥ : X → [0, ∞) with these properties:

• ∥ax∥ = |a| ∥x∥ for every x ∈ X and every scalar a.

• ∥x + y∥ ≤ ∥x∥ + ∥y∥ for every pair of vectors x, y ∈ X.

• ∥x∥ = 0 ⟹ x = 0.

Recall that every norm determines a translation-invariant metric on the underlying space: d(x, y) = ∥x−y∥. If this metric is complete, we call the underlying space (with its norm) a Banach space. Every Hilbert space is a Banach space. Other examples that you have probably seen are: the sequence spaces ℓ^{p} for 1 ≤ p <≤ ∞ and the space C(K), where K is a compact metric space and the norm is the “max-norm”.

3.19. Exercise. Show that the space H^{∞} of bounded analytic functions on the unit disc is a Banach space in the “sup norm”: ∥f∥_{∞} = sup_{z∈U} |f(z)|.

If X and Y are normed spaces and T : X → Y is a linear transformation, then T is said to be bounded if there is a non-negative constant C such that

(3.7) ∥T x∥ ≤ C∥x∥ for every x ∈ X

Note that in order to conserve notation we use the same symbol for the norm in X (on the right side of the equation above) that we use for the norm in Y (on the left side). The infimum of all the numbers C that work in the above definition is called the norm of T.

3.20. Exercise. Show that ∥T ∥ is also a value of C that works in (3.7) above. In other words, if T is a bounded linear transformation between normed spaces, then ∥T x∥ ≤ ∥T ∥ ∥x∥.

Note that the term “bounded” for a linear transformation does not mean that ∥T x∥ is bounded as x runs over the whole space. It is an easy exercise to show that the only zero-transformation has this extreme form of boundedness.

3.21. Exercise. Show that a linear transformation between normed spaces is bounded if and only if the image of every bounded set is bounded( by a bounded subset of a normed space we mean a set A for which sup_{x∈A} ∥x∥ < ∞).

3.22. Exercise. Show that if T is a bounded linear transformation, then

∥T ∥ = sup{∥T x∥ : ∥x∥ ≤ 1},

and that the inequality on the right-hand side of (3.7) can be replaced by both equality and strict inequality.

COMPOSITION OPERATORS 21

3.23. Exercise. Show that every linear transformation on a finite dimensional Hilbert space is bounded.

Examples of bounded linear transformations that we have seen so far are:

(a) The “Taylor transform” T : H^{2} → ℓ^{2} defined just before the statement of Proposition 3.4). Since T is an isometry, ∥T ∥ = 1. Note, however, that non-isometries can also have norm 1. The backward shift operator B that showed up in the proof of Littlewood’s Theorem (§3.15) is a case in point (exercise).

(b) Multiplication operators on H^{2}. The work of §3.13 actually showed that if b ∈ H^{∞} then M_{b} is a bounded bounded operator on H^{2}, and ∥M_{b}∥ ≤ ∥b∥_{∞}. In fact there is equality here (exercise).

(c) Theorem 3.17 showed that for every holomorphic self-map φ of U, the composition operator C_{φ} is bounded on H^{2}, with

∥C_{φ}∥ ≤ (\frac{1 + |φ(0)|}{1 − |φ(0)|})^{1/2} .

There is no known formula for the norm of a general composition operator. The next result shows that all the above-mentioned operators are continuous.

3.24. Proposition. Every bounded linear transformation between normed spaces is continuous.

Proof. Suppose T : X → Y is bounded, so it satisfied (3.7) above. Then for any pair of vectors x, y ∈ X,

∥T x − T y∥ = ∥T (x − y)∥ ≤ C∥x − y∥,

which shows that T is continuous (in fact, uniformly continuous).

In fact, boundedness for a linear transformation between normed spaces is equivalent to continuity.

3.25. Theorem. Every continuous linear transformation between normed spaces is bounded.

Proof. Suppose T : X → Y is continuous. Then the inverse image of the unit ball in Y contains an open ball {x ∈ X : ∥x − x_{0}∥ < ε}, for some x_{0} ∈ X and ε > 0. Thus

∥x − x_{0}∥ < ε ⟹ ∥T x∥ < 1.

Upon setting z = (x − x_{0})/ε, so that x = εz + x_{0}, we see from the implication above that whenever ∥z∥ < 1:

1 > ∥T (εz + x_{0})∥
= ∥εT (z) + T (x_{0})∥
≥ ε∥T (z)∥ − ∥T (x_{0})∥,

22 JOEL H. SHAPIRO

hence

∥T (z)∥ ≤ ε^{−1}(1 + ∥T (x_{0})∥).

For arbitrary z ∈ X (z ≠ 0) we apply the last inequality to the unit vector z/∥z∥ to get ∥T (z)∥ ≤ C∥z∥, where C = ε^{−1}(1 + ∥T (x_{0})∥). Thus T is a bounded linear transformation.

This section closes with some further exercises on the boundedness of composition operators.

3.26. Exercise. Let ℓ^{1}(U) denote the space of absolutely summable complex sequences, but now regarded as a space of analytic functions. That is,

ℓ^{1}(U) = {f ∈ H(U) : ∥f∥_{1} =^{def} ∑_{n=0}^{∞} |f̂(n)| < ∞}

Show that the map T f = f̂ is a linear isometry of ℓ^{1}(U) onto the sequence space ℓ^{1}, and that the following version of Littlewood’s Subordination Theorem holds for ℓ^{1}(U):

Suppose φ ∈ ℓ^{1}(U) and ∥φ∥_{1} ≤ 1. Then φ is a holomorphic self-map of U, and C_{φ} is a contraction on ℓ^{1}(U).

3.27. Exercise. Show that for every 0 < α < 1/2, the function (\frac{1+z}{1−z})^{α} belongs to H^{2}.

3.28. Exercise. Suppose f and g are holomorphic on U, with g univalent and f(U) ⊂ g(U). Show that if g belongs to H^{2}, then so does f. (Hint: first show that g = f ∘ φ for some holomorphic self-map φ of U with φ(0) = 0).

3.29. Exercise. Use the results of the previous two exercises to show that if f is holomorphic on U, and f(U) is contained in an angular sector with vertex angle less than π/2 radians, then f ∈ H^{2}.

COMPOSITION OPERATORS 23

4. Introduction to the Compactness Problem

Recall that a linear transformation between normed spaces is continuous if and only if it is bounded (Proposition 3.24 and Theorem 3.25), i.e., if and only if the image of every bounded set is bounded (Exercise 3.21). The most important subclass of bounded operators are the compact ones.

4.1. Definition. A linear transformation between normed spaces is said to be compact if the image of every bounded set is relatively compact.

To say that a set is “relatively compact” means that it has compact closure. In the language of sequences this says that every sequence taken from that set has a convergent subsequence.

Since compact sets are bounded, it follows immediately that compact operators are bounded.

4.2. Exercise. Show that a linear transformation is compact if and only if the image of the unit ball, or more generally, if the image of some ball is relatively compact.

We have already noted that every linear operator on a finite dimensional Hilbert space is bounded (Exercise 3.23). Because of the Bolzano-Weierstrass theorem, every such operator is therefore compact. In fact, in this argument only the range space is important:

If a bounded linear operator on a Hilbert space has finite dimensional range, then the operator is compact.

On the other hand,

4.3. Exercise. The identity operator on an infinite dimensional Hilbert space is not compact. (Suggestion: Use the existence of an orthonormal basis to show that the closed unit ball is not compact).

The connection between compact operators and finite dimensional ones is, in fact, quite strong. One of the most important results about compact operators is this:

4.4. Theorem. If X is a Banach space and T : X → X is compact, then for each nonzero λ ∈ C, the following are equivalent:

(a) T − λI is invertible.

(b) T − λI is one to one.

(c) T − λI maps X onto itself.

This result, which we won’t prove here, asserts that perturbations of the identity by compact operators have the same invertibility characteristics as operators on a finite dimensional space.

Once you know that a collection of operators between normed spaces are all bounded, it makes sense to try to determine which of these is compact. This is a particularly interesting problem for composition operators on H^{2}. Indeed, “C_{φ} compact on H^{2} means that the operator squeezes the unit ball of H^{2} into a relatively compact (=

24 JOEL H. SHAPIRO

“small”) subset. The question is: how much does φ have to squeeze the unit disc into itself in order for this to happen?

Here are two amusing examples to get us started: If φ(z) = z then C_{φ} is the identity, which, according to Exercise 4.3, is not compact. On the other hand, if φ(z) ≡ constant then C_{φ} has one dimensional range (the constants), and is compact. To examine what lies between we need a “sequential” characterization of compactness that was first observed by H.J. Schwartz (no relation to the Schwarz Lemma) in 1969.

4.5. Theorem. C_{φ} is compact on H^{2} if and only: whenever {f_{n}} is a bounded sequence in H^{2} that converges to zero uniformly on compact subsets of U, then ∥C_{φ}f_{n}∥ → 0.

The proof of this result depends on some deep results about uniform convergence. All of this will be developed shortly, but right now perhaps a better idea would be to give some applications. The first one generalizes the fact that φ(z) ≡ z induces a noncompact operator.

4.6. Proposition. Suppose C_{φ} is compact Then the set of points e^{iθ} ∈ ∂U at which lim inf_{r→1−} |φ(e^{iθ})| = 1 has Lebesgue measure zero.

Note first that, since |φ| < 1 on U, the “liminf” in the definition of E_{φ} may be replaced by “limit”.

Proof. Suppose that E_{φ} = {e^{iθ} : lim inf_{r→1−} |φ(e^{iθ})| = 1} has positive measure. We will show that C_{φ} is not compact.

The monomial sequence {z^{n}}_{0}^{∞} obeys the hypothesis of Theorem 4.5, so by our hypothesis on C_{φ}, the image sequence, which is {φ^{}} converges to zero in H^{2}. Thus the integral representation of the H^{2} norm provided by Theorem 3.9, along with Fatou’s Lemma, yields

∥φ^{n}∥ = lim_{r→1−} \frac{1}{2π} ∫_{0}^{2π} |φ(re^{iθ})|^{2n} dθ
= \frac{1}{2π} ∫_{0}^{2π} | lim inf_{r→1−} φ(e^{iθ})|^{2n} dθ
= \frac{1}{2π} ∫_{0}^{2π} |(lim inf_{r→1−} φ(e^{iθ})|)^{2n} dθ
≥ \frac{1}{2π} ∫_{E_{φ}} |φ(e^{iθ})|^{2n} dθ
= \frac{1}{2π} meas (E_{φ}) > 0.

This calculation shows that ∥C_{φ}(z^{n})∥ does not converge to zero, so by Theorem 4.5, C_{φ} is not compact.

COMPOSITION OPERATORS 25

It is known that for every bounded analytic function, and more generally for any H^{2} function, the radial limit

f(e^{iθ}) =^{def} lim_{r→1−} f(re^{iθ})

exists for almost every e^{iθ} in the unit circle. In particular, the limit that shows up in the definition of E_{φ} in the proof above exists for a.e. point of the unit circle.

4.7. Exercise. Construct a holomorphic self-map φ of U, not the identity map, that induces a noncompact composition operator on H^{2}. (Suggestion: map the unit disc onto the upper half-plane and take a square root.)

The next application of Theorem 4.5 generalizes the fact that constant φ’s induce compact composition operators. Recall the notation ∥φ∥_{∞} = sup_{z∈U} |f(z)|.

4.8. Proposition. If ∥φ∥_{∞} < 1 then C_{φ} is compact on H^{2}.

Proof. Suppose {f_{n}} is a bounded sequence in H^{2} that converges to zero uniformly on compact subsets of U. By Theorem 4.5 it is enough to show that ∥f_{n} ∘ φ∥ → 0. But even more is true: since φ(U) is a relatively compact subset of U, f_{n} → 0 uniformly on φ(U), hence

∥f_{n} ∘ φ∥ ≤ ∥f_{n} ∘ φ∥_{∞} ≤ sup_{w∈φ(U)} |f_{n}(w)| → 0,

as desired.

Observe that we did not need the boundedness of the sequence {f_{n}} for this argument. This suggests that the sufficient condition ∥φ∥_{∞} < 1 for compactness of C_{φ} might not be necessary. We’ll see later on that this is indeed the case: there exist maps φ with ∥φ∥_{∞} = 1 for which C_{φ} is compact on H^{2}.

Before approaching the proof of Theorem 4.5 I present one final application that shows that the necessary condition of Proposition 4.6 does not provide a characterization of compact composition operators on H^{2}.

4.9. Proposition. Let φ(z) = \frac{1+z}{2}. Then C_{φ} is not compact on H^{2}.

Proof. Proof. Consider for 0α < 1/2 the functions f_{α}(z) = (1 − z)^{−α}. By Exercise 3.3 they all belong to H^{2}, and another exercise shows that ∥f_{α}∥ → ∞ as α → 1/2. Let g_{α} = f_{α}/∥f_{α}∥. Then {g_{α}} is a collection of unit vectors that converges to zero uniformly on compact subsets of U as α → 1/2.

Now each f_{α} is an eigenvector of C_{φ}, in fact C_{φ}f_{α} = 2^{α}f_{α}. Thus the same is true of g_{α}, hence as α → 1/2,

∥C_{φ}g_{α}∥ = 2^{α} → \sqrt{2} ≠ 0,

so by Theorem 4.5, C_{φ} is not compact.

26 JOEL H. SHAPIRO

4.10. Proof of Theorem 4.5: necessity. This follows quickly from Corollary 3.7. For suppose C_{φ} is compact on H^{2}, and that {f_{n}} ⊂ H^{2} is bounded and converges to zero uniformly on compact subsets of U. Then the image sequence {f_{n} ∘ φ} is relatively compact, so it has a subsequence {g_{k} = f_{n_{k}} ∘ φ} that converges in the H^{2} norm to some function g ∈ H^{2}. Our goal is to show that g = 0. Since H^{2} convergence implies uniform convergence on compact subsets of U (Corollary 3.7) we know that g_{k} → g in that manner. But f_{n_{k}} → 0 uniformly in compact subsets of U, hence the same is true of f_{n_{k}} ∘ φ = g_{k}, hence g = 0, as desired.

The converse is more difficult. To prove it we need two famous theorems of classical analysis, the first which revolves around the notion of equicontinuity. If A is a collection of functions that are continuous on a set S ⊂ C (or on any metric space, for that matter), we say S is equicontinuous if for every ε > 0 there exists δ > 0 such that whenever x, 4 is a pair of points in S with |x − y| < δ, the |f(x) − f(y)| < ε for every f ∈ A. Thus each f ∈ A is continuous on S, and for each ε the constant δ can be chosen to work for all the functions in A.

4.11. The Arzela–Ascoli Theorem. If K is a compact subset of the plane and A ⊂ C(K) is uniformly bounded and equicontinuous, then every sequence in A has a uniformly convergent subsequence.

Proof. Since K is compact, for each positive integer n there is a finite collection F_{n} of open discs of radius 1/n, with centers in K, that covers K. Let S denote the collection of all the centers of all the discs in all the open covers F_{n}. This is a countable dense subset of K.

Now consider a sequence {f_{n}} in A. Because of the boundedness assumption on A we know that there exists M > 0 such that |f_{n}(z)| ≤ M for each z ∈ K and each n. Since bounded sequences of complex numbers have convergent subsequences, a diagonal argument produces a subsequence g_{k} = f_{n_{k}} that converges pointwise on S. I claim that {g_{k}} is uniformly Cauchy on K, i.e. a Cauchy sequence in the Banach space C(K).

For this, let ε > 0 be given. Use equicontinuity to choose a positive integer N such that if z, w ∈ K and |z − w| < 1/N, then |f(z) − f(w)| < ε/3 for each f ∈ A. Next, use the pointwise convergence of {g_{k}} on S to choose n_{ε} such that whenever k and j are integers > n_{ε} and a is the center of a disc in F_{N} (there are only finitely many discs here!), we have |g_{k}(a) − g_{j}(a)| < ε/3. I claim that if j, k > N_{ε} then for every z ∈ K we have |g_{k}(z) − g_{j}(z)| < ε, which will prove the desired result: {g_{k}} is uniformly Cauchy on K.

For this, fix j, k > N_{ε}, and fix a point z ∈ K. Since F_{n} is a cover of K there is a disc Δ ∈ F_{n} that contains z. Let a be the center of Δ, so that |z − a| < 1/N. Then

|g_{k}(z) − g_{j}(z)| ≤ |g_{k}(z) − g_{k}(a)| + |g_{k}(a) − g_{j}(a)| + |g_{j}(a) − g_{j}(z)|
< \frac{ε}{3} + \frac{ε}{3} + \frac{ε}{3}
= ε,

COMPOSITION OPERATORS 27

where in the second line the first and last inequality come from the proximity of z to a, and and the middle one from the choice of n_{ε}.

4.12. Exercise. Show that the converse of Theorem 4.11 holds also: If every sequence in A has a convergent subsequence, then A is uniformly bounded and equicontinuous. (Suggestion: The hypothesis really asserts that A is relatively compact in C(K). An equivalent formulation of relative compactness thus, for every ε > 0 there is a finite covering of A by balls of radius ε with centers in A. These centers are the key.)

4.13. Theorem. Suppose A ⊂ H(U) is uniformly bounded on a compact subset K of U. Then A is equicontinuous on K.

Proof. Choose 0 < r < 1 so that K ⊂ {|z| < r}, and let γ be the circle {|z| = r}, oriented positively, and let

d = dist (K, γ) =^{def} inf{|w − z| : w ∈ K and z ∈ γ}.

Since K and γ are disjoint and compact, d > 0.

Since γ is compact we know that

M =^{def} sup{|f(z)| : z ∈ γ, f ∈ A} < ∞.

Fix z_{1} and z_{2} in K, and fix f ∈ A. The Cauchy integral formula, along with a little bit of algebra shows that

f(z_{1}) − f(z_{2}) = \frac{z_{1} − z_{2}}{2πi} ∫_{γ} \frac{f(z)}{(z − z_{1})(z − z_{2})} dz

from which it follows quickly that

|f(z_{1}) − f(z_{2})| ≤ \frac{Mr}{d^{2}}|z_{1} − z_{2}|.

Since the constants M,r, and d do not depend on either f ∈ A or z_{1}, z_{2} ∈ K, this shows that A is equicontinuous on K.

4.14. The Osgood-Stieltjes Theorem. If A ⊂ H(U) is uniformly bounded on every compact subset of U, then every sequence in A has a subsequence that converges uniformly on compact subsets of U.

Proof. This is a standard diagonal argument. Let Δ_{j} be the closed disc of radius 1− \frac{1}{j}, centered at the origin, so U is the union of all these subdiscs. Suppose \{f_{n}\} is a sequence of functions in A. By Theorems 4.13 and 4.11 there is a subsequence {f_{1,n}} of {f_{n}} that converges uniformly on Δ_{1}. Similarly there exists a subsequence {f_{2,n}} of {f_{1,n}} that converges uniformly on Δ_{2}. Continuing in this manner we obtain an infinite matrix {f_{m,n}} of functions where:

• Each row is a subsequence of the previous row.

• The first row is a subsequence of the original sequence {f_{n}}.

• The n-th row converges uniformly on Δ_{n}.

28 JOEL H. SHAPIRO

The “diagonal sequence” {f_{n,n}} is therefore a subsequence {f_{n}} that converges uniformly on each of the closed discs Δ_{j}, and therefore uniformly on every compact subset of U.

4.15. Proof of Theorem 4.5: sufficiency. Suppose C_{φ} takes every sequence that is bounded in H^{2} and convergent uniformly on compact subsets of U into a sequence that converges to zero in H^{2}. Our goal is to show that the image of the unit ball of H^{2} is relatively compact.

Fix a sequence {f_{n}} in the closed unit ball B of H^{2}. According to the Growth Estimate §3.6, B is bounded uniformly on compact subsets of U, so by Theorem 4.14 the sequence {f_{n}} has a subsequence g_{k} = f_{n_{k}} that converges uniformly on compact subsets of U to a function g ∈ B. Thus {g_{k} − g} ≤ 2 for every k, and g_{k} → uniformly on compact sets, hence the hypothesis on C_{φ} implies that ∥g_{k} ∘ φ − g ∘ φ∥ → 0. This shows that every sequence in C_{φ}(B) has a convergent subsequence, i.e., that C_{φ}(B) is relatively compact. This completes the proof.