After developing some of the fundamental properties of H2, we show here that every composition operator acts boundedly on Hilbert space. The setting is the simplest in accordance with serious "functional-theoretic operator theory": The unit disk U of the complex plane and the HilbertH2 space of holomorphic functions on U with the coefficients of the summed squared power series.
Some Essential Definition
We denote the set of all such maps by LFT(ˆC), where the notation is intended to draw attention to the fact that, with the obvious convention about the point at∞, every linear fractional transformation as a one-to- holomorphic mapping of the Riemann sphere ˆConto itself. The utility of matrices in dealing with linear fractional transformations comes from the fact that.
Fixed Points
Derivatives at the Fixed Points
Classification
Then the trigonometric series, . ancosnx+bnsinnx) is called the Fourier series corresponding to the function f(x), where. An important consequence of the uniformly convergent theorem is ” if the Fourier series of f converges uniformly to f, then f must be continuous on [−π, π] withf(π) =f(−π).
Dirichlet Kernel and its properties
Some Important Theorem Related to Composition Operators
Each summated square sequence {fˆ(n)} = {fˆ(0),fˆ(1),fˆ(2), ..} of complex numbers is the coefficient sequence of a function H2; if {an}∞n=0 is a summed square, then it is bounded, so the corresponding power seriesP∞. By the uniqueness theorem for power series, the map connecting the function f with the sequence {fˆ(n)} is therefore a vector isomorphism of H2ontol2. Now we can turn H2 into a Hilbert space by declaring the map as an isometry.
Now by applying the Cauchy-Schwarz inequality to this power series representation of f, we obtain for Eachz∈U,. Every norm-convergent sequence in H2 converges (to the same limit) uniformly on a compact subset of U. Proof: Suppose {fn} is a sequence inH2norm-convergent to a functionf ∈H2, i.e.
H 2 via Integrals Means
It is also easy to see from the definition that H2 contains some unbounded functions. So by sending r to 1, we see that every subsum of the series for||f||2 is bounded by M2. To test the usefulness of these results, let us return to the two important facts that linger at the end of the last paragraph.
Littlewood’s Subordination Principle
Proof of Littlewood’s Theorem
Automorphism-induced composition operators: To prove that the Cϕ is bounded even when ϕ does not determine the origin, we use the conformal automorphisms to move points of U from where they are to where we want them to be. We have just seen that Cψ is bounded (in fact a contraction, of the Littlewood's subordination principle), and we know that the product of two bounded operators is always bounded. Relatively compact: A subset of a topological space is said to be relatively compact if its closure in the space is compact.
A linear operator T on a Hilbert space H is said to be compact if it maps every bounded set to a relatively compact one. A subset K of a metric space X is relatively compact if for every ≥0 there is a finite set of points N⊂X such that every point in K is at most distance N.
First Class of Example
The set N is often called "completely bounded." So a set is completely bounded if it is relatively compact. If K is relatively compact, N is obtained by covering the closure of K with open spheres, eliminating the finite subcovering, and choosing the centers as N. Otherwise, if we have N, then for every open covering of a closure K, there is a finite covering subordinate to the original one, from which it follows that this closure is compact.
From our comparison of the norms H2 and H∞, Tn is therefore a rank-bounded operator, bounded on H2, and we can obtain||Tn||. This shows that Cφ is a limit of the operator norm of operators of finite rank, so it is compact in H2.
A Better Compactness Theorem
The title of the Theorem above comes from the fact that its proof shows Cφ to be a Hilbert-Schmidt operator wheneverφ satisfies (3). The Hilbert-Schmidt condition (2) does not depend on the specific choice of orthonormal basis, shows that our proof actually characterizes the Hilbert-Schmidt composition operators as those for which φ satisfies condition (3). To begin the proof, suppose that φ maps the unit disk into one side of the lens Lα, i.e. This shows that anything that maps the unit disk into a lens has induced a Hilbert-Schmidt operator.
Each of these φ extends to a homeomorphism from the closed disk to the closure of the polygon. The inverse of the function on the right is therefore integrable over an interval that is symmetric about θ = 0.
Compactness and Weak Convergence
Now for the general case, the factorization argument above shows that it suffices to consider maps φ that take the unit disk conformally to polygons inscribed on the unit circle. Consider a vertex of the polygon, which, without loss of generality, we can assume is the point +1. Thus, the mapχ= (1 +φ)/2 fixes +1 and takes the disc in a lens Lα for some a sufficiently close to 1, so by the work of the last paragraph the function (1− |χ(eiθ)|2) −1 is integrable over the unit circle.
The function (1− |φ(eiθ)|2)−1 is therefore integrable in the interval centered around the preimage of each vertex of the polygon, i.e. it is integrable in the entire unit circle. With Montel's theorem, we can say∃subsequence{gk=fnk} which converges uniformly on compact subsets of U to the holomorphic function g.
Non-Compact Composition Operators
If a map induces a non-compact operator, then any map whose values approach the unit circle. Corollary: Suppose φ is a univalent eigenmap of U, and that φ(U) contains a disk in U that is tangent to the unit circle. Proof: We can assume, without loss of generality, that the disc (say ∆) is tangent to the unit circle at +1.
If φ(z) approaches the unit circle 'closely', it is not compact since a univalent map contains a disk tangent to a unit circle at U. All these results suggest that Cφ is compact if and only if φ( z) does not approach the unit circle very often.
The H 2 -Norm via Area Integrals
The Theorem
But the method provides what we need most: the "big-oh" condition that stands behind Littlewood's Theorem; it is nothing but the Schwarz Lemma, disguised in the form. According to the Folk Wisdom shared at the beginning of this chapter, the corresponding "small-o" condition should tell a lot about compactness. This intuition is confirmed by the following result, which is the main result of this chapter.
The Univalent Compactness Theorem
Proof of sufficiency
The more subtle part of the theorem is the proof that condition (2) is necessary for compactness.
The Adjoint Operator
All the results developed in this section now show that the adjoint operation preserves compactness. By the approximation theorem, we see that there exists a sequenceFn of bounded finite rank operators such that ||T −Fn|| −→ O. Adjoint composition operators and reproducing kernels: Our second calculation involves the adjoint of a composition operator.
Although there is no good description of the adjoint that works for all composition operators on all H2 functions, we can always account for its action on a special important family of functions on H2: reproductive kernels. It is called the reproductive kernel for the point p, and takes its name from the fact that for each f ∈H2,.
Proof of Necessity
Compactness and Contact
We will construct univalent self-maps of the unit disk by working in the right half-plane Π instead, and then returning to the disk by changing the variable w=τ(z) = 1+z1−z. This result, together with Lemma 2 above, shows that the limit of φ(U) is an α-curve at +1. c) To prove that Cφ is not compact on H2, it suffices, according to the Univalent Compactness Theorem of the last chapter, to show that asx−→1 (in the unit interval) we have. By the principle of comparison, it is sufficient to prove that the Riemann map of the unit disk on the region bounded by this α-curve induces a non-compact composition operator.
Furthermore, f0 has a positive real part in ∆\{0}, so as in the proof of the proposition, f is covalent in . These examples raise the possibility of restating the results of the last chapter in terms of the angular derivative.
The Julia-Carath´ eodory Theorem
The "necessary" part of the program follows immediately from the definitions and, as before, does not require uniqueness. To appreciate the pure function-theoretic power of the Julius-Caratheodory theorem, note how, almost a posteriori, he claims that if on a sequence zn of points in U converging to a limit point w, the images φ(zn) tend to the limit fast enough, then must have a functionφ no matter how sparse or tangential the radial boundary atw is. If a smooth curve in U ends at a point w ∈ ∂U, in which it subtends an angle α < π/2 with a radius to this point, then the same applies to the image curve at the limit point η.”.
In particular, the image of the radius itself meets the unit circle perpendicularly, and if two non-tangential curves intersect at an angle, then their images intersect at the same angle. Due to our requirement that the angular derivative can only exist at points whose (radial) example is on the unit circle, it also does not exist at any of the other points, so Cφ is compact.
The Invariant Schwarz Lemma
This formula will prove very useful in our further study of the pseudohyperbolic distance. With the pseudohyperbolic distance in our corner, we can state a simple but far-reaching generalization of Schwarz's lemma. This form of Schwarz's lemma asserts that holomorphic self-mappings of U that are not automorphisms strictly minimize all pseudohyperbolic distances.
Since ∆(p, r) is the image of a disk U under a conformal automorphism of U, it is a regular open disk. If p= 0, the claim is that φ(rU)⊂rU, which is of course just a geometric interpretation of the original Schwarz lemma.
A Boundary Schwarz Lemma
Here the lim sup of a set of sets is the set of points belonging to infinitely many sets, and the corresponding lim inf is the set of points belonging to all sets from some index onwards. Now we are in a position to use the Disc Convergence Lemma to obtain a limiting version of the Invariant Schwarz Lemma. Note that if φ were to fix the origin, the Schwarz lemma would immediately tell us that δ ≥1.
The same idea works when we apply the Invariant Schwarz Lemma with q= 0, and it yields. Note: Now using this last corollary and Julia Theorem, we can prove that the remaining two parts of the Julia-Caratheodory Theorem.
