Isolation and Component Structure in Spaces of Composition Operators

Christopher Hammond Barbara D. MacCluer

October 24, 2003

Let B_{N} = {z ∈ ℂ^{N} : |z| < 1}, with 𝔻 = B_{1}. Let σ denote normalized Lebesgue measure on ∂B_{N}.

For any 0 < p < ∞, the Hardy space H^{p}(B_{N}) is the set of all analytic functions f : B_{N} → ℂ such that

∥f∥_{p} := ( sup_{0<r<1} ∫_{∂B_{N}} |f_{r}|^{p} dσ)^{1/p} < ∞,

where f_{r}(z) = f(rz).

The space H^{∞}(B_{N}) is the set of bounded analytic functions on B_{N}, with

∥f∥_{∞} := sup_{w∈B_{N}} |f(w)|.

Let φ : B_{N} → B_{N} be analytic. The composition operator C_{φ} on H^{p}(B_{N}) is defined by the rule

C_{φ}(f) = f ∘ φ.

Whenever N = 1 or p = ∞, every analytic φ : B_{N} → B_{N} induces a bounded composition operator from H^{p}(B_{N}) into H^{q}(B_{N}) for all q ≤ p.

Let C(H^{p}(B_{N}), H^{q}(B_{N})) denote the set of (bounded) composition operators taking H^{p}(B_{N}) into H^{q}(B_{N}), with

C(H^{p}(B_{N})) = C(H^{p}(B_{N}), H^{p}(B_{N})).

We would like to know about the component structure of C(H^{p}(B_{N}), H^{q}(B_{N})).

The extreme set of φ : B_{N} → B_{N} is

{ζ ∈ ∂B_{N} : |φ^{∗}(ζ)| = 1},

where φ^{∗}(ζ) := lim_{r↑1} φ(rζ). Consider 1 ≤ p < ∞. For φ ≠ ψ, Berkson [1981] showed that ∥(C_{φ} − C_{ψ}) : H^{p}(𝔻) → H^{p}(𝔻)∥ ≥ [σ(E)/2] ^{1/p} , where E denotes the extreme set of φ.

Shapiro and Sundberg [1990] showed that

∥(C_{φ} − C_{ψ}) : H^{2}(𝔻) → H^{2}(𝔻)∥ _{e} ≥ [σ(E) + σ(F)] ^{1/2} ,

where E denotes the extreme set of φ and F the extreme set of ψ. Their result was extended by Heidler [1998] to C(H^{2}(B_{N})).

Our goal: obtain a similar result that holds for C(H^{p}(B_{N}), H^{q}(B_{N})), for N ≥ 1 and general values of p and q.

Proposition 1. Take p and q to be finite indices. Let T be a bounded operator from H^{p}(B_{N}) into H^{q}(B_{N}); then

∥T : H^{p}(B_{N}) → H^{q}(B_{N})∥ _{e} ≥ lim sup_{n→∞} ∥T (g^{n})∥_{q},

where g is any nonconstant element of H^{∞}(B_{N}) with ∥g∥_{∞} ≤ 1.

Proposition 2. Take q to be a finite index. Let T be a finite linear combination of composition operators; then

∥T : H^{∞}(B_{N}) → H^{q}(B_{N})∥ _{e} ≥ \frac{1}{2} lim sup_{n→∞} ∥T (g^{n})∥_{q},

where g is any nonconstant element of H^{∞}(B_{N}) with ∥g∥_{∞} ≤ 1.

Given a positive measure μ on ∂B_{N} and a number δ > 0, we write Y_{δ}(μ) to denote the set of all nonconstant functions g in H^{∞}(B_{N}) such that

(i) g is continuous on \overline{B_{N}},
(ii) |g(ζ)| ≤ 1 for all ζ on ∂B_{N}, and
(iii) μ({|g(ζ)| = 1}) ≥ μ(∂B_{N}) − δ.

For any μ and any δ, the set Y_{δ}(μ) is nonempty. We call the elements of a particular set Y_{δ}(μ) approximate inner functions.

Let φ : B_{N} → B_{N} be an analytic map with extreme set E; consider φ^{∗} : E → ∂B_{N}. We define the pullback measure σφ^{∗ −1} on ∂B_{N} by setting

σφ^{∗ −1}(A) = σ(φ^{∗ −1}(A))

for any σ–measurable subset A of ∂B_{N}, so that

∫_{E} f ∘ φ^{∗} dσ = ∫_{∂B_{N}} f d(σφ^{∗ −1})

for all f in L^{1}(σ).

Observe that σφ^{∗ −1}(∂B_{N}) = σ(E).

Proposition 3. Let φ and ψ be distinct analytic self-maps of B_{N}. For any ε > 0, there is a nonconstant unit vector g in H^{∞}(B_{N}) such that

lim sup_{n→∞} ∥C_{φ}(g^{n}) − C_{ψ}(g^{n})∥ _{2}^{2} ≥ σ(E) − ε,

where E denotes the extreme set of φ.

Main idea. Take g to be an element of Y_{δ}(σφ^{∗ −1}), where δ = ε/2. Note that

∫ |(C_{φ}(g^{n}) − C_{ψ}(g^{n})) ^{∗}|^{2} dσ
= ∫ |(g^{n} ∘ φ)^{∗} − (g^{n} ∘ ψ)^{∗}|^{2} dσ
≥ ∫ |(g^{n} ∘ φ)^{∗}|^{2} dσ − 2 Re ∫ (g^{n} ∘ φ)^{∗} \overline{(g^{n} ∘ ψ)^{∗}} dσ

for any natural number n. Take the lim sup as n goes to ∞.

Proposition 4. Take q to be a finite index. Let φ and ψ be distinct analytic self-maps of B_{N}. Suppose that σ(E) > 0, where E denotes the extreme set of φ; then, for any ε with 0 < ε < σ(E), there is a nonconstant unit vector g in H^{∞}(B_{N}) such that

lim sup_{n→∞} ∥C_{φ}(g^{n}) − C_{ψ}(g^{n})∥ _{q}^{q} ≥ b(q) [σ(E) − ε]^{c(q)} ,

where

b(q) = \{ \begin{matrix}1/2, & q < 2 \\ 1, & q ≥ 2 \end{matrix}

and

c(q) = \{ \begin{matrix}1, & q ≤ 2 \\ q/2, & q > 2 \end{matrix} .

Theorem 5. Take p and q to be finite indices. Let φ and ψ be distinct analytic self-maps of B_{N} that induce bounded composition operators from H^{p}(B_{N}) into H^{q}(B_{N}); then

∥(C_{φ} − C_{ψ}) : H^{p}(B_{N}) → H^{q}(B_{N})∥ _{e}
≥ (b(q) [σ(E)]^{c(q)} )^{1/q}  ,

where E denotes the extreme set of φ. Similarly,

∥(C_{φ} − C_{ψ}) : H^{∞}(B_{N}) → H^{q}(B_{N})∥ _{e}
≥ \frac{1}{2} (b(q) [σ(E)]^{c(q)} )^{1/q}

for any 0 < q < ∞.

Proof. In light of Proposition 4, we appeal to Propositions 1 and 2 and let ε tend to 0.

Proposition 6. Take 0 < q < p ≤ ∞. Let φ be an analytic self-map of B_{N} that induces a bounded composition operator from H^{p}(B_{N}) into H^{q}(B_{N}). Suppose that σ(E) = 0, where E denotes the extreme set of φ; then C_{φ} : H^{p}(B_{N}) → H^{q}(B_{N}) is compact as long as at least one of the following three conditions holds:

\begin{matrix}(i) & N = 1, \\ (ii) & p = ∞, \\ (iii) & q = 1. \end{matrix}

Proposition 7. Take 0 < p ≤ ∞ and 0 < q < ∞. The compact composition operators form a path-connected set in C(H^{p}(B_{N}), H^{q}(B_{N})).

Theorem 8. Take 0 < q < p ≤ ∞ and suppose that either N = 1, p = ∞, or q = 1. Let φ and ψ be distinct analytic self-maps of B_{N} that induce bounded composition operators from H^{p}(B_{N}) into H^{q}(B_{N}); then the following six conditions are equivalent:

(1) Both φ and ψ have extreme sets with

σ–measure 0.

(2) Both C_{φ} : H^{p}(B_{N}) → H^{q}(B_{N}) and

C_{ψ} : H^{p}(B_{N}) → H^{q}(B_{N}) are compact.

(3) C_{φ} and C_{ψ} belong to the same

path component of C(H^{p}(B_{N}), H^{q}(B_{N})).

(4) C_{φ} and C_{ψ} belong to the same

component of C(H^{p}(B_{N}), H^{q}(B_{N})).

(5) (C_{φ} − C_{ψ}) : H^{p}(B_{N}) → H^{q}(B_{N}) is compact.
(6) The equivalence classes containing C_{φ}

and C_{ψ} belong to the same component of the quotient space Q(H^{p}(B_{N}), H^{q}(B_{N})).

Corollary 9. Take 0 < q < p ≤ ∞ and suppose that either N = 1, p = ∞, or q = 1. The set of compact operators forms a single component in C(H^{p}(B_{N}), H^{q}(B_{N})); every noncompact operator constitutes its own component.

Corollary 10. Take 0 < q < p ≤ ∞ and suppose that either N = 1, p = ∞, or q = 1. The quotient space Q(H^{p}(B_{N}), H^{q}(B_{N})) is totally disconnected.