Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a...

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Continuous Semigroups of Composition Operators on Function Spaces on the Disk Carl C. Cowen IUPUI (Indiana University Purdue University Indianapolis) Session on Function Spaces and Operator Theory Vanderbilt, 14 April 2018

Transcript of Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a...

Page 1: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Continuous Semigroups of Composition Operators

on Function Spaces on the Disk

Carl C. Cowen

IUPUI

(Indiana University Purdue University Indianapolis)

Session on Function Spaces and Operator Theory

Vanderbilt, 14 April 2018

Page 2: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

Page 3: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

Usual spaces: f analytic in D, with f (z) =∑∞

n=0 anzn

Hardy: H2(D) = H2 = f : ‖f‖2 =

∞∑n=0

|an|2 <∞

Bergman: A2(D) = A2 = f : ‖f‖2 =

∫D|f (z)|2 dA(z)

π<∞

weighted Hardy (‖zn‖ = βn > 0): H2(β) = f : ‖f‖2 =

∞∑n=0

|an|2β2n <∞

Page 4: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

Usual spaces: f analytic in D, with f (z) =∑∞

n=0 anzn

Hardy: H2(D) = H2 = f : ‖f‖2 =

∞∑n=0

|an|2 <∞

Bergman: A2(D) = A2 = f : ‖f‖2 =

∫D|f (z)|2 dA(z)

π<∞

For many spaces, including the Hardy and Bergman spaces on the disk,

the operators Cϕ are bounded for all ϕ that map the disk into itself.

Page 5: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

Since ϕ maps D into itself, the iterates of ϕ, that is

ϕ2 = ϕ ϕ, ϕ3 = ϕ ϕ2, etc. make sense.

Page 6: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

Since ϕ maps D into itself, the iterates of ϕ, that is

ϕ2 = ϕ ϕ, ϕ3 = ϕ ϕ2, etc. make sense.

Clearly, Cnϕ is a bounded operator for each positive integer n, and Cn

ϕ = Cϕn

Page 7: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

Since ϕ maps D into itself, the iterates of ϕ, that is

ϕ2 = ϕ ϕ, ϕ3 = ϕ ϕ2, etc. make sense.

Clearly, Cnϕ is a bounded operator for each positive integer n, and Cn

ϕ = Cϕn

We want to ask “When can this be extended to Cϕt for all t > 0?”

Page 8: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

More formally: a strongly continuous semigroup on the Hilbert space H is a

map T : R+ 7→ B(H) such that

• T (0) = I

• T (s + t) = T (s)T (t) for s, t ≥ 0

• For each x in H, limt→0+ ‖T (t)x− x‖ = 0.

Page 9: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

More formally: a strongly continuous semigroup on the Hilbert space H is a

map T : R+ 7→ B(H) such that

• T (0) = I

• T (s + t) = T (s)T (t) for s, t ≥ 0

• For each x in H, limt→0+ ‖T (t)x− x‖ = 0.

The question I want to address is:

“When is there a strongly continuous semigroup T for which T (1) = Cϕ ? ”

Page 10: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

The question I want to address is:

“When is there a strongly continuous semigroup T for which T (1) = Cϕ ? ”

• Operators in semigroups are special and much studied.

Page 11: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

The question I want to address is:

“When is there a strongly continuous semigroup T for which T (1) = Cϕ ? ”

• Operators in semigroups are special and much studied.

• Bad News: I don’t know much!

Page 12: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

If ϕ is an analytic function of D into itself,

and H is a Hilbert space of analytic functions on D,

then the composition operator Cϕ on H is the operator

Cϕf = f ϕ for f ∈ H

The question I want to address is:

“When is there a strongly continuous semigroup T for which T (1) = Cϕ ? ”

• Operators in semigroups are special and much studied.

• Bad News: I don’t know much!

• Goal: This is an interesting problem;

I hope some of you will be interested in working on it and do more than I can!

Page 13: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Denjoy-Wolff Theorem (1926).

If ϕ is an analytic map of D into itself (not an elliptic automorphism),

there is a unique fixed point, a, of ϕ in D such that |ϕ′(a)| ≤ 1.

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Denjoy-Wolff Theorem (1926).

If ϕ is an analytic map of D into itself (not an elliptic automorphism),

there is a unique fixed point, a, of ϕ in D such that |ϕ′(a)| ≤ 1.

Moreover, the sequence of iterates, (ϕn),

converges to a uniformly on compact subsets of D.

Page 15: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Denjoy-Wolff Theorem (1926).

If ϕ is an analytic map of D into itself (not an elliptic automorphism),

there is a unique fixed point, a, of ϕ in D such that |ϕ′(a)| ≤ 1.

Moreover, the sequence of iterates, (ϕn),

converges to a uniformly on compact subsets of D.

This distinguished fixed point a is called the Denjoy-Wolff point of ϕ.

Page 16: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

The iterates of analytic maps of D into itself D have been studied by many

mathematicians for about 150 years!!

Some famous names include E. Schroeder, G. Koenigs, P. Fatou, A. Denjoy,

J. Wolff, C.L. Siegel, I.N. Baker, Ch. Pommerenke, · · ·

Page 17: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

The Model for Iteration [C., 1981]

Let ϕ be an analytic mapping of D into itself, ϕ non-constant and not an elliptic

automorphism of D, and let a be the Denjoy–Wolff point of ϕ.

If ϕ′(a) 6= 0, then there is a fundamental set V for ϕ on D,

a domain Ω, either a halfplane or the plane,

an automorphism Φ mapping Ω onto Ω,

and a mapping σ of D into Ω

such that ϕ and σ are univalent on V , σ(V ) is a fundamental set for Φ on Ω, and

Φ σ = σ ϕ

Moreover, Φ is unique up to conjugation by an automorphism of Ω onto Ω, and Φ

and σ depend only on ϕ, not on the particular fundamental set V .

Page 18: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

The Model for Iteration [C., 1981]

Let ϕ be an analytic mapping of D into itself, ϕ non-constant and not an elliptic

automorphism of D, and let a be the Denjoy–Wolff point of ϕ.

If ϕ′(a) 6= 0, then there is a fundamental set V for ϕ on D,

a domain Ω, either a halfplane or the plane,

an automorphism Φ mapping Ω onto Ω,

and a mapping σ of D into Ω

such that ϕ and σ are univalent on V , σ(V ) is a fundamental set for Φ on Ω, and

Φ σ = σ ϕ

Moreover, Φ is unique up to conjugation by an automorphism of Ω onto Ω, and Φ

and σ depend only on ϕ, not on the particular fundamental set V .

What’s New??

Page 19: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

The Model for Iteration [C., 1981]

Let ϕ be an analytic mapping of D into itself, ϕ non-constant and not an elliptic

automorphism of D, and let a be the Denjoy–Wolff point of ϕ.

If ϕ′(a) 6= 0, then there is a fundamental set V for ϕ on D,

a domain Ω, either a halfplane or the plane,

an automorphism Φ mapping Ω onto Ω,

and a mapping σ of D into Ω

such that ϕ and σ are univalent on V , σ(V ) is a fundamental set for Φ on Ω, and

Φ σ = σ ϕ

Moreover, Φ is unique up to conjugation by an automorphism of Ω onto Ω, and Φ

and σ depend only on ϕ, not on the particular fundamental set V .

What’s New?? One Proof for All Cases!!

Page 20: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

The Model for Iteration [C., 1981]

Let ϕ be an analytic mapping of D into itself, ϕ non-constant and not an elliptic

automorphism of D, and let a be the Denjoy–Wolff point of ϕ.

If ϕ′(a) 6= 0, then there is a fundamental set V for ϕ on D,

a domain Ω, either a halfplane or the plane,

an automorphism Φ mapping Ω onto Ω,

and a mapping σ of D into Ω

such that ϕ and σ are univalent on V , σ(V ) is a fundamental set for Φ on Ω, and

Φ σ = σ ϕ

Moreover, Φ is unique up to conjugation by an automorphism of Ω onto Ω, and Φ

and σ depend only on ϕ, not on the particular fundamental set V .

What’s New?? One Proof for All Cases!! Uniqueness of the Model!!

Page 21: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Let ϕ be an analytic map of D into itself (not an elliptic automorphism),

and let a be the Denjoy-Wolff point of ϕ.

Analytic self-maps of D (not elliptic automorphisms) divide into 5 distinct

classes and the Model for Iteration covers 4 of these cases:

• (Plane/Dilation): |a| < 1 and 0 < |ϕ′(a)| < 1

• (Half-Plane/Dilation): |a| = 1 and 0 < ϕ′(a) < 1

• (Half-Plane/Translation): |a| = 1 and ϕ′(a) = 1, and ϕn(z) interpolating

• (Plane/Translation): |a| = 1 and ϕ′(a) = 1, and ϕn(z) not interpolating

• (no LF model): |a| < 1 and ϕ′(a) = 0

Page 22: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

An Example:

Let ϕ(z) =z + 1

−z + 3

Page 23: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

An Example:

Let ϕ(z) =z + 1

−z + 3

It is easy to check that ϕ(1) = 1 and ϕ′(1) = 1.

This map is in the plane-translation case:

Page 24: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

An Example:

Let ϕ(z) =z + 1

−z + 3

It is easy to check that ϕ(1) = 1 and ϕ′(1) = 1.

This map is in the plane-translation case:

Specifically, for σ(z) = (1 + z)(1− z)−1 and Φ(z) = z + 1, we have

σ ϕ(z) =1 + z+1

−z+3

1− z+1−z+3

=2

−z + 1=

1 + z

1− z+ 1 = σ(z) + 1 = Φ σ(z)

Page 25: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

An Example:

Let ϕ(z) =z + 1

−z + 3

It is easy to check that ϕ(1) = 1 and ϕ′(1) = 1.

This map is in the plane-translation case:

Specifically, for σ(z) = (1 + z)(1− z)−1 and Φ(z) = z + 1, we have

σ ϕ(z) =1 + z+1

−z+3

1− z+1−z+3

=2

−z + 1=

1 + z

1− z+ 1 = σ(z) + 1 = Φ σ(z)

In fact, ϕ is part of the continuous semi-group:

ϕt(z) =(2− t)z + t

−tz + (t + 2)

and

σ ϕt(z) =1 + (2−t)z+t

−tz+(t+2)

1− (2−t)z+t−tz+(t+2)

=−tz + (t + 2) + ((2− t)z + t)

−tz + (t + 2)− ((2− t)z + t)= σ(z)+ t = Φtσ(z)

Page 26: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

This example makes everything look easy!

That is an illusion, but there some things that carry over to more complicated

cases.

Moreover, the conclusion can be partially recovered for all maps ϕ with ϕ′(a) 6= 0!

For simplicity, the result will be stated in the plane-translation case, although

analogous results hold for other cases.

Page 27: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Let ϕ be an analytic mapping on the disk with Denjoy–Wolff point, a,

with |a| = 1 and ϕ′(1) = 1.

In addition, suppose V is a fundamental set for ϕ on D and σ is a map of D into

Ω = C such that ϕ and σ are univalent on V , σ(V ) is a fundamental set for

Φ(z) = z + 1 on C, and

σ ϕ = Φ σ = σ + 1

Page 28: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Let ϕ be an analytic mapping on the disk with Denjoy–Wolff point, a,

with |a| = 1 and ϕ′(1) = 1.

In addition, suppose V is a fundamental set for ϕ on D and σ is a map of D into

Ω = C such that ϕ and σ are univalent on V , σ(V ) is a fundamental set for

Φ(z) = z + 1 on C, and

σ ϕ = Φ σ = σ + 1

For z in D, let ν(z) = minn : ϕn(z) ∈ V

Page 29: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Let ϕ be an analytic mapping on the disk with Denjoy–Wolff point, a,

with |a| = 1 and ϕ′(1) = 1.

In addition, suppose V is a fundamental set for ϕ on D and σ is a map of D into

Ω = C such that ϕ and σ are univalent on V , σ(V ) is a fundamental set for

Φ(z) = z + 1 on C, and

σ ϕ = Φ σ = σ + 1

For z in D, let ν(z) = minn : ϕn(z) ∈ V

Define τ by

τ (z) = inft : ν(z) ≤ t and σ(z) + t1 ∈ σ(V ) for all t1 > t

(Note: τ (z) <∞ for every z in D and τ (z) = 0 for some z in D)

Page 30: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Partially Defined Semigroups [C., 1981]

Let ϕ be an analytic mapping of D into itself that falls into the

Plane/Translation case of the Model Theorem.

There is a function h(z, t) complex analytic in the first argument for z in D

and real analytic in the second argument for t > τ (z) such that:

• h(z, n) = ϕn(z) for n > τ (z)

and • h(h(z, t1), t2) = h(z, t1 + t2) for t1 > τ (z) and t2 > 0.

Moreover, there is a function g meromorphic in D and holomorphic in V that

agrees with the infinitesimal generator of h(z, t) on the set z : τ (z) = 0.

Page 31: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Partially Defined Semigroups [C., 1981]

Let ϕ be an analytic mapping of D into itself that falls into the

Plane/Translation case of the Model Theorem.

There is a function h(z, t) complex analytic in the first argument for z in D

and real analytic in the second argument for t > τ (z) such that:

• h(z, n) = ϕn(z) for n > τ (z)

and • h(h(z, t1), t2) = h(z, t1 + t2) for t1 > τ (z) and t2 > 0.

Moreover, there is a function g meromorphic in D and holomorphic in V that

agrees with the infinitesimal generator of h(z, t) on the set z : τ (z) = 0.

In essence, this result is defining the tail of a continuous semigroup ϕt(z) for each

z in D and t > τ (z).

Page 32: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Corollary: Let ϕ be an analytic function mapping D into itself such that

• ϕ is real valued on the interval (−1, 1)

• ϕ′(x) > 0 for −1 < x < 1

and • the Denjoy-Wolff point of ϕ is 1

then there is a function h(x, t) real analytic in each variable for |x| < 1 and

t ≥ 0 such that h(x, 1) = ϕ(x) and

h(h(z, t1), t2) = h(z, t1 + t2)

for t1 and t2 non-negative real numbers.

Page 33: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Corollary: Let ϕ be an analytic function mapping D into itself such that

• ϕ is real valued on the interval (−1, 1)

• ϕ′(x) > 0 for −1 < x < 1

and • the Denjoy-Wolff point of ϕ is 1

then there is a function h(x, t) real analytic in each variable for |x| < 1 and

t ≥ 0 such that h(x, 1) = ϕ(x) and

h(h(x, t1), t2) = h(x, t1 + t2)

for t1 and t2 non-negative real numbers.

In other words, ϕt(x) is a continuous semigroup on (−1, 1).

Page 34: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

An Example:

Let ϕ(z) = 14(1 + z)2

Clearly, ϕ is an analytic map of D into itself and ϕ(1) = ϕ′(1) = 1,

which means ϕ is in the Plane/Translation case of the Model Theorem.

Page 35: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

An Example:

Let ϕ(z) = 14(1 + z)2

Clearly, ϕ is an analytic map of D into itself and ϕ(1) = ϕ′(1) = 1,

which means ϕ is in the Plane/Translation case of the Model Theorem.

Moreover, ϕ acting on D is univalent and satisfies the hypotheses of

the Corollary above, so ϕt(x) is a real-analytic semigroup on (−1, 1].

Page 36: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

An Example:

Let ϕ(z) = 14(1 + z)2

Clearly, ϕ is an analytic map of D into itself and ϕ(1) = ϕ′(1) = 1,

which means ϕ is in the Plane/Translation case of the Model Theorem.

Moreover, ϕ acting on D is univalent and satisfies the hypotheses of

the Corollary above, so ϕt(x) is a real-analytic semigroup on (−1, 1].

In addition, for each z in D,

the angle between the ray from 1 to z and the real axis

is greater than angle between the ray from 1 to ϕ(z) and the real axis,

so that, in some sense, ϕ is mapping points in D toward the real axis.

Page 37: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

Let ϕ(z) = 14(1 + z)2

Conjecture:

Composition operator Cϕ is part of a continuous semigroup of operators on H2.

Page 38: Continuous Semigroups of Composition Operators on ...ccowen/VandSemigroups1804.pdf\When is there a strongly continuous semigroup Tfor which T(1) = C Operators in semigroups are special

THANK YOU!

A version of these slides is posted on my website:

http://www.math.iupui.edu/˜ccowen/