compositionoperators samos

Composition Operators

on Spaces of Analytic Functions

A. G. Siskakis

Samos, July 2012

Contents

1. General facts about Composition Operators.

2. Spaces of Analytic Functions.

3. Boundedness.

4. Compactness.

5. Membership in Schatten ideals.

6. Invariant subspaces and Cϕ.

7. Cϕ and the Brennan conjecture.

8. Semigroups of Composition Operators.

1. General facts.

Suppose S is a set, and ϕ : S → S a self-map of S.

Let also X be a linear space consisting of function

f : S → C.

The composition operator induced by ϕ is

Cϕ(f)(s) = f(ϕ(s)), f ∈ X.

In addition if ψ : S → C is a function, the weighted

composition operator induced by ψ and ϕ is

Wψ,ϕ(f)(s) = ψ(s)f(ϕ(s)).

We assume Cϕ(X) ⊂ X or Wψ,ϕ(X) ⊂ X whenever

needed.

Initial observations:

1. Clearly Cϕ and Wψ,ϕ are linear maps.

2. Cϕ can viewed as generalizing the translations

f → f(x+ a), on R,

or the rotations

f → f(eiθz) on the unit circle T.

3. If ϕ(s) = s for each s, the identity function,

then Cϕ = I, the identity operator. If ϕ(s) = s0,

constant, then

Cϕ : f → f(s0)

is a point evaluation.

4. Cϕ is multiplicative:

Cϕ(fg) = f ϕ · g ϕ = Cϕ(f)Cϕ(g).

5. We have Cϕ1Cϕ2 = Cϕ2ϕ1. In particular if

ϕn = ϕ ϕn−1, n ∈ N

are the iterates of ϕ then

Cnϕ = Cϕn.

Historical information:

• E. Schroder (1871) asked,

Given ϕ analytic self-map of a domain D, find f

and α such that

f(ϕ(z)) = αf(z), z ∈ D.

• G. Konigs (1884):

Solved Schroder’s equation in case D = D, the unit

disc in C.

• J. E. Littlewood (1925):

If U : D → R is subharmonic and ϕ : D → D analytic

with ϕ(0) = 0 then∫ 2π

0U(ϕ(reiθ)) dθ ≤

∫ 2π

0U(reiθ) dθ, 0 < r < 1.

• B. O. Koopman (1930’s):

Used composition operators in studying Statistical

Mechanics.

• Composition operators were used in Ergodic The-

ory and Dynamical systems. In this setting they are

the adjoints of Perron-Frobenius operators.

• The systematic study on spaces of analytic func-

tions started in the 1960’s with the work of E.

Nordgren, J. Ryff, and H. J. Schwartz.

How wide is the class of Composition Operators

• Consider sequence in l2 as functions on N, i.e.

an = f(n). Then the backward shift operator

(f(0), f(1), f(2), · · · ) → (f(1), f(2), · · · )

is the composition operator f → f ϕ induced by

ϕ(n) = n+1.

• The Banach-Stone theorem (1930’s):

If Q,K are compact Hausdorff spaces and T an iso-

metric isomorphism between C(Q) and C(K) then

T (f)(t) = α(t)f(τ(t)),

where τ : K → Q is a homeomorphism and α ∈C(K) with |α| = 1.

• F. Forelli (1964):

If T : Hp → Hp, (p ≥ 1, p = 2), is an isometric

isomorphism, then

T (f)(z) = eiθ(ϕ′(z))1/pf(ϕ(z)),

where ϕ is a Mobius automorphism of the disc and

θ real constant.

• Consider the multiplication operator

Mg(f)(z) = g(z)f(z)

Which operators commute with Mg? Clearly ev-

ery multiplication operator does. But when ϕ is

such that g(ϕ(z)) = g(z) for every z, then Cϕ also

commutes with Mg:

CϕMg(f) = g ϕ · f ϕ = g · f ϕ =MgCϕ(f).

• There are universal operators among the compo-

sition operators (details later).

• Certain classical operators can be written as av-

erages of composition operators.

Hausdorff means. In the Theory of Summability,

the Hausdorff summability method uses lower tri-

angular matrices H = (cn,k) whose entries are

cn,k =

(nk

) ∫ 10 t

k(1− t)n−k dµ(t), k ≤ n

0, k > n,

where µ is a Borel measure on [0,1).

When applied to sequences or to series, these ma-

trices are expected to “improve convergence”. If

we consider such a matrix H as an operator on

l2, and rewrite it equivalently on H2, the analytic

disguise of l2, then it takes the form

H(f)(z) =∫ 1

0

s

(s− 1)z+1f

(sz

(s− 1)z+1

)dµ(s),

i.e. H is a µ-average of weighted composition op-

erators.

The Hilbert matrix. The Hilbert matrix

H =

1, 1/2, 1/3, · · ·

1/2, 1/3, 1/4, · · ·1/3, 1/4, 1/5, · · ·... ... ...

,which is the prototype for a Hankel operator, when

written on H2, is seen to be an average of weighted

composition operators:

H(f)(z) =∫ 1

0

s

(s− 1)z+1f

(s

(s− 1)z+1

)ds.

Two major directions of study of Cϕ.

1. (S,M, µ) a measure space,

ϕ : S → S a measurable map, and

X = Lp(S,M, µ) or some other space of measurable

functions.

2. S = Ω ⊂ C (or of Cn) a region,

ϕ : Ω → Ω analytic and

X = some space of analytic functions on Ω.

We will concentrate on spaces of analytic functions

in one complex variable, and mostly when Ω = D,the unit disc or Ω = U, the upper half-plane and

will consider mainly Hardy, Bergman and Dirichlet

spaces.

This identifies H2 with l2. In particular H2 is a

Hilbert space, with inner product

⟨f, g⟩ = limr→1

∫ 2π

0f(reiθ)g(reiθ)dσ(θ)

=∞∑k=0

akbk.

Basic properties of Hp:

• If 1 < p < q <∞ then

H1 ⊃ Hp ⊃ Hq ⊃ H∞,

with strict containment in each case.

• If f ∈ Hp then the limit

f∗(eiθ) = limr→1

f(reiθ)

exists for almost all θ ∈ [0,2π]. The resulting func-

tion f∗ is p-integrable on the circle T, and

∥f∗∥Lp(T) = ∥f∥Hp.

• If f, g ∈ Hp and f∗(eiθ) = g∗(eiθ) on a set of

positive measure on T then f ≡ g (this is a form of

the identity principle).

• Consider the polynomials p(eiθ) in Lp(T). Then

f∗ : f ∈ Hp =p(eiθ) : p polynomial

.

Thus Hp can be identified, isometrically, with this

closed subspace of Lp(T).

If 1 ≤ p ≤ ∞ and f ∈ Lp(T) the f = g∗ for some

g ∈ Hp if and only if the Fourier coefficients of f

with negative indices are all zero.

• If 1 ≤ p < ∞ and Φ ∈ Lp(T) then its Poisson

integral

f(z) = (reiθ) =∫TP (r, θ − t)Φ(eit)dσ(t)

is an analytic function in Hp.

• The Riesz Projection P+ : Lp(T) → Hp

P+ :∞∑−∞

f(n)einθ −→∞∑n=0

f(n)einθ,

is a bounded operator for 1 < p <∞.

• Each f ∈ Hp has a factorization

f(z) = B(z)Sµ(z)F (z)

where B(z) is a Blaschke product

B(z) = zm∞∏n=0

|an|an

an − z

1− anz

containing the zeros of f , Sµ(z) is singular inner

function

Sµ(z) = exp

(−∫T

ζ + z

ζ − zdµ(ζ)

)arising from a singular measure µ on T, and F (z)

an outer function.

• Point evaluations are bounded linear functionals

on Hp: For f ∈ Hp,

|f(z)| ≤Cp∥f∥p

(1− |z|)1/p, z ∈ D.

• Let fs(z) = 1(1−z)s. Then

fs(z) ∈ Hp ⇔ s <1

p,

and lims→1

p∥fs∥p = ∞.

Hardy spaces of the half-plane

Let U = z : Im(z) > 0 the upper half-plane.

For 0 < p <∞, Hp(U) contains all analytic

f : U → C such that

∥f∥pHp(U) = sup

y>0

∫ ∞

−∞|f(x+ iy)|p dx <∞.

Hp(U) are Banach spaces for 1 ≤ p < ∞. They

are isometrically isomorphic to Hp(D) via the linear

isometry

V (f)(z) =π−1/p

(i+ z)2/pf(µ−1(z))

where µ : D → U is the conformal map µ(z) = i1+z1−z .

Bergman spaces

Let 0 < p <∞. The Bergman space Ap consists of

all f ∈ A(D) such that

∥f∥pAp =∫D|f(z)|p dA(z) <∞,

where dA(z) = 1πdxdy, the normalized area measure

of D.

Ap are Banach spaces for 1 ≤ p <∞, A2 is a Hilbertspace and

∥f∥2A2 =

∞∑n=0

|an|2

n+1.

Basic properties of Ap:

• If 1 < p < q <∞ then

A1 ⊃ Ap ⊃ Aq ⊃ H∞,

with strict containment in each case. By the defi-nition, if f ∈ Hp then

∥f∥pAp =∫D|f(z)|p dA(z)

=∫ 1

0


)r dr

≤∫ 1

0supr


)dr

= ∥f∥pHp <∞,

so that Hp ⊂ Ap for each p. But in fact the sharpercontainment holds

Hp ⊂ A2p, and ∥f∥A2p ≤ ∥f∥Hp, 0 < p <∞.

• In contrast to Hardy spaces, functions in Ap neednot have boundary values on T.

• Point evaluations are bounded linear functionals.

For f ∈ Ap,

|f(z)| ≤Cp∥f∥Ap

(1− |z|)2/p, z ∈ D.

• For s ∈ R,

fs(z) =1

(1− z)s∈ Ap

if and only if s < 2p, and lim

s→2p∥fs∥p = ∞.

The Dirichlet space

The Dirichlet space D consists of all analytic f on

D for which

∥f∥2D = |f(0)|2 +∫D|f ′(z)|2 dA(z) <∞.

The norm can be written

∥f∥2D =∞∑n=0

(n+1)|an|2, f(z) =∞∑n=0

anzn,

and D is a Hilbert space.

Properties of D:

• For f E V we have IIJIIH2 < ll!llv- In particular V c H 2 , so functions in V have boundary values

a.e. on 1f.

• Notice that

Area(f(illl)) = .~ fJ11dA(z) = k lf'(z)edA(z),

(Jf the Jacobian), so V contains exactly the functions f such that j(]J)) has finite area (counting

multiplicity).

In particular V contains some unbounded functions

and it does not contain all bounded functions: The

function

takes D into itself, but f /∈ D because f covers in-

finitely many times the area of D. In fact D does

not contain any infinite Blaschke product, (it con-

tain all finite ones) and does not contain any non-

trivial ( = to a constant) singular inner function

Sµ(z).

• If f ∈ D then

|f(z)| ≤ C∥f∥D log

(1

1− |z|2

)1/2.

Other spaces on which composition operators have

been studied include, in addition to weighted ver-

sions of the above paces, the spaces BMOA of

analytic functions whose boundary values are of

bounded mean oscillation, the Bloch space, Besov

spaces, spaces of entire functions, spaces of Dirich-

let series, e.t.c.

On each of these spaces questions and results in-

clude

1. Boundedness, compactness of Cϕ.

2. Spectra.

3. Dynamical behavior (cyclicity - hypercyclicity).

4. Semigroups of composition operators.

5. Topology of the metric spaceM = Cϕ : bounded.6. Connections with other questions of operator

theory and classical analysis.

A common characteristic is the relation

function theoreticproperties of ϕ

operator theoreticproperties of Cϕ

Some typical cases of¢

• cp(z) == 1~:z' a E IIJ), a Mobius automorphism, or a finite product of those

n ak- z cp(z) == IT 1 - '

k==l - akz

or ¢(z) == B(z) an infinite Blascke product, or

cp(z) == Sf-L(z) a singular inner function, or com

bination of the above

cp(z) == B(z)Sf-L(z).

In all these cases l¢*(ei8)1 == 1 a.e .

• cp(IIJ)) c IIJ),

• cp(IIJ)) touches 1f in an angle

<f (ID)

<fU)• 1-Vi-t

• ¢(JD)) touches 1r tangentially

i(ll>)

<1 (~) ::: ( + ~ -2

• ¢(JD)) intersects 1r in positive measure

1( m)

• a combination of the above

• One more case

All the above cases can be realized even by univa

lent ¢ by applying the Riemann mapping theorem.

3. Boundedness

Consider this problem first. Suppose ϕ : D → D hasthe expansion

ϕ(z) =∞∑n=1

bnzn,

and let f(z) =∑∞n=0 anz

n be in H2. Then

f(ϕ(z)) =∞∑n=0

anϕ(z)n

= · · ·

=∞∑n=0

Anzn, An = An(ai, bj).

Prove by hand using only the above computationof An that Cϕ is bounded on H2, i.e.

∞∑n=0

|An|2 ≤ C∞∑k=0

|ak|2

with C independent of f .

Littlewood’s subordination theorem:

If U : D → R is subharmonic and ϕ : D → D analyticwith ϕ(0) = 0 then∫ 2π


∫ 2π

0U(reiθ) dθ, 0 < r < 1.

Proof. For 0 < r < 1, find harmonic

h : |z| ≤ r → R, with h = U on |z| = r.

Then U(z) ≤ h(z) for |z| ≤ r and h(ϕ(z)) is har-

monic, so,∫ 2π


∫ 2π

0(h ϕ)(reiθ)) dθ

= 2π(h ϕ)(0)) = 2πh(0) =∫ 2π

0h(reiθ) dθ

=∫ 2π

0U(reiθ) dθ.

completing the proof.

Suppose f ∈ Hp and ϕ(0) = 0. Apply the theo-

rem to the subharmonic function U(z) = |f(z)|p to

obtain

∥f ϕ∥p ≤ ∥f∥p.

Next if ϕa(z) = a−z1−az, a disc automorphism, then

ϕa is 1-1 and maps T onto T. A change of variable

in the integral defining ∥f ϕa∥ gives

∥f ϕa∥p ≤(1+ |a|1− |a

)1/p∥f∥p,

for each f ∈ Hp, 1 ≤ p <∞, and in fact,

∥Cϕa∥ =

(1+ |a|1− |a|

)1/p.

Notice that |a| = |ϕa(0)|.

Finally for a general ϕ, let a = ϕ(0), consider the

automorphism ϕa(z) = a−z1−az and put ϕ0 = ϕa ϕ.

Then ϕ0(0) = 0 and

ϕ = ϕ−1a ϕ0 = ϕa ϕ0,

so that

Cϕ = Cϕ0Cϕa.

Thus Cϕ : Hp → Hp, 1 ≤ p <∞, is bounded and

∥Cϕ∥ ≤ ∥Cϕ0∥ · ∥Cϕa∥

=

(1+ |a|1− |a|

)1/p

=

(1+ |ϕ(0)|1− |ϕ(0)|

)1/p.

Theorem. Cϕ is bounded on Hp and(1

1− |ϕ(0)|2

)1/p≤ ∥Cϕ∥ ≤

(1+ |ϕ(0)|1− |ϕ(0)|

)1/p.

The left estimate is obtained by observing that

C∗ϕ(L0) = Lϕ(0), where Lw is the point evaluation

functional at w.

Boundedness on Bergman spaces

Writing the integral defining the Bergman norm in

polar coordinates and using Littlewood’s subordi-

nation theorem we obtain

Theorem. Cϕ is bounded on Ap and(1

1− |ϕ(0)|2

)2/p≤ ∥Cϕ∥ ≤

(1+ |ϕ(0)|1− |ϕ(0)|

)2/p.

On the Dirichlet space

For Cϕ to be bounded on D it is necessary that

ϕ ∈ D, because

Cϕ(f) = ϕ, for f(z) = z ∈ D.

Thus it must be∫D|ϕ′(z)|2 dA(z) <∞,

so any¢ such that area(¢(ID)) == oo (counting multiplicity) induces unbounded C¢ on V. Such a function is the example

( 1 + z) ¢(z) == exp - . 1-z

Characterization: Given ¢ and wE ID let

n¢(w) == #z: ¢(z) == w

the number of solutions (counting multiplicity) of ¢(z) == w, and

dJ-L¢(w) == n¢(w)dA(w).

Also for ( E 1r and 0 < 6 < 2 let

S((,6) == z E ID: iz- (i < 6

Theorem. C¢ : V--+ V is bounded if and only if

J1q,(S8~(, 8

)) = 0(1), (( E '[', 0 < 8 < 2),

or in the customary terminology, 1-l¢ is a Carleson measure for V.

For future reference we mention that 1-l¢ will be called a vanishing Carleson measure if

M¢(S((, 6)) _ (1

) sup 2 - o , (E'JI' 6

(6--+0).

On Hardy spaces of the half-plane

In contrast to Hp(D), there are ϕ : U → U analytic

which induce unbounded Cϕ on Hp(U).

• No bounded ϕ : U → U induces a bounded Cϕ.

• Consider the linear fractional maps

ϕ(z) =az+ b

cz+ d,

which map U into U. Of those the only ones which

induce bounded Cϕ on Hp(U) are those of the form

ϕ(z) = az+ b with a > 0 and Im(b) ≥ 0.

• If ϕ(z) =√z then Cϕ is not bounded.

• If ϕ(z) = az + b+√z, a > 0, Im(b) ≥ 0, then Cϕ

is bounded.

• There is a characterization of the bounded Cϕ on

Hp(U) in terms of Carleson measures which implies,

(i) either Cϕ if bounded on Hp(U) for all p,

(ii) or Cϕ if not bounded on Hp(U) for any p.

• A better characterization.

Theorem. C¢ is bounded on HP(TJJ) if and only if

¢'(=) = lim ¢(z) z----too z

n.t.

exists and is finite.

Note: z == x + iy ----+ oo non-tangentially (n.t.)

if if z stays inside an

angle lxl < Cy

(

I

I

If the limit ¢'(oo) is finite then necessarily

lim ¢(z) == oo z----+oo

n.t.

and ¢'(oo) E (0, oo). In this case,

1

II c ¢II HP(U)----t HP(1U) == ¢' ( 00)- p.

Remarks about boundednes and norms

1. As a rule of thump, C¢ is bounded on spaces of ~~medium size".

2. Spaces of “small size”, like D, may not containall inducing function ϕ. In such cases bounded-ness is usually characterized by some O(1) condi-tion which may be difficult to apply in practice.

3. Spaces of “large size”can contain fast increas-ing analytic functions, making it difficult for Cϕ tobe bounded. For example for any sequence (βn) ofpositive numbers with βn → 0 we may define theHilbert space

Hβ =

f(z) =∞∑n=0

anzn :

∞∑n=0

β2n|an|2 <∞

.When the convergence βn → 0 is very fast, forexample if

nsβn → 0 for each s > 0,

then the automorphisms ϕ(z) = z+r1+rz, 0 < r < 1,

do not induce bounded Cϕ on Hβ.

4. The norm of Cϕ is difficult to compute in thegeneral case. A few exceptions are,

• If ϕ is a disc automorphism then

∥Cϕ∥Hp→Hp =

(1+ |ϕ(0)|1− |ϕ(0)|

)1/p.

• If ϕ(0) = 0 then ϕ is inner if and only if

∥Cϕ∥Hp→Hp =

(1+ |ϕ(0)|1− |ϕ(0)|

)1/p.

• If ϕ(0) = 0 then ϕ is inner if and only if

∥Cϕ∥Hp0→H

p0= 1, (Hp

0 = zHp).

• If ϕ is univalent then on the Dirichlet space,

∥Cϕ∥D→D =

√√√√L+2+√L(4 + L)

2

where L = − log(1− |ϕ(0)|2).

• If ϕ(z) = sz+ t, |s|+ |t| ≤ 1, then

∥Cϕ∥H2→H2 =

√2

1+ |s|2 − |t|2 +√

(1− |s|2 + |t|2)2 − 4|t|2

4. Compactness

Recall that an operator T : X → X is compact

if the image of the unit ball of X is a relatively

compact set in X, or equivalently, the image of

every bounded sequences in X has a convergent

subsequence.

For Cϕ on Hp(D) this specializes as follows:

Cϕ is compact if and only if,

If fn is a sequence in Hp(D) with ∥fn∥p ≤M and

fn → 0 uniformly on compact subsets of D, then

∥fn ϕ∥p → 0.

• Choosing fn(z) = zn we find: If Cϕ is compact

then the set Eϕ = eiθ : |ϕ∗(eiθ)| = 1 ⊂ T has mea-

sure 0. In particular no inner function can induce

a compact Cϕ on Hardy spaces.

• Either Cϕ is compact on Hp for all p or it is not

compact for any p.

• If ϕ(D) ⊂ D then Cϕ is compact.

• If ¢(JD)) touches 1r only at the vertex of an angle, and otherwise is contained in the angle, then C¢ is compact on HP

• If ¢(z) :.__ 1 tz then C¢ is not compact

As a general rule: C¢ compact rv ¢(JD)) does not touch the boundary 1r too much.

Compactness and angular derivatives:

For ¢: JD)---+ JD) analytic and wE 1r, say that ¢ has a (finite) angular derivative at w, if for some ( E 1r the limit

ql(w) = lim ¢(z)- ( z---+w z- W

n.t.

exists and is finite, where n.t. means z---+ w within an angle. For this to happen it is necessary that

lim ¢(z) == (. z---+w

n.t.

Interpreting the last limit as ϕ(w), we can write

ϕ′(w) = limz→wn.t.

ϕ(z)− ϕ(w)

z − w,

and the finiteness of ϕ′(w) means that ϕ is in “con-

formal”at w.

With this at hand we have:

• If Cϕ is compact on Hp or on Ap then ϕ does not

have a finite angular derivative at any w ∈ T.

• On the Bergman space Ap the converse is also

true. Thus:

Cϕ compact on Ap ⇔ ϕ′(w) does not exist finitely

at any w ∈ T.

• For univalent or bounded-valent ϕ the same char-

acterization is valid on Hp.

• On the other hand there are ϕ that have no finite

angular derivative at any point on T and such that

eiθ : |ϕ∗(eiθ)| = 1 has measure 0, yet Cϕ is not

compact on Hp.

• There are ϕ such that ϕ(D) = D and Cϕ is com-

pact on Hp.

• If Cϕ is compact on Hp then ϕ has a fixed point

in D.

Full characterization of compactness on Hp was

found by J. Shapiro (1987):

For w ∈ D \ ϕ(0) define

Nϕ(w) =∑

z∈ϕ−1(w)log

1

|z|,

if w ∈ ϕ(D), and Nϕ(w) = 0 if w /∈ ϕ(D). This is

the Nevanlinna Counting function.

Note that, since log 1|z| ∼ (1 − |z|) for |z| near 1,

and since ϕ(z)− w is a bounded function of z, the

zeros zj of ϕ(z) − w have the Blaschke property∑(1 − |zj|) < ∞. Thus the above series for Nϕ

converges.

Theorem. Cϕ is compact on Hp if and only if

lim|w|→1

Nϕ(w)

log 1|w|

= 0.

Recall that the essential norm of an operator T is

the distance of T from the ideal of all compact

operators.

Theorem. The essential norm of Cϕ on Hp is

∥Cϕ∥pe = limsup|w|→1

Nϕ(w)

log 1|w|.

Recently it was shown that

lim sup|w|→1

Nϕ(w)

log 1|w|

= limsup|a|→1

∫T

1− |a|2

|1− aϕ(eiθ)|2dθ

so the latter quantity provides another test for

compactness and for the essential norm.

Similar theorems hold for Bergman spaces, with a

modified Nevanlinna counting function.

It is worth noting that although no inner function

can induce a compact Cϕ on Hardy spaces, there

are singular inner ϕ such that Cϕ is compact on Ap.

On the Dirichlet space,

Recall: Cϕ : D → D is bounded if and only if the

measure dµϕ(w) = nϕ(w)dA(w) is Carleson mea-

sure, that is,

µϕ(S(ζ, δ))

δ2= O(1), (ζ ∈ T, 0 < δ < 2)

Theorem. Cϕ : D → D is compact if and only if

supζ∈T

µϕ(S(ζ, δ))

δ2= o(1), (δ → 0)

i.e. µϕ is a vanishing Carleson measure.

The O(1),o(1) analogy for boundedness - com-

pactness of Cϕ appears in many other cases.

On Hardy spaces of the half plane,

Theorem. [Matache] There are no compact com-

position operators Cϕ on Hp(U).

The following general theorem was motivated by

the above result of Matache:

For a simply connected Ω ⊂ C let

h : D → Ω,

a Riemann map. Define Hp(Ω) to consist of all

analytic f : Ω → C such that the integrals of |f(z)|p

over the curves

Γr = h(|z| = r), 0 < r < 1,

remain bounded as r → 1. Then

∥f∥pHp(Ω) = sup

0<r<1

∫Γr

|f(z)|p |dz|

is a Banach space norm for p ≥ 1. For ϕ : Ω → Ω

analytic, Cϕ is defined on Hp(Ω) in the usual way.

Theorem. [Shapiro-Smith] Hp(Ω) supports com-

pact composition operators if and only if the bound-

ary ∂Ω has finite one-dimensional Hausdorff mea-

sure. Equivalently if and only if h′ ∈ H1(D).

This says in particular that there are bounded Ω

such that no composition operator is compact on

Hp(Ω).

Note that for Jordan domains,

∂Ω has finite Hausdorff measure ⇔ ∂Ω is rectifiable.

There is a similar theorem for Bergman spaces. For

simply connected Ω, Ap(Ω) consists of f : Ω → Canalytic such that∫

Ω|f(z)|p dA(z) <∞.

Theorem. Ap(Ω) supports compact composition

operators if and only if Area(Ω) <∞.

5. Membership of Cϕ in Schatten ideals.

Recall that if T : H → H is a bounded operator on

a Hilbert space, its singular numbers (λn) are

λn(T ) =: inf∥T − F∥ : F : H → H is of rank ≤ n

Compact operators are those for which λn → 0,

Finite rank operators are those for which (λn) is

eventually 0.

Between these two lie the Schatten classes Sp(H).

Sp(H) consists of those operators T : H → H such

that (λn) ∈ lp. If 1 ≤ p <∞ then

∥T∥Sp = ∥(λn)∥lp

is a norm on Sp(H), making it a Banach space.

Each Sp is an ideal in the space of all bounded

operators on H.

For p = 2, S2(H) is called the Hilbert-Schmidt

class. An operator T belongs to this class if and

only if∞∑n=1

∥T (en)∥2 <∞,

or equivalently, in case ϕ is univalent and ϕ(0) = 0,

by changing variable,

Cϕ ∈ S2(D) ⇔∫ϕ(D)

1

(1− |z|2)2dA(z) <∞.

Thus, in this special, case Cϕ ∈ S2(D) if and only

if ϕ(D) has finite hyperbolic area.

The complete characterization for Cϕ to be in Sp(H2)

uses the Nevanlinna counting function Nϕ.

Theorem. [Luecking -Zhu] For 0 < p <∞,

Cϕ ∈ Sp(H2) ⇔

Nϕ(z)

log(1/|z|)∈ Lp/2(D, dλ),

where dλ(z) = dA(z)(1−|z|2)2 is the Mobius invariant area

measure on D.

Fort A2 there is a partial result:

Theorem.[Zhu] Suppose p ≥ 2 and ϕ : D → D is

univalent or bounded-valent. Then

Cϕ ∈ Sp(A2) ⇔

∫D

(1− |z|2

1− |ϕ(z)|2

)pdλ(z) <∞.

There are several constructions of maps ¢ such

that Cr_p belongs or does not belong to certain Sp(H2 ).

For example if

1 1 ¢(]J)) = z : Im(z) > 0, lz- -1 < -,

2 2 conformally with ¢(1) = 1,

then Cr_p E Sp(H2 ) :: p > 2. Changing the angle

between the two arcs in the above picture, one can

obtain ¢ such that Cr_p belongs to Sp(H2 ) exactly

when p >Po, for any desired PoE (0, oo).

In addition there are constructions of¢ such that:

(i) Cr_p is compact but Cr_p tf:_ Sp(H2 ) for any p > 0.

(ii) ¢(]J)) = ]J) and Cr_p E Sp(H2 ) for all p > 0.

Similar results hold for composition operators be

longing to Schatten classes of A 2 and D.

6. Invariant subspaces and Composition Op-erators.

If H is a Hilbert space and T : H → H a boundedlinear operator, a subspace K is invariant under Tif T (K) ⊂ K.

The invariant subspace problem asks if every T

has a nontrivial closed invariant subspace. That is,if for every T ,

Lat(T ) = 0, H,

where Lat(T ) is lattice of all invariant closed sub-spaces of T .

Normal operators (T ∗T = TT ∗), Compact opera-tors, Polynomially compact operators, Operatorswhich commute with a non-zero compact opera-tor (and are not multiples of I), have nontrivialinvariant subspaces.

Universal operators:

• An bounded operator U on a Hilbert space H iscalled universal if for every bounded operator Ton H there is a λ ∈ C and M ∈ Lat(U) such that

λT = J−1UJ,

with J : H →M a linear isomorphism, that is λT is

similar to U|M .

For example, Ta : L2(0,∞) → L2(0,∞), a > 0,

Ta(f)(x) = f(x+ a)

are universal.

• If U : H → H is onto and Ker(U) has infinite

dimension then U is universal.

• If U is universal, the following are equivalent:

1. Every linear bounded operator T on H has a

nontrivial closed invariant subspace.

2. Every closed invariant subspace M of U of di-

mension > 1 contains a proper closed invariant sub-

space. i.e. the minimal nontrivial closed invariant

subspaces for U are one-dimensional.

Now consider Cϕ on H2.

• Cϕ is normal if and only if ϕ(z) = az with |a| ≤ 1.

• If U : H → H is onto and Ker(U) has infinite

dimension then U is universal.

Proof: Let K = Ker(U). Construct V,W bounded

operators on H as follows:

V = U−1 where U = U |K⊥ and

W (en) = e′n,

(en), (e′n) orthonormal bases for H and K. Then

1. UV = I, UW = 0

2. Ker(W ) = 0

3. W (H) = K and V (H) = K⊥

Let T be a linear bounded operator on H. Let λ = 0

such that |λ|∥T∥∥U∥ < 1 and define

J =∞∑k=0

λkV kWT k

Then J satisfies J =W+λV JT thus UJ = λJT . In

addition M = J(H) is a closed invariant subspace

of U , and J is an isomorphism onto M .

Concentrate to invertible C¢,

C¢ is invertible if and only if ¢ is a conformal automorphisms of JD),

· a- z ¢(z) == e~r _ , ial < 1, r E (-1r, 1r]

1- az These are classified according to the location of the two fixed points of ¢ as,

Elliptic when ¢ has a fixed point in JD), and the other fixed point outside JD)_ This happens if and only if lal < cos(r/2).

Parabolic when there is one fixed point of¢ on 1f of multiplicity 2. This cor-responds to lal == cos(r/2).

Hyperbolic when there are two distinct fixed points of¢ on 1f. This corresponds to lal > cos(r/2).

• Without loss of generality we may assume that

the fixed points are: 0 for elliptic, 1 for parabolic,

and −1,1 for hyperbolic. This is because in all

cases there is an automorphism ψ that moves the

fixed points to the special ones, and

Cψ−1CϕCψ = Cψϕψ−1

so Cϕ is similar to a composition operator of the

special type.

Invariant subspaces for Cϕ

• Clearly the space C of constants is invariant by

all composition operators on H2.

For invertible Cϕ we have,

• C is the only nontrivial common invariant sub-

space of the set of all invertible composition oper-

ators.

• Moreover the strongly closed unital algebra gen-

erated by the invertible composition operators is

Alg0,C, H2, that is, all operators in B(H2) leav-

ing 0,C and H2 invariant.

• If ϕ is elliptic automorphism of infinite order (i.e.

no iterate of ϕ is the identity) then every strongly

closed algebra A of operators containing Cϕ is re-

flexive, (that is, every operator that leaves invariant

all the invariant subspaces of A is actually in A).

• If ϕ is elliptic automorphism of infinite order and ψ

any automorphism that does not commute with ϕ,

then the only nontrivial common invariant subspace

of Cϕ and Cψ is C.

• If ϕ is parabolic or hyperbolic automorphism then

every weakly closed unital subalgebra of the algebra

generated by Cϕ is reflexive; that is, Cϕ is super-

reflexive.

More concrete results:

For ϕ automorphism define the iterates ϕn : n ∈ Zas ϕ0(z) = z,

ϕn = ϕ ϕn−1, n = 1,2, · · ·

and

ϕ−n = ϕ−1 ϕ−n+1, n = −1,−2, · · ·

• If ϕ is a hyperbolic or parabolic automorphism let

zn = ϕn(0), n ∈ Z,

be the orbit of 0 under all iterates of ϕ. Then

znn∈Z is a Blaschke sequence.

• Therefore we can form the Blaschke product

B(z) =∏n∈Z

λnϕn(z),

where λ0 = 1 and λn = zn/zn for n = 0. This B

satisfies

B(ϕ(z)) = ±B(z),

(+ for parabolic, − for hyperbolic).

• If ϕ is hyperbolic fixing −1,1 then

ϕ(z) =z+ r

1+ rz, for some 0 < r < 1,

and the spectrum of Cϕ on H2 is

σ(Cϕ) =

z :

√1− r

1+ r≤ |z| ≤

√1+ r

1− r

.Furthermore each λ in the interior of this annulus

is an eigenvalue, and for each such λ, Cϕ− λ maps

H2 onto H2.

• Take such an eigenvalue λ, and let f be an eigen-

vector corresponding to λ. Then for every integer

k ≥ 0,

Cϕ(B(z)2kf(z)) = B(ϕ(z))2kf(ϕ(z))

= (−B(z))2kλf(z)

= λB(z)2kf(z)

i.e. B2kf is also an eigenvector corresponding to

λ. Thus Cϕ − λ has infinite dimensional kernel.

Theorem. [Nordgren-Rosenthal-Wintrobe] If ϕ is

a hyperbolic automorphism and λ a point in the

interior of σ(Cϕ) then Cϕ−λ is a universal operator.

Corollary. The invariant subspace problem has a

positive answer if and only if the minimal nontriv-

ial invariant subspaces of Cϕ, for ϕ a hyperbolic

automorphism, are all one-dimensional.

Example: ϕ(z) = 2z+12+z

Note: Cϕ itself is not universal (it has trivial kernel)

but it has the same invariant subspaces as Cϕ − I

which is universal.

In general, a minimal invariant subspace M of an

invertible operator T : H → H is also invariant un-

der T−1.

Fix Cϕ as above. For f ∈ H2 denote

⟨f⟩ = spanCnϕ(f) : n ≥ 0 = spanf ϕn : n ≥ 0.

By the remark above we actually have

⟨f⟩ = spanf ϕn : n ∈ Z.

• Suppose f ∈ H2 is non-constant and limr→1 f(r) =

f(1) exists and is = 0 and f∗ is essentially bounded

on an open arc containing +1 or −1 then ⟨f⟩ is not

minimal.

• If f ∈ H2 has a singular inner factor Sµ(z) such

that µ(1) > 0 or µ(−1) > 0 then ⟨f⟩ is not

minimal.

Some results on non-automorphisms

Theorem. Suppose ϕ is an inner function with

ϕ(0) = 0 and is not an automorphism. Then C∗ϕ is

a universal operator.

Next look at the family of non-automorphisms

ϕa(z) =(2− a)z+ a

−az+2+ a, Re(a) > 0.

Each ϕa maps D into D, and on H2,

σ(Cϕa) = 0 ∪ e−at : t ∈ [0,∞).

In fact Cϕa(ft)(z) = e−atft(z) for

ft(z) = exp(tz+1

z − 1

), t ≥ 0.

Based on this information for the spectrum, the

invariant subspaces of this family of composition

operators were completely determined:

Theorem.[Montes, Ponce-Escudero, Shkarin]

A closed subspace M of H2 is invariant for Cϕa if

and only if there is a closed set F ⊂ [0,∞) such

that

M = spanet1+z1−z : t ∈ F

.

7. Cϕ and the Brennan conjecture.

Let Ω a simply connected open set in C, not the

whole plane, and let

h : Ω → D

a Riemman map. That is, h is 1-1, onto and ana-

lytic on Ω. We may assume that Ω contains 0 and

h(0) = 0.

For p > 0 consider the integral∫Ω|h′(z)|p dA(z).

For p = 2 the integral gives the area of D.

For 4/3 < p < 3 the Koebe distortion theorem from

the theory of Univalent Functions implies that the

integral is finite.

If we take Ω = C \ (−∞,−1/4] and

h(z) = k−1(z) : Ω → D,

where k(z) = z(1−z)2 is the Koebe function, shows

that the integral diverges for p /∈ (4/3,4).

J. Brennan (1978) proved: there is a δ > 0, inde-

pendent of Ω, such that the integral is finite for

4/3 < p < 3+ δ.

The conjecture says: the integral is finite for all

p ∈ (4/3,4).

The conjecture is known to be valid for some spe-

cial cases:

• Ω is convex,

i.e. for z, w ∈ Ω the segment [z, w] ⊂ Ω.

• Ω is starlike at 0,

i.e. for every w ∈ Ω, [0, w] ⊂ Ω.

• Ω is close-to-convex.

• Ω is a basin of attraction to ∞ of fc(z) = z2+ c,

Ω = Ωc = z : (fc)n(z) → ∞ asn→ ∞,

where c ∈ C is such that the orbit

fc(0), (fc)2(0), · · · , (fc)n(0), · · · , n ∈ N,

remains bounded. Each such Ωc is simply con-

nected in the Riemann sphere, and has a fractal

boundary.

• The conjecture is true when log(zh′(z)

h(z) ) has posi-

tive Taylor coefficients at 0.

The range of p for which the conjecture has been

verified is 4/3 < p < 3.42...

Brennan’s conjecture can be restated in terms of

g = h−1 : D → Ω in the following form:∫D

1

|g′(z)|pdA(z) <∞ for − 2/3 < p < 2.

The corresponding interval where it has been veri-

fied is −2/3 < p < 1.42...

Observe that the finiteness of the above integral

can be restated as 1(g′)p/2

∈ A2, so the conjecture is

equivalent to: For each g univalent on D, 1(g′)p ∈ A2

for all p ∈ (−1/3,1).

Connection to composition operators

Consider univalent maps ϕ : D → D, and the weighted

composition operators

Cϕ,γ(f)(z) = (ϕ′(z))γf(ϕ(z)), γ ∈ R.

For f ∈ A2 we have

∥Cϕ,γ(f)∥A2 =∫D|ϕ′(z)|2γ|f(ϕ(z))|2 dA(z)

=∫D|ϕ′(z)|2γ−2|f(ϕ(z))|2|ϕ′(z)|2 dA(z)

=∫ϕ(D)

|ϕ′(ϕ−1(w))|2γ−2|f(w)|2 dA(w),

and if we put f = 1 the last integral is∫ϕ(D)

|ϕ′(ϕ−1(w))|2γ−2 dA(w)

=∫ϕ(D)

|(ϕ−1)′(w)|2−2γ dA(w).

Thus boundedness of Cϕ,γ on A2 implies in partic-

ular the integrability of the derivative of the uni-

valent function h = ϕ−1 on the simply connected

Ω = ϕ(D). Exploiting this idea one obtains,

Theorem.[Shimorin] The following are equivalent

(i) The Brennan conjecture is valid.

(ii) For every ϕ : D → D univalent the weighted

composition operators

Cϕ,γ(f)(z) = (ϕ′(z))γf(ϕ(z))

are bounded on A2 for every γ ∈ (−1,0).

There is different way in which the conjecture re-lates to composition operators,

For g : D → C univalent and ϕ : D → D analytic(not necessarily univalent) consider the weightedcomposition operators

Wϕ,p(f)(z) = (ψϕ(z))pf(ϕ(z)), p ∈ R,

where ψϕ(z) = g′(ϕ(z))g′(z) .

Assuming this operator is bounded on A2 and usingthe reproducing kernels for A2,

Ka(z) =1

(1− az)2, a ∈ D,

we find

W ∗ϕ,p(Ka) = (ψϕ(a))

pKϕ(a).

Suppose now that for some ϕ and p this operatoris compact on A2. Then it can be shown that ϕhas a fixed point b ∈ D, and that ϕ it is not anautomorphism. Then

W ∗ϕ,p(Kb) = (ψϕ(b))

pKϕ(b) = Kb.

Thus 1 is an eigenvalue of W ∗ϕ,p so it is in the spec-

trum of Wϕ,p and since the later is compact, 1 is

also an eigenvalue of Wϕ,p. Thus there is f ∈ A2

such that

(ψϕ(z))pf(ϕ(z)) = f(z)

or equivalently the function κ = (g′)pf satisfies

κ(ϕ(z)) = k(z).

Since ϕ is not an automorphism, its iterates ϕn

converge pointwise to b. It follows that for each

z ∈ D,

κ(z) = κ(ϕ(z)) = κ(ϕn(z)) → κ(b),

i.e. κ(z) = κ(b) is constant = 0 (since f = 0).

Thus (1/g′)p = 1κ(b)f ∈ A2.

Summarizing, if Wϕ,p is compact for some ϕ then

(1/g′)p ∈ A2. The converse of this conclusion is

also true and is verified by choosing ϕ to be a con-

stant, giving a rank 1 operator. So we have

Theorem.[Matache-Smith] The following are equiv-

alent


(ii) For every p ∈ (−1/3,1) there is a ϕ such that

the weighted composition operator Wϕ,p is a com-

pact operator on A2.

The above theorem can be restated in another form

as follows:

For Ω simply connected, h : Ω → D the conformal

map and p ∈ R consider the weighted Bergman

spaces A2ω,p defined by

∥f∥2ω,p =∫Ω|f(z)|2|h′(z)|2p+2 dA(z) <∞.

Let ϕω be an analytic self-map of Ω and let ϕ =

h ϕω h−1 the corresponding self-map of D. One

easily sees that a composition operator

Cϕω : A2ω,p → A2

ω,p

is unitarily equivalent to Wϕ,p : A2 → A2. Thus

Theorem’. The following are equivalent


(ii) The space A2ω,p supports compact composition

operators for each −1/3 < p < 1.

8. Semigroups of composition operators.

For ϕ : D → D analytic consider

ϕ0, ϕ1, ϕ2, · · · , ϕn, · · · , n ∈ N,

the iterates of ϕ, where we put ϕ0(z) = z, the

identity, and ϕ1 = ϕ.

Theorem.

1. If ϕ is an elliptic automorphism of D then there

is b ∈ D with ϕ(b) = b.

2. [Denjoy-Wolff (1926)] Suppose ϕ is not an el-

liptic automorphism.

2-(i). If ϕ has a fixed point b ∈ D then |ϕ′(b)| < 1

and ϕn → b, uniformly on compact subsets of D.2-(ii). If ϕ has no fixed point in D, then there is a

point ω = ω(ϕ) ∈ T such that ϕn → ω, uniformly on

compact subsets of D. Moreover

limr→1

ϕ(rω) = ω, and ϕ′(ω) ≤ 1.

Conversely if there is a fixed point ω ∈ T such that

ϕ′(ω) ≤ 1, then necessarily ϕn → ω.

And if, in addition, ϕ′(ω) < 1 then for each z ∈ D,ϕn(z) → ω nontangentially.

The distinguished point at which ϕn converge, or

the fixed point in the case of elliptic automorphisms

will be called the Denjoy-Wolff point of ϕ.

Fractional iterates

In many cases it is possible to embed the discrete

semigroup of iterates ϕn into a continuous pa-

rameter semigroup

ϕt, 0 ≤ t <∞,

of fractional iterates of ϕ satisfying ϕs ϕt = ϕs+tfor s, t ≥ 0. If ϕ is an automorphism this embedding

is always possible and in fact into a a continuous

parameter group ϕt : t ∈ R.

Thus we assume that we have a family ϕt : t ≥ 0of analytic self-maps of D satisfying

(i) ϕ0 is the identity map of D.(ii) ϕs ϕt = ϕs+t for all s, t ≥ 0.

(iii) ϕtt→0→ ϕ0, uniformly on compact subsets of D.

Such a family will be called a semigroup of func-

tions.

Given a semigroup of functions ϕt : t ≥ 0 let

Ct(f) = f ϕt, f ∈ A(D).

the corresponding composition operators. The fam-

ily Ctt≥0 satisfies:

(i) C0 ≡ I, the identity operator,

(ii) Ct Cs = Ct+s, t, s ≥ 0,

i.e. it forms a semigroup of operators on A(D).Thus if X ⊂ A(D) is a Banach space such that

Ct : X → X

are bounded for all t ≥ 0, then we have an operator

semigroup Ctt≥0 on X.

Examples of ϕt.1. The group of rotations

ϕt(z) = eitz, t ∈ R.

2. A semigroup that shrinks compact sets to 0

ϕt(z) = e−tz, t ≥ 0.

3. A semigroup that shrinks compact sets to 1

ϕt(z) = e−tz+1− e−t, t ≥ 0.

4. A semigroup that shrinks compact sets to 0,

but maintains 1 as a fixed point

ϕt(z) =e−tz

(e−t − 1)z+1, t ≥ 0.

5. Similar as 4., but nontangential at 1

ϕt(z) = 1− (1− z)e−t, t ≥ 0.

6 Group of hyperbolic automorphisms

ϕt(z) =(1+ et)z − 1+ et

(−1+ et)z+1+ et, t ∈ R,

fixing the points −1,1.

Some general facts for ϕt.

[Berkson-Porta (1978)] For a semigroup ϕt of

functions:

• each ϕt is univalent and all ϕt share the same

Denjoy-Wolff point.

• The limit G(z) = limt→0+ϕt(z)−z

t , exists uniformly

on compact subsets of D and it is therefore an

analytic function on D. G is called the infinitesimal

generator of ϕt.

• G(z) satisfies

G(ϕt(z)) =∂ϕt(z)

∂t= G(z)

∂ϕt(z)

∂z,

for each z ∈ D, t ≥ 0.

• G(z) has a unique representation

G(z) = (bz − 1)(z − b)F (z),

where |b| ≤ 1 is the common Denjoy-Wolff point of

ϕt and F (z) is analytic on D with ReF (z) ≥ 0.

The cases |b| < 1 and |b| = 1 for the Denjoy-Wolff

point turn out to be distinctly different as far as

the properties of ϕt and the induced operator

semigroup Ct are concerned. For simplicity and

in most cases without loss of generality we assume:

b = 0, for semigroups with |b| < 1,

b = 1, for semigroups with |b| = 1.

This can be achieved by pre- and post- composing

ϕt with automorphisms of D.

• Furthermore for each semigroup ϕt there is a

unique associated univalent function h : D → C

from which the semigroup is obtained in the fol-lowing way:

Case of DW point b = 0. Let G be the generatorof ϕt. The differential equation

h′(z) =G′(0)

G(z)h(z), h(0) = 0,

has a unique analytic solution h on D. BecauseReF (z) ≥ 0, it can be shown that h is a univa-lent spirallike function, i.e. having the geometricproperty

w ∈ h(D) ⇒ eG′(0)tw ∈ h(D), t ≥ 0,

and satisfies the Schroder functional equation

h(ϕt(z)) = eG′(0)th(z), z ∈ D, t ≥ 0.

In particular

ϕt(z) = h−1(eG′(0)th(z)), z ∈ D, t ≥ 0.

Case of DW point b = 1. The differential equa-tion

h′(z) =G(0)

G(z), h(0) = 0,

has again a unique analytic solution h. This h isa close-to-convex univalent function, having thegeometric property

w ∈ h(D) ⇒ w+G(0)t ∈ h(D), t ≥ 0,

and satisfies the Abel functional equation

h(ϕt(z)) = h(z) +G(0)t, z ∈ D, t ≥ 0.

In particular

ϕt(z) = h−1(h(z) +G(0)t), z ∈ D, t ≥ 0.

Thus, in principle, all information about ϕt is con-

tained in each of the following objects:

• the generator G(z), alternatively in the pair (b, F )

of the DW point and the corresponding function of

positive real part.

• the pair (b, h) of the DW point and the associated

univalent function.

Strong continuity of Ct.

Given a Banach space X of analytic functions on Dsuch that each composition operator Ct(f) = f ϕtis bounded on X, we would like to know if Ct is

strongly continuous on X, i.e. if

limt→0

∥Ct(f)− f∥X = 0, f ∈ X,

or maybe even uniformly continuous i.e.

limt→0

∥Ct − I∥X = 0.

Note: Since ϕt(z) → z as t→ 0 it follows that

Ct(f)(z) = f(ϕt(z)) → f(z)

for each z ∈ D. Thus proving strong continuity

requires that we pass from this pointwise conver-

gence to convergence in the norm of X.

A general argument applying to many spaces X is

as follows: Suppose X contains the polynomials as

a dense set, then for a polynomial P we have,

∥fϕt − f∥ ≤≤ ∥f ϕt − P ϕt∥+ ∥P ϕt − P∥+ ∥P − f∥≤ (∥Ct∥+1)∥P − f∥+ ∥P ϕt − P∥.

If we assume further that supt∈(0,δ) ∥Ct∥ < ∞ for

some δ > 0, a condition that is valid in all spaces

of concern, then the question reduces to showing

limt→0

∥P ϕt − P∥ = 0

for each P , and this will follow if we can show

limt→0

∥ϕt(z)k − zk∥ = 0.

for each k. The latter in many classical spaces

follows from a dominated convergence or a similar

theorem.

Using the above arguments, one can prove strong

continuity of Ct for all inducing ϕt on,

i) the Hardy spaces Hp, 1 ≤ p <∞,

ii) the Bergman spaces Ap, 1 ≤ p <∞,

iii) the Dirichlet space D.

On other spaces of analytic functions different things

can happen. For example,

• No nontrivial ϕt induces a strongly continuous

Ct on H∞.

Note: ϕt is trivial when ϕt(z) = z for each t,

equivalently the generator G ≡ 0.

To prove the assertion we use a general result from

Functional Analysis:

Theorem [Lotz] If a space X

(i) is a Grothendieck space i.e. every weak∗ con-

vergent (xn) ⊂ X∗ is also weakly convergent in X∗,(ii) and has the Dunford-Pettis property i.e. given

any sequences (xn) ⊂ X and (x∗n) ⊂ X∗, weakly

convergent to zero, then ⟨xn, x∗n⟩ → 0,

then every strongly continuous semigroup Tt of

bounded operators on X is automatically uniformly

continuous.

H∞ is such a space. If Ct is strongly continuous

then the resulting uniform continuity would mean

that its infinitesimal generator, which turns out to

be a differential operator of the form

Γ(f)(z) = G(z)f ′(z),

is bounded on H∞, which is impossible unless G =

0.

For the same reason, there are no nontrivial strongly

continuous Ct on the Bloch space B of all analytic

f such that

∥f∥B = |f(0)|+ supz∈D

(1− |z|2)|f ′(z)| <∞.

• On the disc algebra A(D), some ϕt induce strongly

continuous Ct and some other do not. The exact

characterization involves the associated univalent

function h of ϕt. Consider the image Ω = h(D)as a subset of the Riemann sphere and ∂∞Ω the

boundary of Ω in the Riemann sphere, then

Theorem. [M. Contreras - S. Diaz-Madrigal] Ctis strongly continuous on A(D) if and only if ∂∞Ω

is locally connected.

The proof is nontrivial, involves the Caratheodory

extension theorem for conformal maps and prime

ends.

• On Hardy spaces Hp(U) of the upper half-plane,

semigroups ϕt of analytic self-maps of U are de-

fined in analogy with D, but recall that not all

composition operators are bounded on Hp(U). For

semigroups however the following holds

(i) If there is a t > 0 such that Ct(f) = f ϕt is

bounded on Hp(U) then the same is true for all

t ≥ 0.

(ii) In that case Ct is strongly continuous on

Hp(U).

• On the space BMOA:

BMOA contains functions in H2 whose boundary

values f∗ have bounded mean oscillation on T. An

equivalent and more convenient for us definition is:

An analytic function f ∈ H2 is in BMOA if

supa∈D

∥f(ϕa(z))− f(a)∥H2 <∞,

where the sup is taken over all automorphisms

ϕa(z) =a− z

1− az

of the disc. In other words BMOA contains every

function in H2 whose Mobius translates

f(ϕa(z))− f(a) : a ∈ D

is bounded set in H2. With the norm

∥f∥∗ = |f(0)|+ supa∈D

∥f(ϕa(z))− f(a)∥H2,

BMOA becomes a non separable Banach space. Its

size is ∩p<∞

Hp ⊃ BMOA ⊃ H∞,

and it contains unbounded functions such as f(z) =

log( 11−z). In general if f ∈ BMOA then

|f(z)| ≤ C∥f∥∗ log1

1− |z|, z ∈ D.

The subspace of functions f ∈ BMOA such that

∥f(ϕa(z))− f(a)∥H2 → 0 as |a| → 1,

is the space VMOA, of functions with boundary

values f∗ of vanishing mean oscillation on T. With

the same norm ∥ ∥∗, VMOA is separable Banach

space and in fact

VMOA = p : p polynomial ∥ ∥∗.

Every function in VMOA has finite Dirichlet inte-

gral so VMOA ⊂ D. Thus VMOA does not con-

tain all bounded functions, but contains some un-

bounded functions. Note that log( 11−z) /∈ VMOA.

For any ϕ analytic self-map of D, Cϕ is a bounded

operator on BMOA, and

∥Cϕ∥BMOA→BMOA ≤ C log1 + |ϕ(0)|1− |ϕ(0)|

.

But Cϕ is bounded on VMOA if and only if ϕ ∈VMOA.

Now we come to strong continuity.

Since VMOA is the closure of the set of polyno-

mials it follows , one can prove that for every ϕtthe induced operator semigroup Ct is strongly

continuous on VMOA.

On BMOA the situation is much more complicated.

Given ϕt we need

limt→0

∥f ϕt − f∥∗ = 0, for each f ∈ BMOA.

On the other hand an old theorem of D. Sarason

says:

If f ∈ BMOA then the following are equivalent:

(a) f ∈ VMOA.

(b) limt→0 ∥f(eitz)− f(z)∥∗ = 0.

(c) limt→0 ∥f(e−tz)− f(z)∥∗ = 0.

Thus for the semigroups

ϕt(z) = eitz or ϕt(z) = e−tz,

there is no f ∈ BMOA \ VMOA that obeys the

strong continuity requirement.

Consider however the semigroup

ϕt(z) = e−tz+1− e−t,

then for the function

f(z) = log(1

1− z) ∈ BMOA \ VMOA,

we find

limt→0

∥f ϕt − f∥∗ = ... = limt→0

t = 0,

so this ϕt induces a strongly continuous Ct on

V = spanVMOA, log(1/(1− z)),

a space strictly between VMOA and BMOA.

In a similar way suppose

h : D → C, h(0) = 0,

is a starlike univalent function with the property

h ∈ BMOA \ VMOA and

h(z)k ∈ BMOA, 1 ≤ k ≤ k0.

Let

ϕt(z) = h−1(e−th(z)),

then for 1 ≤ k ≤ k0,

limt→0

∥(h ϕt)k − hk∥∗ = limt→0

|e−kt − 1|∥hk∥∗ = 0,

so Ct is strongly continuous on the space

V = spanVMOA, h, h2, · · ·hk0.

We are led therefore to define for each ϕt the

space

V (ϕt) = the maximal subspace of BMOA on which

ϕt is strongly continuous.

From the above examples there are ϕt such that

VMOA & V (ϕt), and there are other for which

VMOA = V (ϕt).

The next theorem gives a condition for equality

VMOA = V (ϕt). Recall that the generator of ϕthas the form

G(z) = (bz − 1)(z − b)F (z)

where b ∈ D and ReF (z) ≥ 0. Assume b = 0 then

G(z) = −zF (z), and recall that functions F of pos-

itive real part satisfy the lower growth inequality

|F (z)| ≥ C1− |z|1+ |z|

, |z| → 1.

Thus we have when b = 0

1− |z|G(z)

= O(1), |z| → 1.

Theorem. Let G be the generator of ϕt. If for

some 0 < α < 1,

(1− |z|)α

G(z)= O (1) , |z| → 1,

then VMOA = V (ϕt).

Thus in addition to

ϕt(z) = e−tz, ϕt(z) = eitz,

there are plenty of ϕt for which VMOA = V (ϕt).

For example,

G(z) = −z(1− z)α, 0 < α < 1.

Infinitesimal generators

For those Ct that are strongly continuous on aspace X, the infinitesimal generator of Ct by def-inition is

Γ(f) = limt→0

Ct(f)− f

t,

the limit taken in ∥ ∥X. In all cases of concern,convergence in norm implies in particular pointwiseconvergence. It follows that for z ∈ D,

Γ(f)(z) = limt→0

f(ϕt(z))− f(z)

t

= limt→0

(f ϕt)(z)− (f ϕ0)(z)t

=∂(f ϕt)(z)

∂t|t=0

= f ′(ϕ0(z))∂ϕt(z)

∂t|t=0

= f ′(z)G(z),

where in the last step we have used the earlier men-tioned fact that

G(z) = limt→0

ϕt(z)− z

t=∂ϕt(z)

∂t|t=0.

Thus the infinitesimal generator of Ct is the dif-ferential operator

Γ(f) = Gf ′,

and is defined on the domain D(Γ) = f ∈ X :

Gf ′ ∈ X. Since differential operators such as Γ

are not bounded on spaces under concern, non-

trivial strongly continuous Ct are never uniformly

continuous.

Resolvent and related operators

The spectrum of the infinitesimal generator Γ is

always contained in a left half-plane and in some

special cases it can be found precisely. For λ not

in the spectrum the resolvent operator is

R(λ,Γ) = (λ− Γ)−1.

We restrict to the case when the DW point of ϕtis b = 0. Calculating the resolvent involves solving

a differential equation of the form

f = λg −Gg′.

Choosing the convenient point λ = −G′(0) we find

R(−G′(0),Γ) to be a constant multiple of

Rh(f)(z) =1

h(z)

∫ z0f(ζ)h′(ζ) dζ,

where h is the univalent function associated to ϕt.

A calculation shows that Rh is closely related to

the operator

Qh(f)(z) =1

z

∫ z0f(ζ)

ζh′(ζ)

h(ζ)dζ,

(for example these two operators belong simulta-

neously to the same operator ideals). Another cal-

culation shows that the same is true between Qhand the operator

Lh(f)(z) =1

z

∫ z0f(ζ)

(log

h(ζ)

ζ

)′dζ.

A theorem from classical analysis says that for ev-

ery univalent function h with h(0) = 0 we have

logh(z)

z∈ BMOA,

and this leads us to consider more general Volterra-

type operators of the form

Tg(f)(z) =∫ z0f(ζ)g′(ζ) dζ

where g is analytic on D. The class of these op-

erators was studied intensively during the last 20

years on various spaces of analytic functions, with

the goal to find conditions on the symbol g which

correspond various operator theoretic properties of

Tg. A sample result is,

Theorem[A. Aleman, A.S.] Let 1 ≤ p <∞, then

(i) Tg bounded on Hp ⇔ g ∈ BMOA

(ii) Tg compact on Hp ⇔ g ∈ VMOA.

(iii) Tg is in the Schatten ideal Sp(H2), 1 < p <∞,

if and only if g ∈ Bp, the analytic Besov space.

As a consequence we have, among other, a com-

plete characterization of those inducing semigroups

ϕt for which the composition operator semigroup

Ct has compact resolvent.

Theorem. Let ϕt be a semigroup of functions

with DW point b = 0 and associated univalent func-

tion h. Let also 1 ≤ p <∞ and Ct be the operator

semigroup induced by ϕt. Then the following are

equivalent:

(i) R(λ,Γ) compact on Hp.

(ii) log h(z)z ∈ VMOA.

(iii) h ∈ ∩p<∞Hp.

Recall that the infinitesimal generator of ϕt has

a representation in terms of the DW point and a

function F of positive real part. Then 1/F has

also positive real part and so it admits a Herglotzrepresentation

1

F (z)=∫T

ζ + z

ζ − zdµ(ζ) + iIm

1

F (0),

where µ is a positive measure on T. It can beshown then that each of the above three conditionsis further equivalent to

(iv) The measure µ in the representation of 1/Fhas no point masses on T.

Similar results about Tg and R(λ,Γ) can be provedon Bergman and other spaces.

Weighted composition semigroups and applications

If ϕt is a semigroup of functions and ψ : D → C isanalytic then the formula

St(f)(z) =ψ(ϕt(z))

ψ(z)f(ϕt(z)),

defines formally a semigroup of operators St.

Various choices of ψ give interesting cases for St.For example if the DW point is b = 0 and we chooseψ = G, the generator of ϕt, then we obtain

St(f)(z) = ϕ′t(z)f(ϕt(z)),

a semigroup of operators related to the Brennan

conjecture. If ϕt is a group of Mobious auto-

morphisms and we choose formally ψ = G1/p we

obtain

St(f)(z) = (ϕ′t(z))1/pf(ϕt(z)),

a group of isometries of Hp.

Many questions about the class of weighted com-

position semigroups remain unanswered. For ex-

ample the strong continuity is not completely char-

acterized even on Hardy spaces.

When St are strongly continuous, their infinites-

imal generators have the form

∆(f)(z) = G(z)f ′(z) + g(z)f(z),

i.e. they are perturbations of the infinitesimal gen-

erators of the corresponding unweighted semigroups,

by the multiplication operator Mg(f) = gf where g

depends on the weight function ψ.

The resolvent operators often are related to classi-

cal integration or averaging operators. A particular

example is the Cesaro operator.

Cesaro operators

The name Cesaro operator is used for the averaging

process on sequences,

(an) → (An), An =a1 + a2 + · · ·+ an

n,

or on locally integrable functions

f → F (x) =1

x

∫ x0f(t)dt.

Observe that both are linear processes. We con-

centrate on the discrete version. The classical

Hardy’s inequality

∞∑n=1

∣∣∣∣a1 + a2 + · · ·+ an

n

∣∣∣∣p ≤(

p

p− 1

)p ∞∑n=1

|an|p,

which is valid for 1 < p < ∞ and for which the

constant(

pp−1

)pis best possible, says that the op-

erator

C : (an) → (An)

is a bounded operator on the sequence spaces lp

with norm

∥C∥lp→lp =p

p− 1.

Using the identification H2 ≃ l2 we see that the

operator mapping f(z) =∑∞n=0 anz

n ∈ H2 to

C(f)(z) =∞∑n=0

(1

n+1

n∑k=0

ak)zn

is bounded and

∥C∥H2→H2 = ∥C∥l2→l2 = 2.

The above transformation of power series can be

shown to map power series with radius of conver-

gence r ≥ 1, to power series with radius of conver-

gence R ≥ 1 (exercise). We would like to know if

it also maps Hp into Hp for p = 2.

To study this question we use a particular weighted

composition semigroup. Let

ϕt(z) =e−tz

(e−t − 1)z+1

and ψ(z) = z, then the operators

St(f)(z) =ϕt(z)

zf(ϕt(z))

are bounded on Hardy spaces, the semigroup Stis strongly continuous and has generator

∆(f)(z) = −z(1− z)f ′(z)− (1− z)f(z).

An involved calculation gives the spectrum,

σ(∆) = z : Re(z) ≤ −1/p.

Thus 0 is in the resolvent set of ∆. Observe that

∆(f)(z) = −(1 − z)(zf(z))′ and we find for func-

tions f(z) =∑∞n=0 anz

n ∈ Hp,

R(0,∆)(f)(z) =1

z

∫ z0f(ζ)

1

1− ζdζ

=∞∑n=0

(1

n+1

n∑k=0

ak

)zn

= C(f)(z),

the Cesaro operator. A detailed study of the semi-

group St leads to

Theorem[A.S.] If 1 ≤ p <∞ then C : Hp → Hp is a

bounded operator. Moreover

(i) If 2 ≤ p <∞ then ∥C∥Hp→Hp = p

(ii) For some p ∈ (1,2), ∥C∥Hp→Hp > p.

It is remarkable that the adjoint C∗ is also the resol-

vent operator for a composition semigroup. Indeed

let

ϕt(z) = e−tz+1− e−t, and ψ(z) = 1− z,

then the operators

Tt(f)(z) =w(ϕt(z))

w(z)f(ϕt(z)) = e−tf(ϕt(z)),

from a semigroup of bounded operators on Hp with

generator

∆(f)(z) =∂(e−tf(ϕt(z)))

∂t|t=0 = (1−z)f ′(z)−f(z).

It turns out that 0 is not in the spectrum of ∆,

and the resolvent at 0 is

A(f)(z) =1

z − 1

∫ z1f(ζ) dζ.

This operator is the adjoint of C. The easiest way

to see this is to find the matrices for C and A with

respect to the usual basis 1, z, z2, · · · of Hp. They

are transposes of each other.

References

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