The Norm of a Composition Operator with Linear Symbol Acting on the Dirichlet Space

Christopher Hammond Connecticut College

April 9, 2005

Let 𝔻 = {z ∈ ℂ : |z| < 1}. Let dA denote normalized area measure on 𝔻.

Let H be a Hilbert space of analytic functions on 𝔻.

Let φ : 𝔻 → 𝔻 be analytic. We define the composition operator C_{φ} on H by the rule

C_{φ}(f) = f ∘ φ.

If C_{φ} takes H into H, then C_{φ} : H → H is bounded.

Every analytic φ : 𝔻 → 𝔻 induces a bounded composition operator on H^{2} or A_{α}^{2} for α > −1.

The Dirichlet space D is the set of all analytic functions f on 𝔻 for which

∥f∥_{D}^{2} := |f(0)| ^{2} + ∫_{𝔻} |f′(z)|^{2} dA(z) < ∞.

Every univalent φ induces a bounded composition operator on D.

Our strategy for determining ∥C_{φ}∥ is to study the spectrum of C_{φ}^{∗}C_{φ}.

Justification:

Let T be a bounded operator on a Hilbert space H. We know that:

• The spectral radius of T^{∗}T equals ∥T^{∗}T ∥ = ∥T ∥^{2}.

• Let h ∈ H; then ∥T (h)∥ = ∥T ∥ ∥h∥ if and only if (T^{∗}T)(h) = ∥T ∥^{2} h.

• If ∥T ∥_{e} < ∥T ∥, then T is norm-attaining.

It is difficult to find a concrete representation for C_{φ}^{∗}.

Cowen [1988] obtained a simple formula for C_{φ}^{∗} : H^{2} → H^{2} when φ is linear fractional. Using this, he showed that ∥C_{φ}∥^{2} equals

\frac{2}{1 + |a|^{2} − |b|^{2} + √(1 − |a|^{2} + |b|^{2}) ^{2} − 4 |b|^{2}}

when φ(z) = az+b. These results were adapted to A_{α}^{2} by Hurst [1997].

Cowen’s adjoint formula was used to find the norm of C_{φ} : H^{2} → H^{2}, for certain linear fractional φ, by [H 2003] and [Bourdon, Fry, H, Spofford 2004].

Let

φ(z) = \frac{az + b}{cz + d} ,

and

σ(z) = \frac{\overline{a}z − \overline{c}}{−\overline{b}z + \overline{d}} .

Gallardo-Gutiérrez and Montes-Rodríguez [2003] showed that C_{φ}^{∗} : D → D can be written

(C_{φ}^{∗}f) (z) = f(σ(z)) + f(0)K_{φ(0)} (z) − f(σ(0)).

Thus

(C_{φ}^{∗}C_{φ}f) (z) = f(τ(z))+f(φ(0))K_{φ(0)} (z)−f(τ(0)),

where τ = φ ∘ σ.

Facts:

(1) Any C_{φ} : D → D, where φ(0) ≠ 0, has norm greater than 1.

(2) Any C_{φ} : D → D, where φ is univalent, has essential norm no greater than 1.

Thus any univalent φ with φ(0) ≠ 0 induces a norm-attaining composition operator on D; that is, ∥C_{φ}∥^{2} is an eigenvalue of C_{φ}^{∗}C_{φ}.

Let φ : 𝔻 → 𝔻 be a linear fractional, with φ(0) ≠ 0. Let λ be an eigenvalue for C_{φ}^{∗}C_{φ}, with eigenfunction g. Then

λg(z) = g(τ(z)) + g(φ(0))K_{φ(0)} (z) − g(τ(0)).

Applying induction, we see that λ^{n}g(z) equals

g(τ_{n}(z))+ n ; ∑ ; j=1 λ^{n−j}[g(φ(0))K_{φ(0)} (τ_{j−1}(z))−g(τ(0))],

where τ_{j} = \underbrace{τ ∘ τ ∘ … ∘ τ}_{j times}.

Take λ > 1. Letting n → ∞, we obtain

g(z) = ∞ ; ∑ ; j=1 (\frac{1}{λ})^{j} [g(φ(0))K_{φ(0)} (τ_{j−1}(z))−g(τ(0))].

Let φ(z) = az + b, with a ≠ 0, b ≠ 0, and |a| + |b| ≤ 1. Note that φ(0) = τ(0).

Then

g(z) = g(τ(0)) ∞ ; ∑ ; j=1 (\frac{1}{λ})^{j} [K_{τ(0)} (τ_{j−1}(z)) − 1]

= g(τ(0)) ∞ ; ∑ ; j=1 (\frac{1}{λ})^{j} log (1/(1 − \overline{b}τ_{j−1}(z))).

Thus λ = ∥C_{φ}∥^{2} is a solution to the equation

1 = ∞ ; ∑ ; j=1 (\frac{1}{λ})^{j} log (\frac{1}{1 − \overline{b}τ_{j}(0)} ).

Furthermore, one can show that this equation has only one (positive real) solution.

If |a|+|b| = 1, then this equation can be rewritten

1 = ∞ ; ∑ ; j=1 (\frac{1}{λ})^{j} log (\frac{1 + (j − 1) |b|}{(1 − |b|) (1 + j |b|)} ).

Take φ(z) = (1/2) z + 1/2; observe that

1 = ∞ ; ∑ ; j=1 (\frac{1}{λ})^{j} log (\frac{2j + 2}{j + 2} ).

Using numerical methods, we see that ∥C_{φ}∥ = √λ ≈ 1.195830076.

In general, one can use our norm representation to obtain the following estimate:

Let φ(z) = az + b, with a ≠ 0, b ≠ 0, and |a| + |b| ≤ 1; then

∥C_{φ}∥ ≤ ; ; ; √1 + log (\frac{1}{1 − \overline{b}w_{0}} ) = √1 + log (\frac{r}{R}) ,

• where w_{0} denotes the Denjoy–Wolff point of τ;

• where R denotes the Euclidean radius of φ(𝔻);

• where r denotes the pseudohyperbolic radius of φ(𝔻).