Topics in Dynamics: I: Flows 9781400872596

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TOPICS IN DYNAMICS I: FLOWS

BY EDWARD NELSON

PRINCETON UNIVERSITY PRESS AND THE

UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 19E$

Copyright

© 1970, by Princeton University Press All Rights Reserved L.C. Card: S.B.N.:

79-108265

691-08080-1

A.M.S. 1968:

3404, 4615

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

I.

FLOWS

1. Differential calculus 2.

(i) Page 1

Picard's method

3· The local structure of vector fields

13 29

4. Sums and Lie products of vector fields 5·

Self-adjoint operators on Hilbert space

6o

6.

Commutative multiplicity theory

77

7·

Extensions of Hermitean operators

99

8.

Sums and Lie products of self-adjoint operators . .

103

Notes and references

112

These are the lecture notes for the first term of a course on differential equations, given in Fine Hall the autumn of 1968. It is a pleasure again to thank Miss Elizabeth Epstein for her typing.

I. FLCWS

In classical mechanics the state of a physical system is repre­ sented by a point in a differentiable manifold

M

and the dynamical

variables by real functions on M . In quantum mechanics the states are given by rays in a Hilbert space H and the dynamical variables by selfadjoint operators on H

In both cases motion is represented by a flow;

that is, a one-parameter group of automorphisms of the underlying struc­ ture (diffeomorphisms or unitary operators). The infinitesimal description of motion is in the classical case by means of a vector field and in quantum mechanics by means of a selfadjoint operator

One of the central problems of dynamics is the

integration of the equations of motion to obtain the flow, given the infinitesimal description of the flow.

1.

Differential calculus In recent years there has been an upsurge of interest in infinite

dimensional manifolds.

The theory has had important applications to

Morse theory, transversality theory, and in other areas.

It might be

thought that an infinite dimensional manifold with a smooth vector field on it is a suitable framework for discussing classical dynamical systems with infinitely many degrees of freedom.

However, classical dynamical

systems of infinitely many degrees of freedom are usually described in terms of partial differential operators, and partial differential operators cannot be formulated as everywhere-defined operators on a

2.

I. FLOWS

Banach space. folds.

We will be concerned only with finite dimensional mani-

Despite this, I will begin by discussing the general case.

this for two reasons:

I do

because the theory is useful in other branches of

mathematics and because the fundamental concepts are clearer in the general context. Let

E be a real Banach space.

space with a function such that

/lx/l ::: 0,

x~

0

/!xll

That is,

mapping

/lxll

only i f

0,

/I x+yll ::: II xii + II yll , and E is complete: an

x

in

E with

/Ix -xII ---;;:. O.

F

=

sup{IIAxll: II xii ::: I} .

be in

E

of

f: U ----;;. F

L(E,F)

o(y)

as

by

(where

The function

Banach space

f

is

with

f: U ----;;. F

case it is differentiable and If

x

F

L(E) . E, and let

x

is a Banach space) is

in case there is an element

l

C

y

I-

O.

0

such that

It is clear that

Df(x)

It is called the (Frechet) derivative of

is differentiable at all points

L(E, F)

L(E,E)

f(x) + Df(x)y + o(y) ,

y ----;;. 0

is unique if it exists. x.

in the norm

is a function defined in a neighborhood of

Ilo(y)II/llyll ---;;:. 0

at

F

the Banach space

such that f(x+y)

where

into

We abbreviate

said to be (Frechet) differentiable at Df(x)

L(E,F)

U be an open subset of the Banach space A function

U

may be an

2 II x II = ( xl2 + ... + x s )2

in the norm

is another Banach space we denote by

Let

there is

1

:IRs

of all continuous linear mappings of IIAII

E

:IR

Ia I/! xl!

=

II x -x II ----;;. 0 n m

For example,

n

s-dimensional Euclidean space If

/! ax/!

if

is a real vector

into the real numbers

E

x

E

then

x

is called differentiable in case it in

U, and it is called

x ~ Df(x) Df

f

l C

is continuous from

is a function from

L(E,F), so it makes sense to ask whether

in

U to

U into the Df

is

l C

1.

The function

f

Ck - l

2

is said to be

and, by recursion, is

DIFFERENTIAL CALCULUS

f

C

in case k C

is said to be

3·

f

l C

is

in case

f

and l C

is

(A trivially equivalent definition is that

Df

f

is

l C

is

and

Df

k C

in

Sometimes one definition and sometimes the other suggests the more convenient way to organize an induction proof to show that be k-l

k

f

is

ck

Similarly, we define

times differentiable in case it is differentiable and times differentiable (or equivalently, in case it is

differentiable and differentiable at

Dk-lf x

is differentiable).

it is continuous at

entiable function is continuous, and a is

.)

k

Df

k-l

Notice that if

x.

to

f

is

times f

is

Consequently a differ-

times differentiable function

Ck - l Let

product

E , ... ,E n l

El X... X En

be Banach spaces, and consider their Cartesian It is possible to give this a Banach space

structure by defining addition and scalar multiplication componentwise and giving an element the norm which is the sum of the norms of its components.

This Banach space is denoted by

El ffi ... ffiE

n

and called

the direct sum of the Banach spaces

E ,· .. ,E n l

Elements of it are

denoted by

is in

Frequently we wish to

Xl ffi ... ffi xn ' where

consider multilinear forms on El X... X En

X.

1

E.

1

El X... X En ; that is, functions on

which are linear in each variable separately.

also a Banach space, we let

L(E

l

X... X En,F)

all continuous multilinear forms on

=

f

is

be the Banach space of

El X... X En

with values in

with the norm IIAII

If

sup{IIA(Yl""'y): "yl",···,"yn II -< I} n

This Banach space may be identified with the Banach space

F,

4.

I.

L(El,···,L(E

FLCMS

n- 1,L(E n ,F)) ... )

under the identification which takes an element

A of the latter into

the form given by

n

The set of symmetric elements of it is denoted by is in

L(E l

X ... X

En,F)

we denote the value

AYl ... Y . n

Also, if

so that

is defined if A

Ayn

U open in

E)

is

k

Y

is in

spaces, let U be open in

is in

yn

means

Ln(E,F)

f(x)·g(x) .

Let

0

E, let

(zl,z2) ~ zl·z2

(fog)(x)

then

A(Yl'·· ."Yn)

Then

be in

L(F

l

X F2 ,G)

fog: U ~ G is

k

(with

takes values in

G be Banach

g: U

~

and define

C

n times,

f: U ~ F

E, Fl , F2 ' and and

A

by 0

Dkf

f: U ~ Fl

If

(y,. o,Y)

If

times differentiable then

Theorem 1 (product rule)

Ck , let

E

L (E,F) sym

F2 fog

be by

and

D(fog)(x)y = Df(x)yog(x) + f(x)oDg(x)y

(1) Proofo

Suppose

f

and

g

are

l C .

f(x+y)

f(x) + Df(x)y + o(y)

g(x+y)

g(x) + Dg(x)y + o(y)

Then

so that f(x+y) g(x+y) 0

Thus

f· g

is

l

C

f(x)og(x) + Df(x)y·g(x) + f(x)oDg(x)y + o(y) and (1) holds, so that the theorem is proved for k = 1

Suppose the theorem to be true for

k-l, and let

f

and

g

be

Ck-l .

1.

Then (1) holds.

is

g

By ~ Ay·z , is continuous and bilinear.

k-l,

k-l C ,and similarly for the other term.

Ck-l ,so

f· g

Ck

is

given by

l

k-l C ,so by the theorem for

are

5·

~: L(E,F ) X F2 ~ L(E,G)

The mapping

(A,z) ~> B ,where and

DIFFERENTIAL CALCULUS

Now

Df

x ~ ~(Df(x),g(x))

Therefore

D(f'g)

is

This concludes the proof.

The same proof shows that the theorem with "Ck" replaced by "k times differentiable" is true. Theorem 2 (chain rule). let

U be open in E, let

and

g: V ~ G be

(2)

k C .

Suppose

Then

f

f(x+y) (go f) (x+y)

=

E, F,

and

G be Banach spaces,

V be open in F, and let

D(gof)(x) Proof.

Let

=

and

gof

k C

is

f: U ~ V

and

Dg(f(x))Df(x) g

l C .

are

Then

f(x) + Df(x)y + o(y) ,

g(f(x+y)) g(f(x)) + Dg(f(x))(Df(x)y + o(y)) + o(Df(x)y + o(y)) g(f(x)) + Dg(f(x))Df(x)y + o(y)

Hence

gof

l C

is

and (2) holds.

Suppose the theorem to be true for Then

Dg

and

The mapping of

are

f

Ck-l , so

L(F,G) X L(E,F)

k

Thus the theorem holds for k-l Dgo f

into

,

and let

is

Ck - l

L(E,G)

and

f

Also

=

1

g

be

k C

Df

is

Ck-l .

which takes two linear

operators into their product is continuous and bilinear, so by Theorem 1, Ck - l

and

(Dgo f) (Df) gOf

is

ck

is

Ck-l .

By

(2),

therefore,

D(go f)

is

, which completes the proof.

The following formulas are easily proved by induction, for

k C

6.

I.

functions

f

and

FLOWS

g

(~)

where the inner sum is over all two sets with

partitions of

and

sets with

q

into

and

where the inner sum is over all into

Y ,· "'Y l k

partitions of

r l' r 2' ... , r q

Y , ""Y

l

k

elements and the natural ordering in

each set. let us define ( Dq f'D k-q g

Xx )Y l · "Y

k

and

We shall see later that if

f

is

Ck

then

Dkf

is symmetric.

denote by Sym the symmetrizing operator; that is, if Sym

cp

in

Lk

sym

(E, F)

€

k

L (E,F)

then

is defined by

where the summation is over all permutations we may write

cp

let us

~

of

1, ... ,k.

Then

DIFFERENTIAL CALCULUS

1.

k

D (f·g)

Sym

7·

k 2:

q=O k

D

(gof)

(The formulas on p·3 of [6] should be corrected to take symmetrization into account.) The following is another proof of a theorem of Abraham [6, p.6]. By

o(yk)

with

f

y

we mean a function such that

o(yk)/I/Ylik -;;.

°

as

y -;;.

°

0 .

Theorem 3 (converqe of Taylor's theorem).

U be open in

Banach spaces, let

Let

E and

E, and suppose that

F

be

f: U -;;. F

satisfies f(x+y) where the f

is

k C

J

and

a.

J

Proof. is true for that

are in

a. (x)

For

k-l.

a. = Djf

for

J

=

L

Djf k

=

j (E, F) sym

for 1

j

=

and each

a.

J

is continuous.

0,1, ... ,k

this is the definition.

Then in (3), since j = 0,1, ... ,k-l.

(~(X)/k!)yk

Suppose the theorem =

o(yk-l) , we know

Now let us expand

f(x+y+z)

two different ways: f ( x+y+ z )

=

) + Df( ) z + ... + ( 1 )' Dk-l f ( x+y ) zk-l f( x+y x+y k-l .

ak(x+y) k k + ---;"k'!- z + o(z ) ,

f(x+y+z)

Then

1

f(x) + Df(x)(y+z) + ... + (k-l)! ak(x) k k + ~ (y+z) + o( (y+z) ) .

in

8.

FLOWS

I.

Fix

x

and restrict

z

so that o( zk ),

matter whether we write

~lIyll

5

IIzll

k

o( (y+z) ),

two equations, collecting coefficients of cient of

zj

by

gj(y).

5 ~lIyll

Then it does not

k

or z

.

o(y)

Subtract the

and denoting the coeffi-

Then k

(4)

o(y ) .

Now

and by the continuity of

a

k

this is

k

o(y ) , so we may drop this term. o(yk )

We claim that each term separately in (4) is ~l'···'~

be distinct numbers, and replace

In this way we obtain

k

by

z

To see this, let for

~.z 1

i=l, ... k.

equations which we write as k o(y )

1

~l

go(Y)

1

~2

gl (~)z

~

. k 1 gk-l (y)z -

II

k o(y )

I

1

~i

Since the

are distinct, the matrix is invertible (it is the Van-

dermonde matrix with determinant is

o(yk ) .

gk-l ( Y) z

ative, this means that k C .

Therefore each j

=

k-l .

But

~)

Dk-1f(x) ka.k (X)yJ k-l k-l _ Dk-l f ( x+y ) - [ (k-l)! - (k-l)! k! z .

Therefore the term in brackets is

is

II (~.-~.)). i ® defined as follows.

be a sequence of irrational numbers decreasing to f(x) = an ; where the

a

εη < J xI < εη+1 ,

0 , and let

Then f

is

f(x) = o(x) but

00 C°° as a function from

Q to

the sense of definitions analogous to those given above for Df = 0 , but Taylor's theorem is not satisfied at

logical space is locally constant in case every point V

such that

f(y) = f(x) for all y

Q (in

]R ) since

χ = 0 . Alsoj f is

not locally constant even though Df = 0 . (A function

borhood

f (o) = 0 ,

χ e Q,

are rational numbers so chosen that

2 not f(x)= o(x ) .

Let

in

f on a topo­

χ

V .

has a neigh­ It follows

that a locally constant function is constant on each connected component of the space.) It is not the incompleteness of

Q which causes the

trouble in the above example, but the fact that

Q is not locally con­

nected.

To proceed further we must make substantial use of the fact

that we are working over the real number field. Theorem iJ-. in

E , let

Then

f

let

be in φ

and

F

be Banach spaces, let

U

be differentiable, and suppose that

be open Df = 0 .

is locally constant.

open ball

Thus

E

f: U —> F

Proof.

x+y

Let

V

Let χ

be in

with center V

and let

χ

U

and let

and radius

φ: [0,l] —> V

is the line segment joining χ

a > 0 be so small that the a

is contained in

be defined by and

x+y .

U .

Let

cp(t) = x+ty . Let

ε > 0 and

10.

I.

S£

(t

E

FLOWS

[0,1]: Ilf(x)- f(cp(s))11 < £s

This is a closed set containing

O.

0 < s::: t}

for

It is also open, for if

to E S£

then f(x + (to+h)y)

f(x + tOY) + hDf(x + toY)Y + o(hY)

=

f(cp(t )) + o(h) O and by the triangle inequality, small enough.

Since

Ilf(x) - f(CP(tO+h))11 ::: c(tO+h)

[0,1].

£

f(x)

Ilf(x) - f(x+y)11 < £

Therefore

f(x+y)

h

is connected (this is an immediate conse-

[0,1]

quence of the least upper bound property of S

for

JR) it follows that Since

£

is arbitrary,

QED

The quickest approach to integration of continuous functions is the following (see [4]). F

be a Banach space.

A step function

a = a

for some partition If

I = [a,b] ,

Let

O

f: I

f: I

< a l b

b

II (l+A(t)dt) a

we define a {II (1 + A( t )dt)-l b

With this convention we always have

2.

P.ICARD'S METHOD

c b II (l+A(t)dt) II (l+A(t)dt)

(3)

b

17· c =

c

We claim that if

II (l+A(t)dt) .

a a < t < b

and A is continuous, then

t t dt II (l+A(s)ds) = A(t) II (l+A(s)ds) a a

d

(4)

To prove this we need only show that t+h II t

(1+ A(s)ds)

=

1 +A(t)h + o(h) ,

for then (4) follows by (3) and the definition of derivative.

(5),

To prove

we use the estimate (2) and find t+h t+h /I II (l+A(s)ds)- II (l+A(t)ds)/i t

t

sup /lA(s) - A(t)/I t

A

Since

A-

l

,

COO

is

Dcp

is a composition of

Ck - l

(by induction on k) and so is a f

is.

x

U is an open subset of E,

~:

such that

X: U

~

I

~

U ,where

s(o) = x

I

k C

is

if

s

s: I

~

is continuous,

Cl

is a

is an open interval containing

0,

and dS(t) = X(s(t)) , dt

Thus

E

U, an integral curve of X starting at x

is in

function

t

is an integral curve of X starting at

€

x

I

if and only if

U is continuous and t

x + fO X(s(s))ds ,

s(t) Theorem 4. let

cp

Thus

QED

If and

function.

X: U

~

Let

integral curve of

X starting at

For each

I

.

V

of

x

x

in

U

E

and

there is an

x, and any two of them agree on the

intersection of their domains of definition. is an open neighborhood

€

U be an open subset of a Banach space

be locally Lipschitz.

E

t

' an

o

For all

Xo

in

U there

a > 0 , and a unique mapping

cp: (-a,a) X V ~ U such that for all X starting at of class

k C

x so is

x

in

V,

t

The mapping cp.

~

cp

cp(t,x)

is an integral curve of

is locally Lipschitz.

If

X

is

2.

Proof.

Let

PICARD I S METHOD

and

~

be two integral curves of X starting

~

at

x, defined on the interval

in

I

contains let

~(t)

such that

Let

I

= ~(t)

Then

is clearly closed in

0, so we need only show that it is open. ~(tO)

W be an open neighborhood of

Lipschitz with Lipschitz constant €~

J

be the set of points

J

< 1 , that

[t o -€' t o+€]

CI

.

~(t)

and

s(t)

For g(t)

~(tO)

=

to

on which

K, and choose

are in W for

Let

€

t and

I

be in

J

X is

> 0 so small that

It-tol

2:

and that

€ ,

we have

It-tol < € t

~(tO) + It

X(s(s))ds 0

~(t)

~(to) + It

to

X(~(s))ds ,

so that

/I~(t) - ~(t)/1 2: III~ [x(g(s)) -X(~(s))]ds/l < o Since this is true for all

and since It-tol

2:

E·

any point

0

E

t

uniformly

for

It I ~ a

there is a

0> 0

such that if sup 11~(t,xl) - ~(t,x2)/1 < 0 Itl~a

(10) then

sup /IDX( ~(t,xl)) - DX( ~(t,x2»/1 < Itl~a

(ll)

E •

(To see this, apply the Heine-Borel theorem to the compact set of ~(t,xl)

It I ~ a .)

with

so there is a

0I > 0

~

But

is Lipschitz in

such that if

sequently (11), holds.

By

II Xl -X 2/1 < 0 I ~

(8) of Theorem 1,

Together with the equicontinuity of

~

in

s

uniformly in t,

then (10), and conis continuous in

t , this shows that

x . ~

is

continuous. Now we want to show that ~(t,x).

By

exists and is equal to

the fundamental theorem of calculus,

d

dt(~(t,x+y) - ~(t,x)} =

D2~(x,t)

=

X(~(t,x+y»

- X(~(t,x»

f~ DX[~(t,x) + r(~(t,x+y) - ~(t,x)}Jdr(~(t,x+y) - ~(t,x)} .

Let A(t)

Then

A and

B

y

=

DX(~(t,x»

,

are continuous mappings of

[-a,aJ

into

L(E) , and

2.

IIA-Byll ~ 0 liBy II -< c.

y ~ 0 ,with

as By

PIaARD'S METHOD

IIByll

27·

remaining bounded, say

(8) of Theorem 1,

sup /J1jr(t,x)y - (cp(t,x+y) - cp(t,xHI < eaca/JA_B IJ/Jy/J y

/tl:::a Therefore both

D2CP

D2CP(t,x) and

/JA/J::: c ,

DIcp

Now let

1jr(t,x).

=

Clearly

~t cp(t,x)

exist and are continuous,

X be

k C

k > 2.

Then

cp

cp

o(y)

X(cp(t,x))

=

is

cl

is

=

Since

Cl . , and as we have

seen d

dt cp(t,x) d

=

d

= DX(cp(t,x))"X(cp(t,x)) ,

dt dt cp(t,x)

Since

X is

k C , the right hand side of this system is a

function of the triple k-l C ,so

in k-l

cp,

d

dt cp,

D2CP.

l Ck -

Notice that we have shown that if X is

k C

d

dt cp

and

D2 cp

are

A function is said to be

Lipk

derivative is locally Lipschitz. Lipk

so is

and

cp

in case it is

QED then

cp

Ck - l

Ck+l

is

and the

A very easy induction shows that

cp, thus giving a very simple proof that if

(which is the same as

Lipk

for all k) so is

surprising how difficult it is to show that if X is l C , but there is no simple proof in the literature, modern proof, see [12].

Ck - l

induction, this triple is

Ck

t.

COO

By

is

if X is is

X(cp(t,x)) ,

cp

l C

X

It is

then

cp

is

For an interesting

28.

3.

I.

FLOWS

The local structure of vector fields We have been discussing mappings

ing, these may be termed vector fields on phism of

U onto an open set

X: U

U.

E.

Y = R*X

schitz it generates a local flow

~(t,x)

F then we

V by

on

The reason for this definition is as follows.

Loosely speak-

R is a diffeomor-

If

V in the Banach space

define the transformed vector field

~t

~

If

X is locally Lip-

U with

on

q:>( t,x) 1t=o = x(x) .

V by setting

This flow may be transported to

The vector field generating this local flow is the above defined Y = by the chain rule.

If

X

is

k C

and

R

k-l

is

C

and this is the best that can be said in general. if

X generates a

if

R

R*X

is

ck

k C

local flow (for example, if

is

then the transformed local flow is also

k C

k C

,

However, notice that X

also generates a

~X

k

C

) and

so that

local flow although it may not be a

k C

vector field. Given a vector field R

such that

R*X

generality, given

X on

U we may seek a diffeomorphism

has a simpler appearance.

Xo

in

There is no loss of

U, in assuming that

E

=

F

and

Thus we are seeking a change of coordinates which simplifies

R(X ) O X.

=

Xo .

3·

A point X(x)

f O,

LOCAL STRUCTURE OF VECTOR FIELDS

χ

in

U

is called a regular point of

otherwise it is called a critical point.

a regular point of X

29-

if and only if

R(x)

X

in case

Notice that

χ

is

is a regular point of RifrX .

The straightening out theorem asserts that we may choose coordinates in the neighborhood of a regular point so that

X

becomes a constant.

Sternberg's linearization theorem (in the finite dimensional case) asserts that in the neighborhood of a critical point we may, in general but not always, choose coordinates so that shall assume throughout this section that Banach space

(lRS

X E

becomes linear.

We

is a finite dimensional

with some norm), although the straightening out

theorem is true without this assumption. We shall achieve a considerable simplification in the proof of the Sternberg linearization theorem by using some ideas which are familiar in quantum mechanics.

Let

U^(t)

and

U(t)

be two one-

parameter groups of transformations on some space (for example, unitary groups on Hilbert space or flows on a manifold)

Suppose that the

ILmit (1)

Iim U(t)UQ(-t) = W t —» OO

exists.

In quantum mechanics this is called the Miller wave operator. Iim u(t)u^(-t) , and the t -> -» S , or S-matrix, of Wheeler and Heisenberg is con­

There is an analogous operator scattering operator

W

=

structed in terms of these wave operators. then U(s )WUQ( -s) = W . If

W

is invertible this means that

Notice that if (l) exists

30.

I.

FLOWS

and the two one-parameter groups are conjugate. by

X and

Xo

this means that

exist, we see that the flow where the flow where

U(t)

(W-1)*X = XO'

UO(-t)

If they are generated In order for

W to

must carry points into a region

is approximately equal to

X is approximately equal to

UO(t) ; that is,

Xo

As a first illustration of this method, we use it to prove the straightening out theorem, although for this simple result the usual proof [6, p.58) is equally easy. Theorem 1 (straightening out theorem). subset of in

JR

s

,let

X: U

-.;> JR

U be a regular point of

diffeomorphism W at hood of

xo

X

s

be

ck

with

Let

k > 1 , and let

Then there exists a local

such that

W*X

k C

is constant in a neighbor-

xO'

IJ (-t) ~ o

Figure 1.

U be an open

Construction of W (Theorem 1).

3·

Proof. X

o

We may assume without loss of generality that

= X(x o ) = (1,0, ... ,0).

mS.

Let

1

> -

1 ~ c:

x

1

Let

X

f

be a

k C

function, 0 < f < 1 , with

in a neighborhood of

"-

I t is clear that

x

x

O

O

'

Then

' agrees with

X and

Xo

morphisms of

X o

generate global flows, which we

k C

'V

U(t)

in

(~X(x))l:: ~~ on U by th e trlang · l e 'lnequa l·t l y.

U ,and

denote by

x

contained in the half-space

U which is 1 in a neighborhood of

k C , agrees with

ou t Sl'de

xo

for all

denotes the first component of x), such that

on U

support in

is

XO(x) = (1,0, ... ,0)

Let

U be a neighborhood of

(where (X(x»

31.

LOCAL STRUCTURE OF VECTOR FIELDS

and

JRs

Thus

UO(t)

U(t)

onto itself, for each

and

lim

diffeo-

t , and they are one-para-

meter groups (by the uniqueness assertion of Theorem Wx

are

UO(t)

4, §2).

Let

"-

U(t)Uo(-t)x

t~oo ~

It is clear that this limit exists, since t

as soon as

t > xl

U(t)Uo(-t)x

is constant in

Furthermore,

W-1x

lim

Uo(t)U(-t)x

t~oo

exists since Thus

W is a

Therefore

Uo(t)U(-t)x k C

-1) 'V ( W *X

proof is complete.

is constant in

diffeomorphism of

= XO

'

Since

~

X

lR s

t

as soon as

t > 2xl .

onto itself, and clearly

X in a neighborhood of

x

o ' the

32.

I.

FLOWS

Our problem now is to show that in general we may choose coordinates at a critical point of a vector field so that it becomes linear. The meaning of "in general" will be made clear later.

Now we show that

this is not always possible; cf. [9, p.812].

Let

Suppose there is a local diffeomorphism

R;X

such that that

is linear.

o

R(O)

=

at

2 (

2)

x + ax + i3xy + )'y ~ 2 y + AxC: + Bxy + Cy

R-l(X) y =

which is

2

C

We may write 2 +o(x) y

2) 2 (

so that

0

,

x - ax - i3xy - )'y ~

~

c:

y - Ax - Bxy - Cy

c:

+

and i3x + 2),y

)

1 + Bx + 2Cy Therefore

Notice that this is also equal to

1 + 2ax+ i3y ( 2Ax+

By

i3x + 2)'y

)

1 + Bx + 2Cy

2

2

2

a(x - Ctx - i3xy - 'YY ) 2 2 ( b (y - Ax - Bxy - Cy

ax + aax + i3bxy + (1 - a), + 2)'b)y 2

and

There is no loss of generality in assuming

DR(O) = 1.

and

R

2

( by + (2Aa - bA)x + Baxy + Cby )

2)

+

3.

LOCAL STRUCTURE OF VECTOR FIELDS

For this to be linear we must in particular have this may be achieved if and only if

2 C

cannot be linearized by a

a

f

2b

33· (1-ay+2Yb) = 0 , and

Thus the vector field

change of coordinates.

More generally, consider the orbits of the flow generated by the linear vector field

for

k > 0

For

k = 2

these are sketched in Figure 2.

orbits are branches of the curves is an integer, each orbit has a

x

k C

= cyk

These

a parameter.

c

k C

diffeomorphisms but

which can be destroyed by adding a perturbing vector field Thus for

k

Xl

------- k).

Let Jr °°

be the set of all

formal power series of the form OO X A.xJ J =I J where each

A. J

is in

S,]RS) Lj (]R symx

and

A, 1

is invertible

This is

a group under formal composition. We have the following commutative diagram of homomorphisms:

Jb'

b

Λ

t

Jj k-l

Jb

k-l Λ

where the mappings in the top row are inclusion, in the bottom row are truncation, and the vertical arrows denote taking Taylor series.

(The

latter depends on the choice of coordinates.) It is evident, except

3·

k =

for

00

,

LOCAL STRUCTURE OF VECTOR FIELDS

~ k ~ ;: k

that c..,

shows that.-u

00

?-OO

~;1-

is surjective.

The following theorem

is surjective, too.

j s Let A. be in L (JRs,JR ) -J--sym

Theorem 2. Then there is a

35·

COO

F: JR s ~ JR s

mapping

for

j = 0,1,2, ...

whose Taylor series at 0

is 00

L:

j=O Proof.

j

F

We need only define

E

we may extend it to all of f

A.x J

S

Let

F.

in a neighborhood of

and has support in the domain

0

JR ~ JR

0::

0, for

by multiplying with a scalar function

which is 1 in a neighborhood of

of definition of

in a neighborhood of

be

0, with support in

0:: 0: :: 1,

Coo,

[-1,1]

0:

= 1

Let

2 A.xJo:(IIA.llllxll ) j=O J J 00.

F(x)

where

Ilxll

L:

is the Euclidean norm (so that

The j'th term is 00

0

except where

IIA.llllxll J

we see that the term-by-term

0:

is 1

< IIAol1 + IIAlllllxll +

)

QED

k

is

COO )

.

so that j 2 L: II xll j=2

IIxll < 1

In the same way

times differentiated series converges

Therefore

in a neighborhood of

as stated.

< 1

-

2

00

C

and the series converges absolutely for

IIxll < 1.

2

~

•

L: IIA.llllxIlJo:(IIA.llllxII j=O J J

absolutely for

x rvv0> IIxll

F

is

COO

on

II xii

0, the Taylor series of

< 1 F

Since at

0

is

I.

Gi ven such that •

J..n

c;-

.,.I-

k

T l

RTR-

in

1J k

FLOWS

we may ask whether there is an

is linear.

If this is true in jb k

since

is surjective.

sufficient conditions for linearizing in ~ k

> O.

ti ve II means

Theorem 3. linear part

Tl

R in)j k

it is also true

The next theorem gives We remark that "posi_

lie igenvalues II we mean complex eigenvalues.

By

Let

for some ~l' ···~s

Let

k

=

1,2, ... ,00

be the eigenvalues of

Tl

,

with

counted

with their algebraic multiplicities as roots of the characteristic equation of

i = 1, ... ,s ,

T , and suppose that for l m ~s

whenever the

are positive integers with 2

Then there is a unique that

RTR-

l

< ml + ... +

R

0

so if

and

- U (t 2 )xll < O

t2

is large enough,

(6) is smaller than

so that

e

Kt2

k k -ckt 2 (k) C Ilxll e J~ e K-c sds

-(Ck-K)t2

e

if

ck >

for

x

Ck-K

But this tends to

K

in a bounded set

ckllxllk

O.

u(-t)uO(t)x

Consequently W

exists, and

converges uniformly.

Hence

W

is continuous. Recall the first variational equation. ~(t,x)

where

d dt where XI

(~(t,X»)

XI

and

= U(t)x , then the pair

*(t,x)

=

~,*

(X(~(t,X» ) DX(~(t,x».*(t,x)

If

*(t,x) = D~~(t,x) C

satisfies

(~(t,X») = XI

*(t,x)

is defined by this equation, and similarly for

Xb

satisfy the same hypotheses as before:

Xb

Xb.

Now

is linear,

3·

LOCAL STRUCTURE OF VECTOR FIELDS

has the same eigenvalues as XIX =

Xbx

OO

+ o(X ) .

Xo

(with higher multiplicities), and

The corresponding flows are

(U(t)x ' \

\DU(t)X.~)

,

Therefore

By what we have already proved,

mapping.

Hence

that

is

W

D(U(-t)UO(t))

l C .

local

COO

so that

in a neighborhood of Theorem of

0

in

DW (0)

lR

s

7. with

Hence (Theorem 3, §2)

W

is a

(W-l)*x = Xo

QED X be a

X(O)

COO

COO

is invertible; in fact,

=

0

COO

vector field in a neighborhood

such that

value condition and each eigenvalue Then there is a local

is

As in the proof of Theorem 1,

O. Let

W

DW (0) = 1

diffeomorphism.

converges to a continuous

converges to a continuous mapping, so

By induction,

It is clear that W x = x + o(XOO)

UI(-t)Ub(t)

A.

of

diffeomorphism

linear in a neighborhood of

0 .

DX(O)

satisfies the eigen-

DX(O) R at

satisfies 0

such that

Re A. < O. R~

is

46.

FLOWS

I.

Proof.

This is an immediate consequence of Theorems

COO

We say that two

4 and 6.

functions are equal to infinite order at

a point if they have the same Taylor series there, and that they are equal to infinite order on a set if they are equal to infinite order at each point in a set. The next theorem is a technical lemma from which the Sternberg linearization theorem will follow easily. Theorem 8.

o

X(o) Let X

DX(O)x, let

Let

E

Xl

vector field on

m> 0

and by

II z-NII < 5

and

IRs, with

satisfies a global Lipschitz condition. uo(t)

X

=

be the flows generated by

Xo + Xl'

N , invariant under

such that for all that if

DjX

U(t)

XO ' and define

linear subspace

COO

X be a

such that each

XOx and

Let

Suppose there is a

X ' and a positive integer O

j = 0,1,2, ...

there is a

5> 0

such

then

be the linear subspace of all

x

in

IRs

such that

lim t

Then for all

j

converges as

t ----;;.

---700

and

0,1,2, ...

00

x

in

E,

and the limit is continuous in

x

for

x

in E.

Let Wx

lim U(-t)UO(t)x t

for

x

in

E.

Then

W

---700

has a

COO extens ion

G to

lR s

which is

3.

47·

LOCAL STRUCTURE OF VECTOR FIELDS

the identity to infinite order in a neighborhood of

Proof.

Since

N

quotient space

E/N,

Uo(t) ~ 0

stants

C
0

and

and

and choose

X ' O

t ~

as

00

,

t > 0

and such

to infinite order. NC E.

On the

so there are con-

Let

K

be a Lipschitz constant for

E

K

+ iK

and let

t2

be large enough so that

E

N

4.

N

m so that

K be a compact set in

Figur~

in

such that

mc > Let

(G-l);K = Xo

0 in E,

that in a neighborhood of

0

Construction of W

(Theorem 8).

48.

I.

s ~ t2

and

x

of the theorem).

Let

tl ~ t 2 ,

whenever

FLOWS

is in

in the proof of Theorem 6, for

(where

K

tl x

=

5

is as in the hypothesis Then by Theorem 5, as

t2 + t

K

in

Ilu(-tl)uO(tl)X - u(-t 2 )U O(t 2 )xll

=

/lU(-t 2 )U(-t)u (t)u (t 2 )x - u(-t 2 )U (t )x/i < o O O 2

Kt e

2

J~ eKsCme

-mc(s+t~) c:: Ilxllm

e

K£(S+t~) c:: Ilx/l£ds

e -(mc-K-£K)t2 1Ixllt+mcm mc- K -iK ~ 0 . Therefore W

exists and is continuous on

E.

Notice also that since

m may be chosen arbitrarily large, this shows that if W

is

COO

(and we shall prove this below) then it is the identity to infinite order on

N. Now consider the first variational equations of X and

on

E s EB L( JR s).

We let XI

NI

Xb,

They are given by

be the space of all and

NI

X ' O

(~)

with

x

in

N.

We see that

again satisfy the hypotheses of the theorem, with

3.

£

replaced by

x

in

Xb

By

£+1.

Let

E.

The space

U'(t)

and

E'

is the space of all

Ub(t)

on

to a continuous function of

E

By

in E

with and

X'

what we have already shown, we know that

proof of Theorem 6, this implies that 00

(~)

be the flows generated by

converges to a continuous mapping for all

t ~

49.

LOCAL STRUCTURE OF VECTOR FIELDS

induction,

and W

is

x

in E, and as in the

D(U(-t)UO(t))x x

in

Dj(U(-t)Uo(t))x

converges as

E, so that

converges as

W

is

l C

t~oo for x

COO

Let

for

x

in

and

E

j

proof of Theorem 2 (0: of

0, and

0:

E,

space to

is

COO

o


K

and

,

If

diffeomorphism Similarly, if there is a

e

k

is linear near

j

/,

For the constant

0

=

Then

K

u(-t)uO(t)

Theorem 10.

~I

, with

R such

/'+1

c

we may

(after modifying

converges on

converges on

E+

E+

at each step of the

is linear to order

(G-l)*R*X

> (k+I)K , where

< min{ IRe ~I: Re

c

pro-

provided

satisfies this inequality, there is a local

~

j

e

j

on

E+

> OJ

I l (H- )*( G- ) *R*X

H such that

local diffeomorphism

o.

s

and for the Lipschitz

> max IRe

is replaced by

G such that jc

lR

Now consider the proof of Theorem 8

Dj(U(-t)Uo(t)) /,

in

local diffeomorphism

c+ < min{IRe ~I: Re ~ < OJ

mc+ > (j+I)K , since induction.

em

we may take any number

K

0

satisfies the eigenvalue condition to

away from the origin).

vided

~

nx(O)

is linear to order m.

take any number

R*X

vector field defined near

Elf Theorem 4 there is a

applied to

constant

em

In this way we obtain the following theorem: Let

m and

k

be integers

vector field in a neighborhood of

0

in

lR

~

s

I , let

with

X be a

X(O) = 0

such

3·

that

DX(O)

LOCAL STRUCTURE OF VECTOR FIELDS

53·

satisfies the eigenvalue condition to order m, and

suppose that k < minflRe >-.1: Re >-. > O} (minflRe >-.1: Re >-. < O} m-l\ -1 . max Re >-.1 max IRe >-.1 L) Then there is a local

k C

diffeomorphism

is linear in a neighborhood of

F

at

0

such that

F~

O.

The Sternberg linearization theorem may be formulated and proved in an entirely analogous manner for mappings instead of vector fields: Theorem 11 (Sternberg linearization theorem for mappings). Let with

T be a

COO mapping defined in a neighborhood of

T(O) = 0

such that

value condition. such that

FTF-

l

DT(O)

Then there is a

0

in

IRs

satisfies the multiplicative eigen-

COO

local diffeomorphism

is linear in a neighborhood of

F

at

0

0 .

The proof is simpler in some inessential respects than the proof for vector fields.

Theorem

4 may be omitted, and Theorem 5 is

replaced by the following: Theorem 12. Then

Let

T be Lipschitz with Lipschitz constant

K.

54.

I.

Proof.

FLOWS

The first inequality is trivial.

inequality, write

Tn _ Tn

To prove the second

as the telescoping surn

o

(Tn-IT

Tn-ITo) +

2) (n-l n-l) ( Tn-2TTO - Tn -TOTO + ••• + TTO - TOTO This identity makes the second inequality obvious.

QED

From a topological point of view, the only invariant of a vector field at an elementary critical point (elementary means that no eigenvalue of the derivative is purely imaginary) is the index We may use wave operators to give a simple proof

dim E+ - dim E

of this fact in the special case that

Re

A


0 , (x,Xox) ::: -c(x,x) ,

(For diagonalizable

Xo

x

€

JRs

this is clear, and for the general case we

may consider the Jordan canonical form with the l's replaced by

£' S .)

3·

LOCAL STRUCTURE OF VECTOR FIELDS

55·

o•

Figure 6.

COO

Choose a

Construction of W (Theorem 12).

function

hood of

0

and let

'" X =

f

o < f < 1

with

which is 1 in a neighbor-

and has compact support in the domain of definition of -1 + f· (X+l)

Then

'"X = X

near

o ,

'"X =

X

outside a

-1

compact set, and

'" < -c(x,x) , (x,Xx) for some

c > 0 Let

U(t)

if we choose and

f

uO(t)

appropriately, since

Xx

be the flows generated by

=

X

XOx + o(x). and

Consider 'V

Wx

lim U(t)UO(-t)x t

~ 00

This limit clearly exists and is u(t)UO(-t)x

is eventually constant in W-lx

TIl. s _ (O} , since

on t

lim Uo(t)U(-t)x

=

t

~ 00

Also,

WO

=0

, and

-1.

56.

FLOWS

I.

""'X , '"U(-t)x

has the same properties, since by the construction of (for

x

f

0) eventually enters the region where

clear that

W and

morphism of since

:ill s

X =X

W- l

are continuous at

onto

near

0

o,

"-

X

=

-1

so that

It is also

W is a homeo-

:ill s .

and

the proof is complete.

This topological classification is too crude to be of much See the examples of vector fields in [8, pp 372-375J.

interest.

4.

Sums and Lie products of vector fields If

X

is a locally Lipschitz vector field in an open subset

of a Banach space, we denote by Theorem 1.

Let

X

defined in an open subset in

and

E

the local flow it generates.

Y be locally Lipschitz vector fields

U of the Banach space

U there is a neighborhood

and an

UX(t)

V of

Xo

with

E .

For all

V contained in

U

> 0 such that lim n

uniformly for Proof.

and for each

x

in

V and

--7 00

It I < E .

We have

this holds uniformly for

x

in a neighborhood of Therefore

4.

SUMS AND LIE PRODUCTS OF VECTOR FIELDS

uniformly in a neighborhood of

57·

If we write

xO.

as a telescoping sum (as in the proof of Theorem 12, §3) we see that if K

is a Lipschitz constant for

x

uniformly for

k

C

Xo

xO.

then

QED

vector fields, with

U in the Banach space

in an open set

[X, y]

near

in a neighborhood of

X and Y be

Let

X+Y

E.

k > 1 , defined

We define the Lie product

by

= DY(x) ·X(x)

[X, Y](x)

This is a

k l C -

vector field.

- DX(x) ·Y(x) ,

j

~

E

U .

Recall that we are abusing the term

"vector field", and it is necessary to show that if diffeomorphism, with

X

R

is a

j

C

2 , then

This is a consequence of the symmetry of the second derivative, as we now show.

We have DRoR

Let

Rx

y.

-1

·XoR

-1

Then

by the chain rule, and similarly for

y.

Therefore

58.

I.

FLOWS

[R*X,R*Y] (y)

= D(DR'Y)(x)'DR-l(y)DR(X)X(X) - D(DR,X)(X)'DR-l(y)DR(X)Y(x) = D(DR'Y)(x)'X(x) - D(DR'X)(x),Y(x) . But

D(DR'Y)'X - D(DR·X)·Y =

2

2

D R'Y'X + DR·DY·X - D R·X·Y - DR·DX·Y

= DR· (DY·X -

nx·y)

Let

Y be

= DR· [X,Y]

so that

Theorem 2. an open set

U

neighborhood

X

and

of a Banach space

V

of

E.

2 C

vector fields defined in

For all

contained in

Xo

U

in

U

and an

there is a

> 0 such

Xo

with

V

lim

(U (-

i!)ux(- In i!)uyin ( I!)ux/n ( /!))nx In

E

that

U[X y](t)x , uniformly for

x

Proof.

=

n~oo in

Since

V X

Y

and is

§2), and we have

so that by Taylor's formula

0 < t < E .

c2

so is the local flow (by Theorem

4,

4.

SUMS AND LIE PRODUCTS OF VECTOR FIELDS

h

2

59.

2

l+hX + 2" DX·X + o(h ) uniformly for

x

in a neighborhood of

x

O

' and similarly for

Y.

Therefore

h2 h2 h2 h2 ~ C (l-hY + 2" Dy,y)o(l-hX + '2 DX·X)o(l+hY + '2 DY,Y)o(l+hX + 2 DX'X) +o(h ) . 2 h DY'X

When we expand this we must not forget terms like

in

We obtain 2 2 222 h h I-hY + -2 DY'Y + h DY'X - h DY'Y - h DY'X - hX + -2 DX·X 2 2 2 2 h 2 h 2 h DX'Y - h DX'X + hY + ~ DY'Y + h DY'X + hX + ~ DX'X + o(h ) C C 2

2

1 + h [X, y] + o(h ) .

Therefore u[

X,Y

](-) nt

= U (-

Y

A

-)u (n X

uniformly in a neighborhood of

Jtn Y An X finn

xO'

-)u (

-)u (

-) + 0(-) nt

When we write the difference of

the nlth powers as a telescoping sum, as in the proof of Theorem 1, we obtain the desired result.

QED

. Theorem 2 shows that the Lie product has an invariant meaning, independent of the choice of coordinates, so that (1) also follows from Theorem 2.

60.

5.

I.

FIAJWS

Self-adjoint operators on Hilbert space We shall develop briefly those aspects of Hilbert space theory

which are of greatest relevance to dynamics.

A knowledge of integra-

tion theory is assumed. Let

i=t

is a mapping

;+

X ~ ~ C which takes each ordered pair

into a complex number in

u

A sesquilinear form on ~

be a complex vector space.

, such that

and linear in

v

(u,v)

is conjugate linear

(we follow the physicists' convention).

For

a sesquilinear form, computation shows that the polarization identity holds: 4

=

- - i + i .

This means that any sesquilinear form is determined by the associated quadratic form in case

~

.

A sesquilinear form is called Hermitean

= , so that a sesquilinear form is Hermitean if

and only if case

u

~ 0

is always real.

The form is called positive in

(so that a positive form is necessarily Hermitean),

and strictly positive in case

> 0

unless

u

=

0 .

For a positive form we have the Schwarz inequality: 1

(1)

1

l1 < 22

To prove this, observe that

o<

=

+ + + ,

so that -2Re < + .

:J.

We may multiply

v

61.

SELF-AlliOINT OPERATORS

by a number of absolute value

1

to ensure that

-Re = l1 , so that 211 ~ + .

( 2) = 0

Suppose

of degree 1 in 2 is

in

u

Then the left hand side of (2) is homogeneous while the right hand s ide is homogeneous of degree

u, so both are

= O.

0, and similarly if

If neither

1, and (2) implies (1).

0, we may take them both to be

For a strictly positive form we define 1

IIull = 2 . The triangle inequality IIu+vll ~ IIull + IIvll holds, so

u ~ IIull

is a norm.

1+

A Hilbert space

is a complex vector space with a strictly

positive sesquilinear form which is complete in the associated norm. (Thus a Hilbert space is a Banach space together with a sesquilinear form such that If

m

II ull 2

= for all u.)

is any subset of the Hilbert space

to be the set of all

u

r+

in

such that

1+

=0

,we define

m.l

for all

in

v

"1YL • This is clearly a closed linear subspace of 1+ . Theorem 1 (projection theorem). subspace of the Hilbert space

Proof.

eN- .

n "fYL..l

Clearly 'Yil. d

inf x

Em

Then

=

0

IIu-xll.

Let

~

1-1

Let

be a closed linear

'YY1. EEl 1'\'\...1

u

and

be in '}f and let

62.

Let xn

FLOWS

I.

x

be a sequence in

n

'tr/.

is a Cauchy sequence.

II u-x II ~ d

such that

We claim that

n

Computation shows that the parallelogram

law 2 Ilx -x II n m holds.

But

211x _ul/ n

=

(x +x )/2 n m

c~

+ 211x _ul/ m

m ,

is in

c~

X

_ 411 n

+x c~ m _ ull 2

so that

and Ilx -x II n m

H-

Since

2

< 211x -ull n

is complete,

is closed, and Since minimum at

IIx-ull

in

y

m

m1 m

t = 0

+ 211x -ull m

2

~ 0 .

which is in

x

for

in 'in- and

y

its derivative there is 2 Re

em ED

'm 11 ED

"m 1

')n 11

- 4d

m since m

d

"l+

Thus

1

2

has a limit

=0

t

real has a

0, so that

This remains true if we multi-

by a complex number, and so

,It =

=

n

+ ply

x

2

= O.

"YYt 1.

By

Since

That is,

em em 11

, we have

QED

is a continuous linear functional on

unique vector

u

in

r+

such that

=

f(v)

(Apply the projection theorem to the null space of If

is

the same result applied to

A corollary is the Riesz representation theorem: v rvvu> f(v)

u-x

A is in

L( '}f )

then for each

u ,

r+

then there is a

for all f

If

v

in

f+

.)

v~

is

a continuous linear functional, so by the Riesz representation theorem there is a unique vector, which is denoted by

* such that Au,

5.

SELF-ADJOINT OPERATORS

~ for all v.

called the adjoint of self· adjoint in case normal in case

AA *

A*

An operator

A A

Clearly

~

A*

is in

where

E ~ E*

= E2

It is

is called ~

-A

*

u = x+y

with

1+ )

L(

ilL

and

'1+

the mapping

in 1'Y1. and

x

y

in

E = E* = E2

, and

is

It is called the

E

Conversely, an operator

em 1 L(}f )

in

with

is the projection onto ~ where ~ is the range of'

An operator bijective and as saying that

L( ~)

U in

=

is unitary in case

for all

u

and

.*

-1

U =U

U is invertible and

v .

,defined for all real

This is the same

A strongly continuous

is clear that

u

U(O) = 1,

in

U(t)

on

t , such that

u(t+s) and such that for all

E

U is

one-parameter unitary group is a f'amily of unitary operators

1+

,

* AA

easily seen to be in projection onto

)

A

skew-adjoint in case

If' ~ is a closed linear subspace of' E: u rvvW- x

"'H-

L(

A in

~ )

L(

u(t)U(s)

1+

t

~>

U(t)u

is continuous.

It

U(-t) = U(t)* = U(t)-l , and that the

U(t)

commute. If' A

is a self-adjoint operator in

L(~)

then

U(t)

=

e

"tA

l

(defined by the power series expansion) is clearly invertible with U(-t)

= U(t)* = U(t)-l

, and it is a one-parameter unitary group with

the stronger continuity property that JR

to

L( 1+)

t ~> U(t)

is continuous from

(norm continuity or 1U1iform continuity).

Three topologies on

L(;+)

are especially useful: the norm

topology in which a basic set of neighborhoods of

0

is given by

64.

I.

{A

N

E

E

FLOWS

L(

1+ ): IIAII
0 , the strong topology with

where

E>O

and

where

E

>

and

0

u l '.··' vn

are in

'f+ ,

are in

J+

E}

and the weak topology with

The norm topology (also

called the uniform topology) is stronger than the strong topology, which is stronger than the weak topology. We will be concerned mainly with unbounded operators. operator

A on -~

is a linear transformation from a linear subspace

'H-

jJ (A) , called the domain of A, to

Jd

(A)

graph on

An

Its range is denoted by

It is convenient to introduce a more general notion:

rt

is a linear subspace

A of

1+

EEl

"H

a

We may identify u EEl Au ,

an operator with its graph, the set of all vectors of the form

)J(A) ,

U E

in "H-EEli+

the set of all its range A for some A

u

implies that

n

E

rt

in

Q(A)

graph is closed. U

u

If

uEElv

such that

is the set of all A graph v

=

0

A

jJ(A)

is a graph its domain

A

v

is in in

f+

A for some such that

u EEl y

is an operator i f and only if

An operator

Thus an operator Au

A n

A

y

and

is in

o EEl v

in

is called closed in case its

is closed if and only if

---;;.. v

,

is

implies that

u E

5. and

= v.

Au

SELF-ANOINT OPERATORS

An operator is pre-closed if its closure (as a graph)

is an operator, called the closure of A and denoted by

A

operator or graph

.lJ (A)

dense in

'l+ .

A

is called densely defined in case

Wee define the operators p( u ffiv)

vffiu

-r( u ffiv)

(-v) ffiu

If A is any graph, its inverse is any graph, its adjoint Theorem 2.

Let

p

A*

A-

l

and

..

on

is defined to be

is defined to be

t+

ffi

An is

it

pA.

by

If

A

(-rA)l

A be a graph on the Hilbert space

f+

Then A*-l = A- 1 *

( i)

( ii)

A- l = A- l -* A = -A~ = A*

( iii)

( A-1)-1

(iv)

A

=

(v)

-A

= A**

(vi)

A*

is an operator if and only if

(vii)

A*

is densely defined if and only if

If

A is a densely defined operator,

if

v ~ A*u

all

15(A).

is

in

A* , and Proof.

is in

is the unique vector such that

A**

f)(A *)

if and only

JJ (A)

, in

= for

If A is a densely defined closed operator, so A .

The statements (i)--(v)

(vi), notice that

A is an operator.

is a continuous linear functional on

which case v

u

A is densely defined,

A*

are trivial to verify.

is an operator i f and only i f A* n (0 ffi

To see

'H- ) = 0 ,

66.

I.

if and only if A

J_

FLOWS

.

Π ( iH" ® θ) = 0 (since

«£Γ(A)^ = 0 , if and only if

if and only if

by the projection theorem.

defined and

is densely defined,

* A

The remaining statements are

QED

An operator A A

A

τ =-l),

** A is an operator if and only if

is densely defined, which proves (vii). trivial to verify.

2

is unitary with

Ey the projection theorem again,

— 11 ** A = A =A , but by (,vijj

operator

τ

is called self-adjoint in case

*

A=A

.

An

is called Hermitean (or symmetric) in case it is densely A /C_ A* .

Thus a self-adjoint operator is Hermitean, but

the converse is false in general. In our proof of the spectral theorem we will use the RieszFischer theorem, which asserts that

2

L

of a measure space is com­

plete and hence a Hilbert space, and the other Riesz representation theorem which says that if

I

is an interval,

of continuous functions on

I in the supremum norm, and

positive linear functional on also denoted by

C(l) the Banach space μ

is a

C(l) , then there is a measure on

I,

μ , such that μ(ΐ) = J1 ΐ{χ)άμ(χ)

for all

f

in

C(l) .

If f

is a measurable function on a measure

space, the corresponding multiplication operator is the operator Mf: g fg *. fg

· · is m

on the domain

-0"(Mf) of all

g

in

if

such that

2

T L .

A linear operator from one Hilbert space to another is called unitary in case it is bijective and preserves inner products.

5·

Theorem 3 (spectral theorem). space ^j-

67.

SELF-ADJOINT OPERATORS

An operator

A

on a Hilbert

is self-adjoint if and only if there is a unitary operator

from

Jc A A~

to 1

L^(M,μ) , for some measure space

(Μ,μ) , such that

is a multiplication operator by a real measurable

function. Proof. operator in

Suppose to begin with that

L( Ή" ) .

A

is a self-adjoint

Let

Ι = E-II AII,||A||] and let

ρ

be a polynomial.

(3)

IIp

To see this, let E

η

We claim that

(A)II


+ :: Therefore

i-A

is injective and since

fd. (i-A)

this also shows that is dense.

is closed.

domain of A*

=

We also claim that

orthogonal to it, by

By the definition of adjoint,

This holds for all

u

in

o= which is impos sible since

=

z

is in the

- + i .

.lJ(A) , so we may set

(i_A)-l

z

u

and obtain

- + i ,

is real.

being closed and dense, is all of ~

JJ(A) ,

z

ti< (i-A)

A and

o

in

is a closed operator,

If not, there is a non-zero vector

the projection theorem.

u

i-A

is in

L(

1+)

Therefore

and since with norm

ti< (i-A)

II (i-A)ull < 1.

:: lIuli

, for

We claim

5.

that it is normal.

SELF-ADJOINT OPERATORS

Now

L( 1+

),

(i-A) * = i * -A *

i-A

and

-i-A

. )-1* ( l-A

=

. )*-1 (l-A

-i-A, so that

=

73· and since

is in

i

(.l-A )-1* = (-l-A . )-1

But

commute, so

. )-1(.l-A )-1* ( l-A

=

(i-A)- 1 (-i-A)- 1

(( -i-A)( i-A) )-1

((i_A)(_i_A))-l

(_i_A)-l(i_A)-l

(i_A)-l*(i_A)-l

and

(i_A)-l

is normal.

unitary mapPing;: :

'1+ ~ L2(M,~)

taking

0

C

almost everywhere because

annihilate any non-zero vector. able function, and so is

A

A

Therefore "'_I

•

= l-C

*"

into multi-

The function C

a-I

and a

C

= (i_A)-l does not

is a complex measur·

c-

.

Thus

A

plication by

(i_A)-l

C

plication by some complex measurable function is different from

(M,~)

Hence there is a measure space

takes

A

into multi-

A

A.

Since multiplication by

real almost everywhere.

A

A

is self-adjoint,

A is

This proves the spectral theorem for an

unbounded self-adjoint operator. To conclude the proof of the theorem, we need only show the converse.

More generally, if

the measure space

(M,~)

f

is a complex measurable function on

we will show that

Then there is an

h

in

2

L (M, ~)

*

Mf

=

Mr .

Let

(in fact,

g

be in such

that

for all that

k

* MfC

in

Mr'

h

(M ). f

From this it follows that

The reverse inclusion is obvious.

h QED

fg

a. e., so

74.

FLOWS

I.

As in the case of a self-adjoint operator in is an arbitrary self-adjoint operator and

f

L(

if

A

is a Baire function we

may use the spectral theorem to define the operator

f(A) , and the

definition is independent of the choice of unitary map space

ft) ,

~

and measure

(M,~).

Theorem 4 (Stone's theorem).

Let

U(t)

be a strongly con-

tinuous one-parameter unitary group on a Hilbert space there is a unique self-adjoint operator

1+

Then

A such that

u(t) = e itA

(4) Conversely, if

A

is a self-adjoint operator then

(4) is a strongly

continuous one-parameter unitary group. Proof. Let

The converse follows easily from the spectral theorem.

U(t)

be a strongly continuous one-parameter unitary group.

Define its infinitesimal generator lim

Bu

B by

U(h)-l h

u

h~O

~(B)

on the domain show that

B

Let

of all

R~

for which the limit exists.

is skew-adjoint; that is, that Re

~

B* =-B

> 0 and let R~u =

Clearly,

u

is in

Joo0

L(1+)

e -~tU( t ) udt , with norm

< liRe ~.

U €

irr .

We have

We will

5.

U(~)

- 1 RAu =

~ {~

/I.e -At

A --7>

A real,

00

AHA

75·

e -AtU( t )udt} ( ) e- A t-h U(t)udt

Jh0

converges strongly to 1 since

has integral 1 and becomes concentrated at

is bounded). operator

Since each

AHAu

J)(B)

is in

and

0

(and

IIu( t )ull

AHAu --7> u , the

B is densely defined. u

If

and

J~

e -At U( t+h)udt _

( e -A(t-h) -e -At) () h u t udt _ ~ h

Jooo

As

SELF-ANOINT OPERATORS

BU(t)u

entiable.

fJ (B)

is in

= U(t)Bu

it is clear that

Hence if

U(t)u

is in .!J(B),

u

is in .fJ(B)

U(t)u

is differ-

We have

~t lt=o = + = 2 Re .

But

Re = O.

is a constant since

Consequently

Now let

u

U(t)

is unitary, so that

B is skew-Hermite an; that is,

be in g(B)

Be -B* .

Then

RABu = JOO0 e -AtU( t ) Budt = Joo0 e -At ddt U() t udt

so that with

RA(A-B)U = u

(iI.-B)R

A

for

u

in

J/(B)

and

Re A> O.

Together

= 1 , this implies that Re A> 0 .

76.

FLOWS

I.

If we replace

Jo OO

t

by

-t

we find in the same way that

) - lu e -AtU(-t) udt = (A+B

*u

But the left hand side is

, so that

R~

* RA

= (-A+B )-1 .

Thus ( A-B )

-1*

( -A+B )-1 ,

so that (A-B) * = -A+B and since

A is in

sequently A = -iB

L(

i+ ) ,

-B*

= Band

B is skew-adjoint.

Con-

is self-adjoint.

Let Vet)

= e itA

To conclude the proof we need only show that to do this is to observe that for

JOOo e -AtV( t ) udt

= U(t).

One way

Re A > 0 ,

( A-iA )-1 u

=

Vet)

= Jroo

O

e -AtU( t ) udt

By the uniqueness theorem for Laplace transforms,

Vet)

= u(t)

.

A

more direct proof is the following. For n

c
x from A

t1.-0

u [-/lAIi,IiAIiJ A

such that for each

A

E

Q,o

in Q, 0 '

is in

[-/lAIl, IlAIl J .

With the

to

6.

COMMUTATIVE MULTIPLICITY THEORY

product topology, theorem [20] .

is a compact Hausdorff space by the Tychonoff

I

For

79·

A in

a0 '

let

be the function in

I\.A

C(I)

given by I\.A (x) and let I\.A .

Cf(I)

By

Any element of in

AI' ... ,An

(1),

from

be the subalgebra of

C(I)

generated by the functions

By the Stone-Weierstrass theorem [20, p .8],

C(I) some

= xA '

the

Cf(I)

is of the form

Cf(I)

Qo

Cf(I)

is dense in

P(I\.A , .. ·,I\.A ) for 1 n p in n variables.

and some polynomial

* homomorphism

to

CL

is norm-decreasing, and so has a unique extension

*

to a norm-decreasing

homomorphism

cp: C(I) ~ Q,. Let

n

be the kernel of

cp

and let

it

X be the "hull" of

that is, X = (x Now

n

C C(I-X)

E

I: f(x)

o

for all

f

in

n} .

; that is, the restrictions of functions in

11..

to the

looally compact Hausdorff space

I-X

are continuous and vanish at

infinity.

I-X

at which all of the functions in

I'l. I-X

There is no point in

vanish, and since By

is an ideal in

the Stone-Weierstrass theorem,

Since it is closed in

n

0 , so for some

then

for all

g

in

If

f

is in

C(X)

C(X)

II fll2 - ff = gg . Consequently IIfll 2 - ep(f)ep(f) * = ep(g)ep(g) * ~ 0 , so that

Ilep(f)lI:::. II fll .

A is in

a.

then

We may work the same argument backwards.

IIAII2 = IlAA*1I , and

~

0 , so for some

CL

B in

BB*

IIAII2 - AA* (We know that

B = (IIAII2 -

function is continuous on mable by polynomials.)

ep

AA *)!

is in Q since the square root

[0, II All 2]

and therefore uniformly approxi-

Consequently

Thus both

so that and

IIAII2 - AA*

If

ep

and

ep

-1

are norm-decreasing,

is an isometry. By the Riesz representation theorem every continuous linear

functional on

C(X)

is given by a complex measure

~

on

X, and if

6.

81.

COMMUTATIVE MULTIPLICITY THEORY

the functional is multiplicative it is easy to see that unit mass at some point

x

in

X.

The topology of

must be the

~

X agrees with

the weak-* topology of the corresponding multiplicative linear functionals.

Thus the compact Hausdorff space

of the Banach algebra morphism.

C(X) , so that

p

space, is a * homomorphism is a Hilbert space.

p

space of show that

~

decreas ing.

1+

spaces

1

f

a2

is unique up to homeo-

C(X) , for C(X) ~ L(

The space;+

X a compact Hausdorff

"It)

with

p(l)

=

1, where

is called the representation

and is denoted by ~ (p).

The argument used above to

was an isometry shows that any representation is normTwo representations and

it 2

in

of

and

are unitarily equivalent, U: ;.t 1 ~ 'i-/-- 2

C(X).

C(X)

on Hilbert

Pl '" P2 ' in case

such that

We will classify all representations of

up to unitary equivalence. C*

of p:

there is a unitary operator

for all

is describable in terms

QED

A representation

~

X

X

In this way we will find all commutative

algebras, up to unitary equivalence. on Hilbert spaces

14 1

and

'H 2

in case there is a unitary operator

/)

1.A.

2

C(X)

=U

Notice that two commutative C*

U:

Two

C*

algebras ~ 1

and

are called unitarily equivalent

~1 ~

'7+ 2

such that

Q,-l lU, algebras can be * isomorphic and

isometric without being unitarily equivalent; for example, the algebras of all scalars on Hilbert spaces of different dimensions.

82.

A representation

in /+(p)

in which case

If

f P~

Theorem 2.

If

representation of C(X)

z

p(f)z

X then

,

C(X)

p

C(X)

'"V'V0>

is dense p

Mf ' where

is a representation of

M f

is

C(X)

X then

P~

We

is a cyclic

~

for some measure

be a cyclic representation of

X .

To prove

P~

C(X)

on

P

with cyclic

= is a positive linear functional

and so is a measure on

Uf = p(f)z

in

Conversely, every cyclic representation

~(f)

Then

C(X)

f

The function 1 is a cyclic vector for

the converse, let

z

f

is a measure on

~

with

is called a cyclic vector for

is unitarily equivalent to Proof.

is cyclic in case there is a z

C(X)

L2(~)

on

shall denote it by

on

of

is a measure on

~

multiplication by

vector

P

such that the set of all

in 1+(p)

of

FLOWS

I.

X

For

f

in

C(X)

define

We have IIUfll

2

In particular, if

f

=0

a.e.

[~]

then

isometry from the dense linear subspace

Uf

= O.

C(X)

Thus

L2(~)

of

U is an onto a dense

linear subspace of 1+(p) , and so extends uniquely to a unitary Clearly U is a unitary equivalence

QED

between

are representations of

and PI

is a subrepresentation of

P2

C(X) , we say that

in case 0 a.e. [V]

IUll

2

I-l ~ V

and

dl-l/dV QED

Next we shall construct the Hilbert space ~(X) on

X.

We write

measure on for some

X.

(f,l-l)

for a pair with

We say that

A with

I-l«

(f,~)

and

V«

A,

A and

g /dV

If this holds for some V«

A'

f

2

in

(g,v)

L (1-l)

of a-functions and

~ a

are equivalent in case

a e. [A]

dA

and

(so that

C(X)), 2

J hdl-l = Jlfl 2dl-l Consequently,

Ll(v)

FLOWS

A and we also have a measure

A'

with

I-l«

then we claim that we also have

f

To see this, let

A"

/dV dX'

a.e. [A'] .

be a measure with

A«

A"

and

A' «A"

A+A' )

~ dA

g

~ dA

a.e. [A] ,

dA dA" ~m

g

dr." ~ dA ~

a.e. [A"]

f

dA" ~

g

11*11

a.e. [A"] ,

~ dA'

~' dA"

g

~ dr.'

f

~ dA'

g

~ dA'

f

f

= g /

A"

example, we may take

f

~

/dX'

Then

jdA' dA"

a.e. [ A"] a.e. [r.'] ,

(for

A'

6.

COMMUTATIVE MULTIPLICITY THEORY

with each equation implying the succeeding equation.

It follows

:f 0

for

By

Suppose not. all

lJ. a .

)'

< t3

Clearly 1ft3

Theorem 6, the

;.t

a

is in all

i{! i{! 1

f

0

Then

We claim that they span

such that

0

').t

rna

a< t3 .

Suppose

are orthogonal.

In particular, i{!

.l.(

is in

Then there is a

not happen that

nl 1t3

n frL.)' n

so that

i{! i{!

H- (X) .

is orthogonal to is in

tYL

It can-

0

since they are eventually

0

6.

93·

COMMUTATIVE MULTIPLICITY THEORY

'ffi.n,.

I

•

,

..

+--+---1--

c.m . . +

I

· .. n Wd,

I

.,(l ~ ,

Then we may define product.

/I III

tt- 1

JY

to be the completion of

Any sequence in

/I I ,

ttl

~ i+

natural norm-decreasing linear mapping

JJ .

}t

functions with compact support,

=

L2( lR),

b" all

defined linear operator on the Hilbert space

.

be the completion of

~

F+

.JJ (A)

1

A

be a densely

such that U

€

.!J(A)

in the inner product

rt

The operator

J

t+ 1

O

1

extends by conti-

as a dense linear subspace of

Hilbert space in a larger norm.

An

continuous

Let

nui ty to an injective norm-decreas ing linear map

H

which is the

,

Then the identity mapping ~(A) ~

we may identify

so that we have a

+ i:i.{6)v(O)

l

Theorem 3 (Friedrichs extension theorem).

Ii 1

in this inner

However, this mapping need not be inj ecti ve.

example is the Hilbert space

Let

./).

Jj which is a Cauchy sequence in the norm

is also a Cauchy sequence in the norm

identi ty on

€

U

-;>

1+

7+

,so that

which is a

of Theorem 2 is a

102.

FLOWS

I.

self-adjoint extension of Proof.

A .

Since the identity mapping

.b (A) -;;.1+-

is norm-

decreasing, it extends by continuity to a unique norm-decreasing mapping

L(

in

'f+ 1 ,ft)

.

For all

By continuity, this holds for all

is

o.

Lu =

Thus

0

The operator

J

O

u

in

1+

1

and

is orthogonal in }+ 1

u

L is injective.

The operator

in .(J(A) ,

v

= =

l

Therefore if

and

u

v to

It is clear that

in

.!JCA)

JJ (A)

ACJ

O

.

and so QED

is called the Friedrichs extension of

A

A may have other self-adjoint extensions, but the

Friedrichs extension is constructed in a canonical way and is of great importance in many applications. semi-bounded in case for some

c <

If

A

is semi-bounded then

Theorem 3·

If

J

O

A Hermitean operator

~

00

U

E

JJ (A)

satisfies the hypotheses of

is the Friedrichs extension of

is a self-adjoint extension of

is called

,

-c ,

A+c+l

A

A+c+l

then

JO-c-l

A, called its Friedrichs extension.

Thus every semi-bounded Hermitean operator has a natural self-adjoint extension.

8.

8.

SUMS AND LIE PRODUCTS OF OPERATORS

103·

Sums and Lie products of self-adjoint operators A theorem of Paul Chernoff [23] gives us the result needed in

order to discuss the one-parameter unitary group generated by the sum or Lie product of two self-adjoint operators.

The natural context for

the discussion is given by the notion of a contraction semigroup on a Banach space. Let

*

a family of operators p

o

A contraction semigroup on

be a Banach space. pt

in

= 1 ,

L(

P

X) ,

for

0 < t
0 , where A is a self-adjoint operator on a Hilbert space. The infinitesimal generator of a contraction semigroup

thE operator

pt

is

defined by

A

Au =

ph_l -h- u ,

lim

h -70

on the domain

)J(A)

Theorem 1. imal generator for all

~

Let

pt

Then

A

with

of all

Re

~

u

in

1

for which the limit exists.

be a contraction semigroup with infinitesA

> 0,

is a closed, densely defined operator and ~

is in the resolvent set of

A,

104.

FLOWS

I.

-f...tPt udt ( f...-A ) -1u = Joo 0 e

(2 ) for all

u

X

in

Proof (cf. the proof of Stone's theorem, Theorem 4, §5). Let

Re f... > 0

an operator

Then the integral in (2) clearly converges and defines Rf...

L(

in

X)

with

I/R"II:: liRe f....

We have

{~ e-f...(t-h)ptudt _ J~ e-f...tptudt}/h =_~ Jh e-f...tptudt + Joo e h

0

h

Therefore .G«Rf...) Also, if and

Thus

u

-f...(t-h)

C

is in

J7(A)

-Cr(A)

and

h

- e

-f...t

ptudt

ARf... =-1 + A.Rf... ; that is,

then, as is easily seen,

ptu

(f...-A)Rf... = 1 .

is in ~

(A)

APt u = P t Au , so that

Rf...(f...-A)u = u

(f...-A)Rf...

=1

for

u

in ~(A).

Together with the fact that

, this means that

Re f... > 0 . Now lim f...~oo

A.Rf...u = lim f...~oo

f...

J~ e-f...tptudt

u,

U E }.

,

8.

and each

AR~u

(~_A)-l

is in

so

A

SUMS AND LIE PRODUCTS OF OPERATORS

is in ~(A).

1:. ),

L(

is closed.

Thus

(~_A)-l

A

105·

is densely defined.

~-A

is closed, so

Since

is closed, and

QED

The Hille-Yosida theorem [26] asserts that if densely defined operator on a Banach space such that resolvent set of

A for

t.. > 0

II (t..-A) -111 :::

with

A

is a closed, is in the

~

lit..

then

A

is the

infinitesimal generator of a unique contraction semigroup. Let

JI

is a linear subspace of

*

A be an operator on the Banach space of

J1(A)

such that

A

A to.tJ have the same closure:

=

A core of

A and the restriction

Alh.

For example, if

is a self-adjoint operator on a Hilbert space, a core of

JJ

linear subspace

of

rIJ' (A)

A

A

A

is any A to J)

such that the restriction of

is essentially self-adjoint (for if one self-adjoint operator is contained in another, they are equal). Theorem 2.

Let

An' for

n

=

1,2,3, ... , and

A be the infin-

itesimal generators of the contraction semigroups

pt n

and

pt

Let

JJ'

in

olJ,

u

is in

be a core of

./J(A ) n

for

A

,

and suppose that for all

sufficiently large and

n

A

n

Then for all

u

u

in

u~Au

J:. ,

(4) uniformly for Proof.

t

in any compact subset of Let

Re t.. > O.

[0,00) .

We claim that for all

u

in

J

106.

I.

Since the

(A-A )-1 n

FLOWS

are bounded in norm uniformly in

we need only show that (5) holds for is dense because

.tJ

is a core and

may assume that

u

(A-A)v

Let

u

by (3).

0

and this tends to

with

be in

JJ

u

(by liRe A) ,

n

in a dense set.

Now

(A-A) ,}J(A) = ~ v

JY.

in

(A-A)J)

Therefore we

Then

Thus (5) holds.

and let

> 0

t

t < 0

Since

A

is densely defined and ~ (t)

fore if we show that

n

.tr is

ff

a core,

is dense.

converges uniformly in

t

0, we are

to

are bounded in norm uniformly in

through, since the

There-

n

(by 1) .

Now

for

t

>

0 , and this is bounded in norm uniformly in

Thus the

~

n

order to show that

(t)

~ (t) ~ 0

(~*p)(t) ~ 0

p: JR

~

JR

uniformly in

n

that

n

are equi-uniformly continuous.

uniformly in

t , for all

with compact support, where

(~n *p)(t)

~n*P

= Joo ~n (t-s)p(s)ds _00

t

nand

t , by

Therefore, in

we need only show

C~

functions

is the convolution

8.

SUMS AND LIE PRODUCTS OF OPERATORS

~n*P

The Fourier transform of

~nP'

is

107·

where

and

is in

Ll(IR).

the Lebesgue dominated convergence theorem and the

By

Fourier inversion formula,

uniformly in

t.

QED

Theorem 3.

Let

T

be in

L(

1.)

with

/lTII < 1

Then

t'VV\J> et(T-l) is a contraction semigroup.

(6)

For all

u

in

1 ,

II (en(T-l) - Tn)ull :::.[n II (T-l)ull Proof.

For

lie t(T-l)11

The function

t

~>

t

~

0 ,

=

lie -t

k~=O t~k,.kll

e t(T-l)

< e -t

k t k=O k!

~

is continuous from

= 1 .

[ ] 0,00

to

L (Y.. ~

)

and a fortiori it is strongly continuous, so it is a contraction semigroup. For any

u

in

*- ,

108.

By

I.

FLOWS

Ik-n I

the Schwarz inequality, applied to the sequences

the weights

and 1 with

nk/k! ,

where cp(n)

e

-n

k

co

~ (k-n) k=O k!

2

L:

We see that

and

cp(O) =

° , so that

This proves

cp(n) = n

Theorem 4 (Chernoff's theorem). IIF(t)11 < 1

for all

t

in

traction semigroup on ~ be a core of

A.

h

uniformly for

u

in

t

and

QED

F: [0,00) ~ L( ~)

F(O) = 1

Let

with infinitesimal generator

pt

F(h) -1 h

U

= Au ,

U E

~O

*

in any compact subset of

[0,00) .

with

be a con-

A, and let

Suppose that lim

Then for all

[0,00)

Let

(6).

ff.

J:r

8.

SUMS AND LIE PRODUCTS OF OPERATORS

Fix

Proof.

By Theorem 3,

!n

> 0 and let

t

C

109·

is the infinitesimal generator of a contraction

n

tin ~ 0 ,

semigroup, and since

C

n

is itself the infinitesimal

generator of the contraction semigroup tC t "VV'0 e

By Theorem 2, for all

u

in e

uniformly for

t

n

Jt

tCn

t

u~Pu

[0,00).

in any compact subset of

But by Theorem

3,

t n

F(-) - 1

0 (uniformly for t

which converges to for

u

in

.J:r.

all

u

in

'*- ,

Theorem

in any compact subset of

Thus (7) holds for a dense set of u I s and hence for F(! ) n n

since the

5

are bounded in norm by 1.

(Trotter product formula). A, B,

and

Let

A+ B

be the infinitesimal generators of the contraction semigroups

on the Banach space

(8)

[0,00))

X

Then for all

u

in

)l ,

QED

no.

I.

uniformly for

i~

t

Proof.

any compact subset of

)) (MB)

=

J)(A) n JJ(B)

t

F(t)u Therefore

FLOWS

P (u+tBu+o(t))

(8) holds by Theorem 4.

[0,00)

is a core for

MB

Let

u + tAu + tBu + o( t)

QED

We state a special case of this explicitly. Theorem 6.

Let

A

and

B be self-adjoint .operators on a

Hilbert space

1+ ,

and suppose that

Then for all

u

l+-

in e

it(A+B)

t

case

iA

00

in any compact subset of

An operator

(_00,00) .

A on a Hilbert space is called skew-adjoint in

is self-adjoint; that is, in case

essentially skew-adjoint in case that is, in case

is essentially self-adjoint.

lim

u

n.~

uniformly for

A+B

-A = -A *

If

iA A

A = -A *

It is called

is essentially self-adjoint; is skew-adjoint then

are infinitesimal generators of contraction semigroups

A e

tA

and

-A

and

e

-tA

which together make up the strongly continuous one-parameter unitary group

e

tA If

for A

-00 < and

t