com pop 0905 slides a

7/30/2019 Com Pop 0905 Slides A

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Composition Operators on

Spaces of Analytic Functions

Carl C. Cowen

IUPUI

(Indiana University Purdue University Indianapolis)

Spring School of Functional Analysis, Rabat, 19 May 2009


2/31

Functional analysis began a little more than 100 years ago

Questions had to do with interpreting differential operators

as linear transformations on vector spaces of functions

Sets of functions needed structure connected to the convergence

implicit in the limit processes of the operators

Concrete functional analysis developed with results on spaces

of integrable functions, with special classes of differential operators,

and sometimes used better behaved inverses of differential operators


3/31

The abstraction of these ideas led to:

Banach and Hilbert spaces

Bounded operators, unbounded closed operators, compact operators

Spectral theory as a generalization of Jordan form and diagonalizability

Multiplication operators as an extension of diagonalization of matrices

Concrete examples and development of theory interact:

Shift operators as an examples of asymmetric behavior possible

in operators on infinite dimensional spaces

Studying composition operators can be seen as extension of this process


4/31

The classical Banach spaces are spaces of functions on a set X: if is map

ofX onto itself, we can imagine a composition operator with symbol ,

Cf = f

for f in the Banach space.

This operator is formally linear:

(af + bg) = af + bg

But other properties, like Is f in the space? clearly depend on the

map and the Banach space of functions.


5/31

Several classical operators are composition operators. For example, we may

regard p(N) as the space of functions ofN into C that are pth power

integrable with respect to counting measure by thinking x in p as the

function x(k) = xk. If : N N is given by (k) = k + 1, then

(Cx)(k) = x((k)) = x(k + 1) = xk+1, that is,

C : (x1, x2, x3, x4, ) (x2, x3, x4, x5, )

so C is the backward shift.

In fact, backward shifts of all multiplicities can be represented as

composition operators.


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Moreover, composition operators often come up in studying other operators.

For example, if we think of the operator of multiplication by z2,

(Mz2f)(z) = z2f(z)

it is easy to see that Mz2 commutes with multiplication by any bounded

function. Also, Cz commutes with Mz2:

(Mz2Czf)(z) = Mz2f(z) = z2f(z)

and

(CzMz2f)(z) = Cz(z2f(z)) = (z)2f(z) = z2f(z)

In fact, in some contexts, the set of operators that commute with Mz2

is the algebra generated by the multiplication operators and the

composition operator Cz.


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Also, Forelli showed that all isometries ofHp(D), 1 < p < , p = 2, are

weighted composition operators.

In these lectures, we will not consider absolutely arbitrary compositionoperators; a more interesting theory can be developed by restricting our

attention to more specific cases.


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Definition

Banach space of functions on set X is called a functional Banach space if

1. vector operations are the pointwise operations

2. f(x) = g(x) for all x in X implies f = g in the space

3. f(x) = f(y) for all f in the space implies x = y in X

4. x f(x) is a bounded linear functional for each x in X

We denote the linear functional in 4. by Kx, that is, for all f and x,

Kx(f) = f(x)

and if the space is a Hilbert space, Kx is the function in the space with

f, Kx = f(x)


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Examples

(1) p(N) is a functional Banach space, as above

(2) C([0, 1]) is a functional Banach space

(3) Lp([0, 1]) is not a functional Banach space because

f f(1/2)

is not a bounded linear functional on Lp([0, 1])

We will consider functional Banach spaces whose functions are analytic on

the underlying set X;

this what we mean by Banach space of analytic functions


10/31

Examples (contd) Some Hilbert spaces of analytic functions:

(4) Hardy Hilbert space: X = D = {z C : |z| < 1}

H2(D) = {f analytic in D : f(z) =

n=0

anzn with f2H2 =

|an|

2 < }

where for f and g in H2(D), we have f, g =

anbn

(5) Bergman Hilbert space: X = D

A2(D) = {f analytic in D : f2A2 =

D

|f()|2dA()

< }

where for f and g in A2(D), we have f, g =

f()g() dA()/

(6) Dirichlet space: X = D

D = {f analytic in D : f2D = f2H2 +

D

|f()|2dA()

< }

(7) generalizations where X = BN


11/31

IfH is a Hilbert space of complex-valued analytic functions on the domain

in C or CN and is an analytic map of into itself,

the composition operator C on H is the operator given by

(Cf) (z) = f((z)) for f in H

At least formally, this defines C as a linear transformation.

In this context, the study of composition operators was initiated about 40

years ago by Nordgren, Schwartz, Rosenthal, Caughran, Kamowitz, and

others.


12/31

IfH is a Hilbert space of complex-valued analytic functions on the domain

in C or CN and is an analytic map of into itself,

the composition operator C on H is the operator given by

(Cf) (z) = f((z)) for f in H

At least formally, this defines C as a linear transformation.

Goal:

relate the properties of as a function with properties ofC as an operator.


13/31

Today, well mostly consider the Hardy Hilbert space, H2

H2 = {f =

n=0

anzn :

n=0

|an|2 < }

which is also described as

H2 = {f analytic in D : sup0


14/31

For H2, the Littlewood subordination theorem plus some easy calculations

for changes of variables induced by automorphisms of the disk imply that

C is bounded for all functions that are analytic and map D into itself

and the argument yields the following estimate of the norm for composition

operators on H2:

1

1 |(0)|2

12

C

1 + |(0)|

1 |(0)|

12

This is the sort of result we seek, connecting the properties of the operator

C with the analytic and geometric properties of .


15/31

When an operator theorist studies an operator for the first time, questions

are asked about the boundedness and compactness of the operator,

about norms,

spectra,

and adjoints.

While the whole story is not known, much progress has been made


16/31

When an operator theorist studies an operator for the first time, questions

are asked about the boundedness and compactness of the operator,

about norms,

spectra,

and adjoints.

While the whole story is not known, much progress has been made

and we expect the answers to be given in terms of analytic and geometric

properties of.


17/31

Very often, calculations with kernel functions give ways to connect the

analytic and geometric properties of with the operator properties ofC.

For a point in the disk D, the kernel function K is the function in

H2(D) such that for all f in H2(D), we have

f, K = f()

f and K are in H2, so f(z) =

anz

n and K(z) =

bnzn

for some coefficients. Thus, for each f in H2,

an

n

= f() = f, K =

anbn

The only way this can be true is for bn = n = n and

K(z) =

nzn =

1

1 z


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For a point in the disk D, because the kernel function K is a function in

H2

(D

), we have

K2 = K, K = K() =

1

1 =

1

1 ||2

These ideas show that H2(D) is functional Hilbert space and that

K = (1 ||2

)1/2

For each f in H2 and in the disk,

f, C K = Cf, K = f , K = f(()) = f, K()

Since this is true for every f, we see C (K) = K()

Further exploitation of this line of thought shows that C is invertible if and

only if is an automorphism of the disk and in this case, C1 = C1


19/31

In addition to asking When is C bounded? operator theorists would

want to know When is C compact?

Because

analytic functions take their maxima at the boundary

compact operators should take most vectors to much smaller vectors

expect C compact implies (D) is far from the boundary in some sense.

Ifm({ei : |(ei)| = 1}) > 0, then C is not compact.

If < 1, then C is compact.


20/31

In H2 and similar spaces, as || 1, then 1KK 0 weakly.

C is compact if and only ifC is compact, and in this case, we must have

C

1

KK

=K()

K=

1 ||2

1 |()|2

is going to zero.

Now if non-tangentially with || = 1 and the angular derivative ()

exists, then the Julia-Caratheodory Theorem shows that 1||2

1|()|2 1()

In particular, C compact implies no angular derivative of is finite.


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Theorem (1987, J.H. Shapiro)

Suppose is an analytic map ofD into itself. For C acting on H2(D),

C2e = limsup

|w|1

N(w)

log |w|

where N is the Nevanlinna counting function.

Corollary

C is compact on H2(D) if and only if lim sup

|w|1

N(w)

log |w|= 0

In some spaces larger than the Hardy Hilbert space, like the Bergman space,

C is compact if and only if has no finite angular derivatives


22/31

Caughran and Schwartz (1975) showed that if C is compact,

then has a fixed point in D

and found spectrum ofC in terms of data at the fixed point.

This was the first of many results that show how the behavior ofC

depends on the fixed points of . Digress to talk about fixed points.

If is a continuous map ofD into D, then must have a fixed point in D.

Only assume is analytic on D, open disk!

Definition

Suppose is an analytic map ofD into itself.

If |b| < 1, we say b is a fixed point of if(b) = b.

If |b| = 1, we say b is a fixed point of if limr1

(rb) = b.


23/31

Julia-Caratheordory Theorem implies

Ifb is a fixed point of with |b| = 1, then limr1 (rb) exists

(call it (b)) and 0 < (b) .

Denjoy-Wolff Theorem (1926)

If is an analytic map ofD into itself, not the identity map,

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

For not an elliptic automorphism ofD, for each z in D, the sequence

(z), 2(z) = ((z)), 3(z) = (2(z)), 4(z) = (3(z)),

converges to a and the convergence is uniform on compact subsets ofD.

This distinguished fixed point will be called the Denjoy-Wolff point of .


24/31

The Schwarz-Pick Lemma implies has at most one fixed point in D

and if has a fixed point in D, it must be the Denjoy-Wolff point.

Examples

(1) (z) = (z+ 1/2)/(1 + z/2) is an automorphism ofD fixing 1 and 1.

The Denjoy-Wolff point is a = 1 because (1) = 1/3 (and (1) = 3)

(2) (z) = z/(2 z2) maps D into itself and fixes 0, 1, and 1.

The Denjoy-Wolff point is a = 0 because (0) = 1/2 (and (1) = 3)

(3) (z) = (2z3 + 1)/(2 + z3) is an inner function fixing fixing 1 and 1

with Denjoy-Wolff point a = 1 because

(1) = 1 (and

(1) = 9)

(4) Inner function (z) = exp(z+ 1)/(z 1) has a fixed point in D,

Denjoy-Wolff point a .21365, and infinitely many fixed points on D


25/31

Denjoy-Wolff Thm suggests looking for a model for iteration of maps ofD

Five different types of maps ofD into itself from the perspective of iteration,

classified by the behavior of the map near the Denjoy-Wolff point, a

In one of these types, (a) = 0, (e.g., (z) = (z2 + z3)/2 with a = 0),

the model for iteration not yet useful for studying composition operators

In the other four types, when (a) = 0, the map can be intertwined with

a linear fractional map and classified by the possible type of intertwining:

intertwines and in the equality =

We want to do this with linear fractional and univalent near a, so that

is, locally, a change of variables. Using the notion of fundamental set, this

linear fractional model becomes essentially unique [Cowen, 1981]


26/31

A linear fractional model in which maps D into itself with a = 1 and

(1) =1

2, maps D into the right half plane, and (w) =

1

2w


27/31

Linear Fractional Models:

maps D into itself with (a) = 0 ( not an elliptic automorphism)

is a linear fractional automorphism of onto itself

is a map ofD into with =

I. (plane dilation) |a| < 1, = C, (a) = 0, (w) = (a)w

II. (half-plane dilation) |a| = 1 with (a) < 1, = {w : Re w > 0},

(a) = 0, (w) = (a)w

III. (plane translation) |a| = 1 with (a) = 1, = C, (w) = w + 1

IV. (half-plane translation) |a| = 1 with (a) = 1, = {w : Im w > 0},

(or = {w : Im w < 0}), (w) = w + 1


28/31

Linear Fractional Models:

maps D into itself with (a) = 0 ( not an elliptic automorphism)

is a linear fractional automorphism of onto itself

is a map ofD into with =

I. (plane dilation) |a| < 1, = C, (a) = 0, (w) = (a)w

II. (half-plane dilation) |a| = 1 with (a) < 1, = {w : Re w > 0},

(a) = 0, (w) = (a)w

III. (plane translation) |a| = 1 with (a) = 1, = C, (w) = w + 1

{n(0)} NOT an interpolating sequence

IV. (half-plane translation) |a| = 1 with (a) = 1, = {w : Im w > 0},

(or = {w : Im w < 0}), (w) = w + 1

{n(0)} IS an interpolating sequence


29/31

Theorem (Koenigs, 1884)

If is analytic map ofD into itself, (0) = 0, and 0 < |(0)| < 1, then

there is a unique map with (0) = 0, (0) = 1, and = (0)

Moreover, iff is analytic (not the zero map) and is a number so that

f = f, then = (0)n for some n = 0, 1, 2, 3, and f = c n

for some c

This is the plane dilation case (I.) with a = 0 and (w) = (0)w


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there is a unique map with (0) = 0, (0) = 1, and = (0)



for some c

Proof:

Starting with the second part, suppose f satisfies f((z)) = f(z) for some

and all z in D. Consider the Taylor series for f, f(z) = akzk with firstnon-zero coefficient an, that is, an = 0 and ak = 0 for k an integer, k < n.

Since any non-zero multiple of f will work just as well, we suppose an = 1.

Compare the Taylor series for f and f

get f unique and = (0)n


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there is a unique map with (0) = 0,

(0) = 1, and =

(0)



for some c

Proof:

For the first part, define by (z) = limk

k(z)

(0)kwhere 2(z) = ((z)),

3(z) = (2(z)), etc. Establish convergence; observe satisfies functional

equation, (0) = 0, and (0) = 1.

To finish, since the solution f from the earlier part was unique and n

also satisfies the conditions, f = n.

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