Spectra of weighted composition operators with automorphic symbols - - PowerPoint PPT Presentation

Spectra of weighted composition operators with automorphic symbols

Mikael Lindstr¨

• m

Department of Mathematical Sciences University of Oulu Joint work with Olli Hyv¨ arinen, Ilmari Nieminen and Erno Saukko Gothenburg, July/August, 2013

We denote by H(D) the space of analytic functions on the open unit disc D of the complex plane. For ϕ an analytic selfmap of D and u in H(D), the weighted composition operator uCϕ is defined by

(uCϕ)f(z) = u(z)f(ϕ(z)),

where f ∈ H(D). There are two particularly interesting special cases of such operators: on one hand, taking u = 1 gives the composition operator Cϕ, and on the other, putting

ϕ = id, the identity function on D, gives us the

multiplication operator Mu.

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The study of composition operators on analytic function spaces was started about 40 years ago by Nordgren, Kamowitz, Shapiro, Cowen and MacCluer and others. There are at least two main goals by studying weighted composition operators.

• Relate properties of the functions u, ϕ to properties
• f the operators Cϕ and uCϕ.
• Relate properties of the operators Cϕ and uCϕ to

properties of other operators, in order to better understand other operators.

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For invertible uCϕ, we aim to

• calculate the spectral radius of uCϕ; and
• determine its spectrum.

The methods of proof should be general, so that the results hold on many large spaces of analytic

• functions. In fact, we have designed a unified

approach to determine the spectra of invertible weighted composition operators on a broad class A of analytic function spaces. The spectrum of uCϕ was studied by Gunatillake (2011) on the Hardy-Hilbert space H2(D) and by Kamowitz (1978) on the disc algebra A(D).

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Spectral properties of uCϕ depend on the type of the symbol ϕ. These types are:

• elliptic - ϕ has fixed point in D;
• parabolic - ϕ has a unique fixed point in ∂D;
• hyperbolic - ϕ has two fixed point in ∂D.

The hyperbolic case is the most interesting one. Fact (1987, Nordgren, Rosenthal, Wintrobe) : every linear bounded operator T has a closed non-trivial invariant subspace ⇔ the minimal non-trivial closed invariant subspaces for Cϕ on H2(D) are

• ne-dimensional, where ϕ is a hyperbolic

automorphism of D.

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Let A ⊂ H(D) be a Banach space; its norm is denoted by || · ||A. Suppose there is a constant s > 0 such that the following hold:

1) For each f ∈ A and z ∈ D we have |f(z)| ||f||A(1 − |z|2)−s. 2) For each z ∈ D there is some fz ∈ A with ||fz||A ≤ 1 such that fz(z)(1 − |z|2)s = 1. 3) For ϕ an automorphism, ||Cϕ|| (1 − |ϕ(0)|2)−s. 4) For w ∈ H∞(D) we have ||Mw|| ≤ ||w||∞.

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The following spaces satisfy all the stated conditions.

• Hardy spaces Hp(D), 1 ≤ p < ∞, with s = 1

p.

• Weighted Bergman spaces Ap

α(D), p ≥ 1, α > −1,

consisting of all analytic functions f on D such that

• D

|f(z)|p(1 − |z|)αdA(z) < ∞.

Here s = (α + 2)/p.

• The weighted Banach spaces of analytic

functions H∞

p (D), 0 < p < ∞, consisting of analytic

functions f on D with

sup

z∈D

|f(z)|(1 − |z|)p < ∞.

Here s = p.

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Theorem 1. For such A and ϕ an analytic selfmap of

D, the weighted composition operator uCϕ is

Fredholm on A if and only if Mu is Fredholm and ϕ is an automorphism of the unit disc. In the rest of the talk we will assume that

uCϕ : A → A is invertible and therefore ϕ is an

• automorphism. We have:

Theorem 2. For such A and an automorphism ϕ the

• perator uCϕ : A → A is invertible if and only if u is

bounded and bounded away from zero. The inverse is given by

(uCϕ)−1 = 1 u ◦ ϕ−1Cϕ−1.

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Concerning the boundary fixed points of ϕ, one result

• f particular importance is the celebrated Denjoy-Wolff

theorem. If ϕ is either a parabolic or a hyperbolic automorphism of D, this theorem guarantees that there is a (unique) fixed point a ∈ ∂D such that

lim

n→∞ ϕn(z) = a

uniformly on compact subsets of D; the point a is called the Denjoy-Wolff point of ϕ. Moreover,

• if ϕ is parabolic, then ϕ′(a) = 1; and
• if ϕ is hyperbolic and its other fixed point b ∈ ∂D,

then 0 < ϕ′(a) < 1 and ϕ′(a) = 1/ϕ′(b).

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The parabolic case

Theorem 3. Let A be any of the spaces Ap

α(D), p ≥ 1, α > −1, and

s = α+2

p ; Hp(D), p ≥ 1, and s = 1 p; or H∞ p (D), 0 < p < ∞, and

s = p. Suppose that u ∈ A(D) is bounded away from zero on D and

let ϕ be a parabolic automorphism of D whose Denjoy-Wolff point is

a ∈ ∂D. Then the spectral radius r(uCϕ) = |u(a)|

and the spectrum σA(uCϕ) is the circle

σA(uCϕ) = {λ ∈ C; |λ| = |u(a)|}.

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The hyperbolic case

Theorem 4. Let A be any of the spaces Ap

α(D), p ≥ 1, α > −1, and

s = α+2

p ; Hp(D), p ≥ 1, and s = 1 p; or H∞ p (D), 0 < p < ∞, and

s = p. Let ϕ be a hyperbolic automorphism of D with fixed points a

(attractive) and b (repulsive) in ∂D. If u ∈ A(D) is bounded away from zero, then

r(uCϕ) = max

• |u(b)|

ϕ′(b)s, |u(a)| ϕ′(a)s

• and if, moreover, |u(b)/ϕ′(b)s| ≤ |u(a)/ϕ′(a)s|, then

σA(uCϕ) =

• λ ∈ C; |u(b)|

ϕ′(b)s ≤ |λ| ≤ |u(a)| ϕ′(a)s

• .

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The elliptic case

Theorem 5. Let A be any of the spaces Ap

α(D), p ≥ 1, α > −1, and

s = α+2

p ; Hp(D), p ≥ 1, and s = 1 p; or H∞ p (D), 0 < p < ∞, and

s = p. (a) Suppose that u ∈ A(D) and ϕ is an automorphism of D such

that there is a positive integer j with ϕj(z) = z for all z ∈ D. If n is the smallest such integer, then

σA(uCϕ) = {λ ∈ C; λn = u(n)(z), z ∈ D},

where u(n) = n−1

m=0 u ◦ ϕm ∈ H(D).

(b) Suppose u ∈ A(D) is bounded away from zero on D and let ϕ be

an elliptic automorphism such that ϕn(z) ≡ z for all positive integers

• n. If a ∈ D is the unique fixed point of ϕ, then

σA(uCϕ) = {λ ∈ C; |λ| = |u(a)|}.

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Let us now consider two examples in which we determine the spectrum of a weighted composition operator uCϕ.

(1) Let uCϕ be a unitary weighted composition operator on H2(D),

where ϕ is an automorphism. Then s = 1/2. Since uCϕ is unitary, it can be shown that

u(z) = c

• 1 − |z0|2

1 − z0z ,

where ϕ(z0) = 0 and |c| = 1. Moreover, r(uCϕ) = 1 and

σ(uCϕ) ⊆ ∂D. (a) Suppose that ϕ(z) = µz, where |µ| = 1. Then u(z) = c. If µj = 1 for some positive integer j, then if n is the smallest such

integer, the previous Theorem 5 gives

σH2(D)(uCϕ) = {λ; λn = cn} = {µkc; k = 0, 1, . . . , n − 1}.

If there is no such integer, we get σH2(D)(uCϕ) = {λ; |λ| = 1}. 13

(b) Let ϕ(z) = (1+i)z−1

z+i−1 , that is, ϕ is a parabolic automorphism of D.

Then z0 =

1 1+i. By Theorem 3 r(uCϕ) = |u(1)| = 1 and the

spectrum

σH2(D)(uCϕ) = {λ; |λ| = 1}. (c) Consider the hyperbolic automorphism ϕ(z) = z+r

1+rz where

0 < r < 1, that is, ϕ is a hyperbolic automorphism of D with

Denjoy-Wolff point a = 1, and the other fixed point b = −1. Then

z0 = −r. In this case |u(−1)| > |u(1)|, and |u(1)| ϕ′(1)1/2 = |u(−1)| ϕ′(−1)1/2 = 1 = r(uCϕ).

Theorem 4 gives us that the spectrum of uCϕ is the unit circle. 14

(2) Let us consider the space H∞

p (D), which is non-Hilbert. Now

s = p. Let ϕ(z) = z+r

1+rz with 0 < r < 1, so again ϕ is a hyperbolic

automorphism of D with fixed points a = 1 and b = −1. Put

u(z) = (ϕ′(z))p = (1 − r2)p/(1 + rz)2p. Then u ∈ A(D) is bounded

away from zero, and so uCϕ is invertible on H∞

p (D). By Theorem 3,

the spectral radius of uCϕ is

r(uCϕ) = max

• |u(−1)|

ϕ′(−1)p, |u(1)| ϕ′(1)p

• = 1

and

σH∞

p (uCϕ) = {λ ∈ C; |λ| = 1}.

Finally we briefly discuss the proof of

r(uCϕ) = max

• |u(a)|

ϕ′(a)s, |u(b)| ϕ′(b)s

• ,

with hyperbolic automorphism ϕ(z) = z+r

1+rz, 0 < r < 1.

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One only need to show that r(uCϕ) ≤ max |u(a)|

ϕ′(a)s, |u(b)| ϕ′(b)s

• . We

proceed as follows. For any n ∈ N we have

||(uCϕ)n|| = ||

n−1

• j=0

u◦ϕjCϕn|| = ||

n−1

• j=0

u ◦ ϕj (ϕ′ ◦ ϕj)s·

n−1

• j=0

(ϕ′◦ϕj)sCϕn||.

Notice that n−1

j=0 ϕ′ ◦ ϕj = (ϕn)′. Further, w =: u (ϕ′)s and observe

that w ∈ A(D) is also bounded away from zero. By condition 4), we

• btain

||(uCϕ)n||1/n ≤ ||w(n)||1/n

∞ ||(ϕ′ n)sCϕn||1/n.

Since it can be shown that

lim

n→∞ ||w(n)||1/n ∞ = max

• |w(a)|, |w(b)|
• = max
• |u(a)|

ϕ′(a)s, |u(b)| ϕ′(b)s

• and limn→∞ ||(ϕ′

n)sCϕn||1/n ≤ 1, the claim follows.

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