Fattorini H.O., Kerber A. The Cauchy Problem

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6. Normal, Symmetric, Self-Adjoint, and Unitary Operators in Hilbert Space

A is called self-adjoint if

A = A*. (6.6)

It is evident that if A is everywhere defined and symmetric, then it is

self-adjoint (note also that, since A = A*, A must also be closed, thus

bounded by the closed graph theorem). Finally, a bounded, everywhere

defined operator B is called unitary if B* = B-'.

A projection is a bounded symmetric operator P with

p2 = p. (6.7)

A spectral measure (denoted P(dX)) is a map from the Borel sets in the

plane to (H) whose values are projections and which satisfies P(f) = 0,

P(C) = I,

P(Uej)u=Y_P(ej)u (uEE) (6.8)

for every sequence e e2, of disjoint Borel sets, and

P(e, rl e2) = P(el)P(e2)

(6.9)

for any two Borel sets e1, e2.

6.1

Theorem.

Let A be normal. Then there exists a spectral measure

P(dX) such that P(e) = 0 if e rl a(A) =0, P(e)H C D(A), and AP(e) D

P(e)A for any bounded Borel set e (moreover, P(e) commutes with all

bounded operators that commute with A), the domain of A consists of all u with

IAI2IIP(dX)uII2 <ao

(6.10)

a(A)

and

(Au,v)=f

X(P(dX)u,v) (uED(A),vEH).

(6.11)

a(A)

Note that

EIIP(e,)uII2 = E(P(ej)u, P(er)u) = E(P(e,)u, u) =

(P(e)u, u) = (P(e)u, P(e)u) = IIP(e)II2, where the ej are disjoint Borel sets

and e = U ej so that

I I P(dX) u 112

is a measure; the same is

true of

(P( d A) u, v). Moreover, it follows from the definition of P(d A) that

IIP(dA)uII2 = IIuII2

(6.12)

a(A)

for every u E E. Observe next that,

since (P(e)u, u) = IIP(e)u112 >_ 0,

(P (u) u, v)

a (semidefinite)

scalar product so

that (P(e) u, v) <_

IIP(e)uII21IP(e)v112 _< IIP(e)uII2IIvII2 By the Cauchy-Schwartz inequality,

If A(P(dX )u, v)I _< IIvII2 f IXI2IIP(dA)u112 so that the right-hand side of (6.11)

exists and defines an element of H if u satisfies (6.10).

Theorem 6.1 particularizes to self-adjoint and to unitary operators as

follows: a normal operator A is self-adjoint if a(A) is contained in the real

Elements of Functional Analysis

axis and, conversely, if A is self-adjoint, then A is normal and every element

in the spectrum is real. It follows that the spectral measure provided by

Theorem 6.1 vanishes for any e that does not intersect the real axis.

Therefore, condition (6.10) takes the form

X2IIP(dX)uII2 <00

(6.13)

(Au, v) = f \(P(dX)u, v) (u E D(A), v E H). (6.14)

- o0

We note that a(A) is contained in X >_ w if and only if (Au, u) >_ cIIuII2 for

u E D(A). Finally, in the case where A = U is unitary, a(A) is contained in

the unit circle I X I =1 and (6.11) becomes

(Uu,v)= f

X(P(dX)u,v);

(6.15)

1x1=

the spectral measure P(A) vanishes outside of IXI =1. Observe that when A

is bounded (which is automatically true in the unitary case) condition (6.10)

is always satisfied since P must vanish in the complement of a(A).

The spectral measure in Theorem 6.1 (which is uniquely determined

by A) is called the resolution of the identity associated with A.

Theorem 6.1 is the basis of the following functional calculus for A.

Let 3(A) be the set of all complex-valued Borel measurable functions

defined in a(A) and essentially bounded with respect to the resolution of

the identity P(A) (I f(X)I < C except in a set e with P(e) = 0). Define

'Df(u,v)= f f(X)(P(dX)u,v) (u,vEH). (6.16)

a(A)

Then (Df is linear in v, conjugate linear in u, and bounded (IDf(u, v)I <

C I I u I I

II v11). Therefore, there exists a unique operator B E (H) with (Bu, v) _

4if(u, v) (u, v e H). We define then

f (A) = B.

(6.17)

The functional calculus enjoys the following properties:

(f+ g)(A) =f(A)+g(A), (fg)(A) =f(A)g(A),

(6.18)

IIf(A)II =1181,

(6.19)

where 11f 11 indicates the essential supremum of f with respect to P(dX).

Moreover, if f (X) = 1, f (A) = I. We also have

f (A)* = f (A). (6.20)

The functional calculus can be defined as well when f is unbounded, in

which case the operator f (A) will be in general unbounded. We shall not

6. Normal, Symmetric, Self-Adjoint, and Unitary Operators in Hilbert Space

find occasion to make use of this general version. For a proof of Theorem

6.1 and for further details see Dunford-Schwartz [1963: 1].

A problem of considerable interest is that of extending a symmetric

operator A to a self-adjoint operator, if such an extension exists. To this end

we define the deficiency indices of a symmetric operator A as follows: the

positive deficiency index d+ of A is the dimension of the orthogonal comple-

ment of (A + J) H (which, of course, may be infinite), and the negative

deficiency index d_ is similarly defined with respect to (A - it )H. Then A

can be extended to a self-adjoint operator if d+ = d_ and, in particular, A is

self-adjoint if and only if d+ = d_ = 0. If d_ and d+ are different, this

extension is not possible, although we can always extend A to a symmetric

operator such that one or the other of the deficiency indices are zero; these

operators are maximal symmetric (that is, they are not properly contained in

any other symmetric operator).

We note, finally, an important particular case of Theorem 6.1 and of

the functional calculus. When the spectrum of A consists of isolated points

A1, A

2, ... , then each Xi is an eigenvalue of A with eigenspace Hj = PjH,

Pj = P({Aj}). Here (6.12) becomes

EIIP,u112

= IIuI12

(u E E) (6.21)

while, by virtue of (6.9), (Pi u, Pj u) _ (P, Pj u, u) = 0 if i * j. Condition (6.10)

describing the domain of A becomes

EIX;I2IIP;u112 < 00

and

Au = EA1PPu.

In particular, it results from (6.19) and the comments following that u can

be developed in a series of eigenvectors of A, any two terms in the

development being orthogonal (in fact, eigenvectors corresponding to differ-

ent eigenvalues are always orthogonal).

6.2

Example. When A is a bounded self-adjoint operator, both the "ho-

lomorphic" functional calculus in Example 3.12 and the "measurable"

calculus described above apply. The latter contains the former: if f E `3C (H),

the operator f(A) can be defined by (6.16) and (6.17). A similar observation

holds in relation to the functional calculus for unbounded operators in

Example 3.12.

6.3

Example.

Let A be a symmetric operator. Then its deficiency index

d+ (resp. d_) is the dimension of the subspace Hx = (u E D(A*); A*u = Au)

for any complex A with Im A > 0 (for any complex A with Im A < 0). See

Dunford-Schwartz [1963: 1, p. 1232].

Elements of Functional Analysis

6.4

Example. From the operators in Example 3.8, none is self-adjoint or

even symmetric. However, iB is symmetric and both deficiency indices equal

1. For a complex, Ial = 1, let Aa be the restriction of A defined by the

boundary condition u(1) = au(O). Then each iAa is a self-adjoint extension

of iB. If a = e'9', then a(iAa) consists of the sequence of eigenvalues

-((p +2kir), k =..., -1,0,1, ... .

7. SOME SPECIAL SPACES AND THEIR DUALS

We denote as customary by R' m-dimensional Euclidean space, the ele-

ments of R' indicated by x = (x1,...,xm), y = (yt,...,y), and so on. Let 0

be a Borel subset of R' and µ a positive Borel measure in Q. Given p,

1 < p < oo, LP(Q; µ) is the space of all µ-measurable functions u in 9 with

1/P

IIuII = IIuIIp = II UII LP(st,µ)

(fIu(x)IP(dx))

< oo.

(7.1)

Here II'IIp is a norm and LP(Q; it) is a Banach space with respect to it.

(Strictly speaking, the elements of LP(Q; µ) are not functions but equiva-

lence classes of functions, the equivalence relation being equality almost

everywhere with respect to µ.) The space L°°(S2; µ) consists of all µ-measur-

able, µ-essentially bounded functions u with the norm

IIull = p.-ess.sup{lu(x)I; x E 0)

(7.2)

and is likewise a Banach space; the same observation on equivalence classes

applies. When µ is the Lebesgue measure, we write simply LP(Q; µ) = LP(S2).

7.1 F. Riesz Representation Theorem. Let 1< p < oo. Then the

dual space LP(Q; p.)* can be linearly and metrically identified with LP'(2; µ),

pi-1 + p-1 =1, an element u* of

LP.

acting on LP as follows:

U) =

(7.3)

For a proof see Dunford-Schwartz [1958: 1, p. 286] (note that

inequality (2.1) becomes the Holder inequality Ju*udµ <_ Ilu*Ilp-IIullp)

The spaces LP(Q; µ) are always separable if 1< p < oo. The space

L2(SZ; µ) is a Hilbert space; the scalar product is

(u,v)= fu(x)v(x)µ(dx) (7.4)

(compare with (7.3)). We shall make occasional use of LP spaces of

vector-valued functions; we limit ourselves here to the finite-dimensional

case. The space LP(Q; µ)P consists of all vector-valued functions u(x) =

7. Some Special Spaces and Their Duals

(u 1(x ), ... , u ,(x )), where each of the components is µ-measurable and

IIuIIp- (f

Iuj(x)I°µ(dx)

<oo

Uj=1

in the case 1< p < oo; again LP(S2; µ)° is a separable Banach space, and the

duals can be identified as in the scalar case. The space L2(S2; µ)" is a Hilbert

space, the scalar product being

(u, v) =

(u (x), v(x))µ(dx)

uj(x)vj(x))p(dx). (7.5)

The other spaces where subsequent developments take place are

spaces of continuous functions. Let K be a compact subset of R m. The space

C(K) consists of all continuous functions u in K endowed with the

supremum norm

lull = sup(Iu(x)I; x E K).

Then C(K) is a Banach space and its dual can be identified as follows. Let

E(K) be the space of all finite Borel measures defined in K. For µ E E(K )

define the measure 1µl, the total variation of µ by the formula

ItLI(e) = supElµ(ej)l,

(7.6)

where e is any is-measurable set and the supremum is taken over all possible

decompositions of e into a disjoint union of a finite number of [L-measurable

sets (ej ). Then l µ I E E(K) and we can define a norm in E(K) that makes it

a Banach space thus:

Ilµll = Iµl(K)

7.2

F. Riesz Representation Theorem.

The dual space C(K )* can

be linearly and metrically identified with E(K ), an element u of E(K) acting

on C(K) as follows:

(µ, u) = j u(x)µ(dx).

(7.7)

See Dunford-Schwartz [1958: 1, pp. 97, 127 and 265] for the proof of

a more general result where K may be a compact Hausdorff space.

Subspaces and extensions of C(K) will also be used. If J is a closed

subset of K, we denote by Cj(K) the subspace of C(K) consisting of all u

that vanish in J; the dual Cj(K)* can then be identified with the quotient of

E(K) by the equivalence relation: It, - P2 if µ, and A2 coincide outside of

J-that is,

if µ1(e) = µ2(e) for every Borel subset e of K such that

Elements of Functional Analysis

e fl J = 0. This space can in turn be linearly and metrically identified with

the subspace E,, (K) of E(K) consisting of all measures µ with ,t (J) = 0. To

see this it suffices to notice that for every µ E E(K ), there exists another

µJ E E(K) such that It - µ j and js (J) = 0, the measure µk being defined by

µj(e) = µ (e fl (K \ J )); note that µ j is the element of least norm in its

equivalence class. In applications, however, we will most of the time ignore

the equivalence relation and represent linear functionals by (7.7), the value

of µ on J being irrelevant. The set J will usually be the boundary of K or a

subset thereof.

If K is closed but not compact, define C0(K) to be the space of all

continuous functions in K with

lim u(x) = 0.

lxI -,ao

The space E(K) is defined and normed in the same way as when K is

compact. The identification of the dual of C0(K) is provided by

7.3 Theorem. The dual space C0(K)* can be linearly and metrically

identified with E(K), an element µ E E acting on C0(K) according to (7.7).

Theorem 7.3 is an immediate consequence of Theorem 7.2. In fact,

let K U(oo) be the Alexandroff or one-point compactification of K (Royden

[1968: 1, p. 168]). Then C0(K) coincides with the subspace C,.)(K U(oo))

consisting of all functions in C(K U(oo)) that vanish at oo; accordingly,

C0(K)* can be linearly and metrically identified with E(.)(K U(oo)), which

is linearly and metrically isomorphic to E(K ).

The same observations made in connection with Cj(K) apply to the

space C0,j(K) consisting of all u E C0(K) that vanish in a closed subset J

of K: the dual space Co,

(K) * can be identified with E, (K) linearly and

metrically.

Although we will work most of the time with complex LP, C, Co, and

E spaces (i.e., the functions and measures take complex values), we shall

also find occasion to use spaces of real-valued functions. When necessary to

distinguish between the two cases we shall do so by means of the subindices

C and R. (For instance, we shall write LP(K; µ)c, C(K)R, etc.)

We note finally that the spaces C(K) and C0(K) (and of course any

of their subspaces) are separable.

Let now fl be a domain (open connected set) in Rm. The space 6D (2)

(of Schwartz test functions in I) consists of all infinitely differentiable

functions p whose support supp p = (x; (p x $ 0) is compact and con-

tained in Q. When SZ = R ', we write simply 6 (R m) = 6D and we introduce as

well the space S (R = S (whose elements are also called Schwartz test

functions), consisting of all infinitely differentiable functions >L dying down

at infinity faster than any power of jxj together with all their derivatives.

61. (k)(S2) consists of all functions qp, k times continuously differentiable and

8. Convolution and Mollifiers. Sobolev Spaces

having compact support contained in U. Clearly 6D(9) = rl O(O(SZ). On the

other hand, 'l (k)(S2) consists of all functions qp defined and k times

continuously differentiable in SZ, vanishing outside a bounded set and such

that all partial derivatives of order < k are continuous in 9. The space

C(k)(S2) is defined in the same way but omitting the requirement of

bounded support; obviously, if SZ is bounded 61 (k) (D) and C(k)(SZ) coincide.

We also define 6D(Z) = rl k6l. and C(°°)(S2) = rl kC(k)(SZ) and write

simply 6D "k), C(k), and so on when SZ = R '. More often than not these

spaces will be given no topology or notion of convergence, with the partial

exception of 61. and other spaces of test functions in Chapter 8; also the

functions in them may be real or complex, the distinction indicated with a

subindex C or R as in LP or C spaces.

Some liberties with the notations just introduced (and with others)

will be taken in subsequent material. For instance, C(2) will be written

C[a, b] (not C([a, b])) when SZ = [a, b], C(Q) will be abbreviated to C

whenever SZ has been previously identified, and so on.

CONVOLUTION AND MOLLIFIERS. SOBOLEV SPACES

We consider the spaces LP(Rm) = LP(Rm; µ), where µ(dx) = dx is the

ordinary Lebesgue measure and 1 < p < no. If f E Lp(R m), g e L(m), we

define their convolution by

(f * g) (x) = f f(x - Y)g(Y) dy.

(8.1)

(All the integrals in this section are over R'.)

8.1

Theorem of Young. Iff ELP and gELq, 1<p,q<oo,p-'+

q-' >>-1, then the integral in (8.1) exists for almost all x E R ', f * g belongs to

Lr(r-'=p-'+q-'-1) and

if * gllr<IIflIpilgllq

For a proof see Stein-Weiss [1971: 1, p. 1781. As a particular case of

this result we obtain that if /i is a function in L', the operator Ju = $ * u is

bounded and IIJII <- II/II1.

An important instance of this kind of operator is the following. Let $

be a nonnegative function in 6D with support in (say) I x 1 < 1 and j/3 dx = 1.

For n >-1 define nm/i(nx), and

U. (8.2)

8.2 Theorem. (a) The operator J is a bounded operator in LO(Rm),

1 < p < oo, with norm < 1 (n >-1) (b) if 1:5 p < oo,

IIJu-uiip- O as n-*oo

for any u E L".

Elements of Functional Analysis

Part (a) results from the preceding comments; see Stein-Weiss [1971:

11 for a proof of part (b).

The interest of this result lies in that each Jnu is infinitely differentia-

ble; in fact, if D" = (D')"-

(Dm)"- (a=

aDj = d/dxj) is an

arbitrary differentiation( monomial, then

D"(Jnu) = D"(Nn * u)

= D"Nn * u

so that we can approximate an arbitrary u E LP by infinitely differentiable

functions in the LP norm. Moreover, if the support of u is bounded,

supp(Jnu) c (x; dist(x, supp u) < 1/n) is also bounded. Since any function

u in LP can be approximated by a function with bounded support (say, its

restriction to Ixi < n for n large enough), we obtain in particular that 6D is

dense in LP(Rm), that is,

61 =LP(Rm) (1<p<oo).

We shall call the sequence (an) a mollifier, or mollifying sequence, or a

8-sequence.

Given an arbitrary domain 12 c R "', an integer j >_ 0 and a real

number p, 1:5 p < oo, we define Wj"P(S2) as the space of all functions

u E LP(Q) such that

Dau E 09)

for j al = a, +

+ an

j (the derivatives understood in the sense of distri-

butions). Then the space Wj,P(S2) is a Banach space with the norm

Hull = IIuII j,P = II uII W1.P(5 )

I/P

= I fID"u(x)IPdx) . (8.3)

Clearly

Hilbert space for any j, the scalar product defined by

(u, v) = (U, V)j,2 = (U, V) Wj-'(Q)

D"u(x)D"v(x) dx.

(8.4)

I"Isj

For proofs of these facts and those that follow, as well as for additional

details, the reader may consult Dunford-Schwartz [1963: 1, Ch. 16], Lions-

Magenes [ 1968: 1, Ch. 11, or Friedman [ 1969: 1, Part 11. When p = 2 the

notation Hj(S2) is often used instead of

In case S2 = R "', the space Hj(R ') = Wj, 2 (R') can be easily char-

acterized using the Fourier-Plancherel transform in L2(Rm),

dx (8.5)

qu(a) = u(a) = lye

(2zr) m/2

Ix1 <

8. Convolution and Mollifiers. Sobolev Spaces

where a = (o, x) = a1x1 + + a x,,,, and the limit in (8.5) is

understood in the sense of the norm of L2(Rm).' The operator F is an

isometric isomorphism from L2 (D m) onto L2 (U8 m) (subindices are self

explanatory). The inverse of T is given by

do,

(8.6)a

u(x) _ i 'u(x) = limo

Ia s a

the limit understood as in (8.5). A function u E L2 satisfies D' u E L2 if and

only if a' Y u E LQ, where we have set a ° = a l a2 2

a"; moreover,

(D"u)(a) _ (-

It follows that u E HJ(F m) if and only if (1 + l a l2) i/2 u E L2 (here 1012 =

a + + o,?) and the norm 11U11;,2 = II(1

is equivalent to the

norm of HJ(R'). These observations make natural the introduction of the

spaces HS(Rm) (here s is a nonnegative parameter) that consist of all u E L2

such that (1 + I a 12 )sl2 u belongs to L2. These are Hilbert spaces under the

scalar product

(u, v) = (U, v)s,2 = (U, V)Hr(R')

fRm(1+IaI2)Su(a)v(a) do

(8.7)

corresponding to the norm

(lull = Ilulls,2 = IIUIIHS(R'") =

11(1+1012)S12

UIIL2(Rm)

We shall also make use of HS spaces of vector-valued functions u =

(u , , ... ,

These spaces are denoted by HS(R m)

the scalar product is

defined by (8.7) with (u (o ), v (a)) in the integrand.

Sometimes, functions that belong to LP of WW' only locally will

appear. For instance, let 2 be a domain in Rm. We write u E Lfa(S2) or

u E Lf in Sl to indicate that the function u, defined in Sl, belongs to LP(K)

for every compact set K C S2. Similarly, u E W p(Sl) means that u E

for every bounded domain Sl' with Q' c Q.

'Note that, due to the factor (277)- /2 in our definition of the Fourier transform, the

convolution formula takes the form flu * v) = (217)m/2 Y u

j-V.

Chapter 1

The Cauchy Problem for Some

Equations of Mathematical Physics:

The Abstract Cauchy Problem

The purpose of this chapter is to introduce the notions of properly

posed Cauchy problem in t > 0 (Section 1.2) and in - oo < t < oo

(Section 1.5) for the equation u'(t) = A u(t) in an arbitrary Banach

space E. These definitions will be fundamental in the rest of this

work. In the case A is a differential operator in a function space E,

there are relations with Cauchy problems that are well posed in the

sense of Hadamard; these relations are explored in Section 1.7. The

rest of the chapter is understood as motivation for the central idea of

well posed Cauchy problem: several equations of mathematical physics

are examined by ad hoc Fourier series and Fourier integral methods

with the aim of discovering properties of solutions which will be

generalized later to wide classes of equations.

Sections 1.1 and 1.3 deal with the heat-diffusion equation in a

two-dimensional square. We find that (a) the equation produces a

properly posed Cauchy problem in the spaces LP (I s p < oo) and in

spaces of continuous functions; (b) nonnegative initial data give rise

to nonnegative solutions; (c) the L' norm of nonnegative solutions is

preserved in time; (d) solutions become extremely smooth in arbi-

trarily small time. All of these properties are instances of general

theorems on parabolic equations to be found in Chapter 4.

Section 1.4 treats the Schrodinger equation. Its properties are

in a sense opposite to those of the heat-diffusion equation; for

instance, the Cauchy problem is well posed in the space Lz (but not in