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

Подождите немного. Документ загружается.

3. Unbounded Operators; The Resolvent. Closed Operators

operator A is closed if and only if whenever (u,) is a sequence in D(A) such

that un - u and Aun -> v for some u, v E E, it follows that u e D(A) and

Au = v. Clearly a bounded, everywhere defined operator is closed. An

equivalent definition of closedness is the following. Consider the Cartesian

product E X E endowed with any norm that imparts to it the product

topology (for instance, IKu, v)IIEXE = IIuII+IIvID Then E X E is a Banach

space. The graph of the operator A is

I'(A) = {(u, Au); u E D(A)).

Clearly I'(A) is always a subspace of E X E and it is closed if and only if A

is closed according to the previous definition.

A fundamental result is:

3.1

Closed Graph Theorem. Let A be closed and everywhere de-

fined. Then A is bounded.

This theorem is equivalent to the following result, which is of great

importance in its own right:

3.2

Open Map Theorem. Let B E (E; F). Assume that B (E) = F.

Then if U is an open set in E, B(U) is an open set in F.

3.3 Corollary.

Let B E (E; F). Assume that B is one-to-one and

B(E) = F; that is, assume that B has an inverse B-1. Then B- 1 E (F; E).

For proofs of these and more general results, see Banach [1932: 1, p.

38], Hille-Phillips [1957: 1, pp. 46-47], or Dunford-Schwartz [1958: 1, pp.

55-58].

3.4 Lemma. Let A be a closed operator in E. Assume that is

defined in a (bounded or unbounded) interval J and takes values in D(A);

suppose, moreover, that u(-), Au(-) are continuous (or only piecewise continu-

ous) and that 11u(-)II, IIAu(-)II are integrable in J. Then ffu(t) dt E D(A) and

A f u(t) dt = f Au(t) dt. (3.1)

The proof is immediate for compact intervals; in fact, we only

have to approximate fu dt by a sequence of Riemann sums un =

F-k (tk,

n - tk - 1,

n) u(4k n) and observe that Aun is a sequence of Riemann

sums approximating f Au dt. If J is not compact, we approximate f, u dt by

f fnu dt, where J,, is a sequence of compact subintervals of J, and apply the

previous observation.

Naturally, this result can be greatly generalized; the integral may be

multidimensional or understood in the sense of Lebesgue-Bochner. For a

very general statement the reader may consult Hille-Phillips [1957: 1, p. 83].

Given an operator A (not necessarily densely defined) in E, we

define, as in the bounded case, the resolvent set of A (denoted again p(A))

Elements of Functional Analysis

as the set of all complex A such that Al - A has a bounded inverse

R(A) = R(A; A). In the present situation this means

(AI - A)R(A) = I (3.2)

and

R(A)(AI-A)u=u (uED(A)). (3.3)

Note that, in view of the definition of the domain of the composition of two

operators, (3.2) includes the requirement that R(A)E C D(AI - A) = D(A).

The spectrum of A, a(A), is again defined as the complement of p(A).

Unlike as in the case where A is bounded, there are examples where p (A) or

a(A) are empty, but it is still true that p(A) is open and that is an

analytic function in p(A). Its Taylor development about A E p(A) is given

as before by

R(µ) =

-µ)nR(A)n+1

0" (3.4)

n=0

the series convergent in (E) for

l u - A I < I I R (A )I I

(in particular, every

such ,s belongs to the resolvent set p(A)). For a proof see Dunford-Schwartz

[1958; 1, p. 566]. An important ingredient in the proof is the following

observation: if B E (E; E) is one-to-one, then its inverse B-' (which may

not be everywhere or even densely defined) is closed. This shows that

Al - A = R(A)-1 is closed if A E p(A), a fortiori A itself is closed. The

converse is not true: a closed operator may very well have an empty

resolvent set. It follows immediately from (3.4) that powers and derivatives

of the resolvent are related by the following formula:

R(A)(n)=(-l)nn!R(A)n+1

(AEp(A),n>1).

(3.5)

An immediate consequence of formula (3.4) and following comments is:

3.5 Lemma.

Let the sequence (An) belong to p(A); assume that

An - A and that IIR(An)II remains bounded (or, more generally, that IIR(An)II

_ o(IA - Anl) Then

A E p(A).

In fact, the disk Iµ - Anl

IIR(An)II-1 must be contained in p(A) for

all n; for sufficiently large n the disk contains A.

In intuitive terms, Lemma 3.5 states that "the resolvent must blow

up (rapidly enough) when we approach the spectrum."

The next result, equally elementary, establishes that the resolvent

may be constructed by analytic continuation.

3.6 Lemma. Let Sl be an open connected set in C such that

Sl rl p(A) * 0, and let Q(A) be a (E)-valued analytic function in Sl such that

Q(A) = R(A) in Sl rl p(A). Then SZ c p(A) and Q(A) = R(A) in 2.

To prove Lemma 3.6 use Lemma 3.5 as follows. Take A0 E Sl rl p(A),

A E 0 and join them by a curve F in Q. Let F' = F n p(A). Then F' is open

3. Unbounded Operators; The Resolvent. Closed Operators

in r; but R(X) = Q(X) remains bounded in r, so that by Lemma 3.5 T'

must as well be closed in p (A). By connectedness F'= F.

Equalities (3.2) and (3.3) are sometimes called the first resolvent

equation(s). The second resolvent equation, which results rather easily from

the definition, is

R(A)-R(µ)_(t-A)R(A)R(s) (A,aEp(A)).

(3.6)

Another useful equation involving the resolvent can be obtained

taking u e D(Am), writing (3.3) in the form R(X)u = A-'u + A-'AR(A)u,

replacing the left-hand side into the right-hand side, and iterating. The

result is

R(A)Amu (A E p(A), A * 0).

We have observed in the previous section that if A is bounded, then

R (X) exists for I A I large enough. This property is shared by the resolvents of

some unbounded operators. However, we have:

3.7 Lemma. Assume R (A) exists for I A I> a and has a pole at

infinity. Then A is bounded.

In fact, we must have

R(A) XnB

(IAI > a),

(3.8)

n=-oo

where B E (E) (- oo < n< m). The coefficients are given by the usual

formulas

2Iri

A-to+'>R(A) dA,

jAJ=a+'

and it follows then from Lemma 3.4 that

c D(A). Making use of

(3.2) in (3.8) and equating coefficients in the power series thus obtained we

arrive at the relations Bm = 0, B = AB,,+ , (n * -1), B-1 = ABo + I, which

show that Bo = B I=

= Bm = 0, B_ I = I, B-2 = A, so that A E (E) as

claimed.

3.8

Example. (a) Let E = L2(0,1) (see Section 7), A the operator

Au(x) = u'(x), (3.9)

D(A) defined as the set of all u E L2(0,1) that are absolutely continuous in

0 < t < 1 and such that u' E=- L2(0,1). Then A is densely defined and closed

but p(A) = 0 (the equation (Al - A)u = v has multiple solutions.) (b) Let

B0 be the restriction of A defined by u(0) = 0. Then B0 is densely defined

Elements of Functional Analysis

and p(B0) = C. R(A) has an essential singularity at infinity. (c) Let B be the

restriction of A defined by u(0) = u(l) = 0. Then p(B) =0 (the equation

(Al - B)u = v has no solution except for certain right-hand sides.)

It is sometimes useful to classify complex numbers in the spectrum of

an operator. We shall only single out eigenvalues -that is, complex num-

bers A such that there exists u E D(A), u - 0 such that (Al - A)u = 0, or

Au = Au. (3.10)

Solutions u 0 of (3.10) are called eigenvectors (corresponding to the

eigenvalue A); when A is an eigenvalue, the set of all solutions of (3.10) is a

nontrivial subspace E(A) of E called the eigenspace corresponding to A. If A

is closed, its eigenspaces (if any) are closed. Given an eigenvalue A, a vector

u e D(A') is called a generalized eigenvector if there exists an integer m >-1

such that

(AI-A)"'u=0 (3.11)

for some u E D(Am), u x 0; generalized eigenspaces Eg(A) are defined in the

obvious way. A generalized eigenvector has degree m if it satisfies (3.11), but

(Al - A)"u * 0 for m' < m; the degree of an eigenvalue A is the maximum

of the degree of all its generalized eigenvectors (which may be infinite).

Equally infinite may be m(A), the multiplicity of A, defined as dim E(A) and

dim Eg(A), the generalized multiplicity of A, denoted mg(A). Obviously

E(A) c Eg(A), so that m(A) < mg(A).

3.9

Example. Spectral theory of finite-dimensional operators.

Let E _

C m (C m is m-dimensional unitary space), A an operator from E into E.

Then (a) a (A) consists of a finite number of eigenvalues A1,.. . , A" (n::5 m),

(b) E = Eg(A i) ® ®Eg(A"), where ® indicates direct sum (so that

mg(Ai)+ ... + mg(A") = m).

*3.10

Example. Spectral theory of compact operators.

An everywhere

defined operator A in a Banach space E is compact if and only if (Au")

contains a convergent subsequence whenever (u,,) is bounded. Then (a)

a(A) consists of a (finite or countable) sequence (A,, A2,...) of complex

numbers having an accumulation point at zero when the sequence is infinite,

and (b) Each nonzero A E a(A) is an eigenvalue with mg(A) < oo (hence

with finite multiplicity). However, unlike in the finite-dimensional case, the

space E may not be spanned by the generalized eigenvectors, even in an

approximate sense. Consider the Volterra operator

VU (X) = f 'u (t) dt (3.12)

in L2(0,1); A is compact and a(A) = (0). Since 0 is not an eigenvalue of V,

V has no generalized eigenvectors at all (see Dunford-Schwartz [1958:

Sec. 7.4] or Kato [1976:

1, Sec. 3.7] for complete treatments of spectral

properties of compact operators).

4. Adjoints

3.11

Example. Let A be an operator in a Banach space with p (A) * 0

and such that R(A) is compact for some A E p(A). Then: (a) R(A) is

compact for all A E p(A) (use (3.6) and the fact that the sum of two

compact operators, and the composition of a compact operator and a

bounded operator is compact). (b) a(A) is empty or consists of a (finite or

infinite) sequence A1, A2,... of eigenvalues with mg(Xk) <cc. If the se-

quence is infinite, then IAkI - oo. The case a(A)=0 is exemplified by the

operator Bo of Example 3.8(b); here B6-1 = V (V the Volterra operator

defined by (3.12)).

*3.12 Example. A functional

calculus for bounded operators.

Let

A E (E), KC (A) the class of all (scalar-valued) functions f holomorphic in

some open set of D a(A) (depending in general on f ). For every f E 9C (A)

we define

f (A) = 21- f f (A)R(A)

dX,

(3.13)

where r is a finite union of simple closed curves contained in of and

containing a(A) in their interiors. Then: (a) f(A) E (E). (b) If f, g E 9C (A),

then f + g and fg belong to 5C (A) and (f + g)(A) = f(A) + g(A), (fg)(A)

= f(A)g(A). (c) If f(A) =1, g(A) = A, then f(A) = I, g(A) = A. (d) If

f(A) =

the series convergent in a disk IAI < a containing a(A), then

f (A) = Ea"A", the series convergent in (E). (e) Spectral mapping theorem: if

f E ¶C(A), then a(f(A)) = f(a(A)) (see Dunford-Schwartz [1958:

1, p.

566]). The functional calculus can be extended to unbounded operators A

with "sufficiently small spectrum"; the algebra 3C(A) consists now of all

functions f holomorphic in some open set of D a(A) and at infinity. For-

mula (3.13) becomes

f(A)= f(oo)I+

2Iri

f f(A)R(A)dX,

(3.14)

with r described as before. The properties are rather similar to those for the

case A E (E); for additional details see Taylor [ 1958: 1 ] or Dunford-Schwartz

[1958: 1].

4. ADJOINTS

If B E (E), its adjoint is defined by the well-known formula (B*u*, u) =

<u*, Bu) (u e E, u* (=- E*). Then B* E (E*) (moreover

IIBII(E))

and

(A+B)*=A*+B*, (AB)*=B*A*,

(4.1)

if A belongs as well to (E) (see Hille-Phillips [1957: 1, p. 43]). The definition

Elements of Functional Analysis

of the adjoint is more complicated when we consider unbounded operators.

Let A be a densely defined operator in E. Then u* E E* belongs to the

domain of the adjoint A* if and only if u - (u*, Au) is a continuous linear

functional of u-that is, if there exists v* E E* such that

(v*, u) = (u*, Au), (4.2)

and we set

A*u = v*. (4.3)

Since we assume D(A) dense in E, (4.2) is sufficient to identify v*, thus A*

is well defined. It may well be the case, however, that D(A*) reduces to (0).

As we shall see below, this rather unfortunate situation can be avoided if we

impose some mild assumptions on A. To clarify this we begin by identifying

the dual of E X E endowed with the norm Ku, v}II = Hull + 110.

4.1 Lemma. The dual space (E X E)* can be linearly and isometri-

cally identified with the space E* X E* normed with

II(u*, v*}IlE*xE* _

max(Ilu*II,IIv*ID, an element of E* X E* acting as a functional on E X E

through the formula

((u*, v*), (u, v)) = (u*, v) - (v*, u). (4.4)

The proof is simple. It is clear that (4.4) defines a linear functional in

E X E whose norm does not surpass max(IIu*Ii,lly*Il). These two numbers

must in fact coincide; if, say, llu*ll ? Ilv*ll, we can insert in (4.4) elements of

the form (0, v) with llvll =1 and Ilu*(u)ll arbitrarily close to Ilu*II. On the

other hand, let F be a continuous linear functional in E X E. Then 4D((u, v))

= (u*, v) - (v*, u), where u*, v* are defined by (u*, v) =1'((0, v)) (v E E)

and (v*,u) _ -4((u,0)) (uEE).

The reasons for choosing the identification (4.4) over the far more

symmetric ((u*, v*),(u, v)) = (u*, u) + (v*, v) (which works equally well)

will be clear in a few lines.

We digress now briefly. If K is a subset of a Banach space E, we

define K' C E*, the orthogonal of K, as follows:

K' = (u*EE*; (u*, u) = 0, u E K).

It is easy to check that K 1 is always a closed subspace of E*. If E is a

reflexive space, then K 1 1 = (K 1)1 may be considered as a subspace of E.

We have:

4.2

Lemma.

Let E be reflexive, K a subspace of E. Then K 1 1 = K

= closure of K; in particular, if K is closed, then K 11 = K.

To prove this we observe that K'' D K even if K is not a subspace.

On the other hand, if u OE, we may construct through Corollary 2.2 a

functional u* such that u*(K) = (0), u*(u) = 1, which proves that u 44 K' 1

4. Adjoints

have

I'(A*) = r(A)1.

(4.5)

The proof is an immediate consequence of (4.2) and the definition of

orthogonal, and is left to the reader.

If E is a reflexive space, it is easy to see that the product E X E is as

well reflexive; that is, E X E can be identified with the dual of (E X E)* _

E* X E* using formula (4.4). We obtain combining the previous results:

4.4 Theorem. Let A be densely defined and closed and E reflexive.

Then D(A*) is dense in E*.

In fact, if this were not true, we could find a nonzero u E E with

(u*, u) = 0 for all u* E D(A*). But then (0, u) E I'(A*)1, and because of

Lemma 4.2 (0, u) E I'(A), which is impossible (AO = 0 * u).

If E is not reflexive, D(A*) may not be dense in E*; we note,

however, that A* is always closed. This follows directly from the definition

or from (4.5).

The operator A is called closable if r (A) is the graph of an operator

A, which is then called the closure of A. Clearly A is closable if and only if

whenever a sequence (u,,) in D(A) is such that u -* 0 and Au,, - v, then

v = 0 (observe than an arbitrary subspace of E X E is the graph of some

linear operator if and only if it does not contain any element of the form

(0, v), v * 0).

Since an arbitrary set and its closure have the same orthogonal, we

obtain from Lemma 4.3 that

F(A*)

17(,T) 1

r(A)1

F(A)1

= r(A*)

that is, j-* = A*; the adjoint of a closable operator coincides with the adjoint

of its closure (so that it is densely defined if E is reflexive).

Then

A** = A. (4.6)

In fact, F(A**) = I'(A*)1 = F(A)11 = r(A)= r(A) (we have used

Lemma 4.2 in the third equality). Note that A** = (A*)* is well defined

since A* is densely defined.

Computations with adjoints of unbounded operators are consider-

ably more complicated than those involving bounded operators. For in-

stance, instead of (4.1) we have

A*+ B* c (A+ B)*, A*B* c (BA)*,

(4.7)

The relation with adjoint theory is established by:

4.3 Lemma. Under the identification established in Lemma 4.1 we

4.5 Lemma. Let A be densely defined and closable and E reflexive.

Elements of Functional Analysis

where A C B means r(A) c F(B) (that is, B is an extension of A). To prove

(4.7) (which is immediate), we assume of course that both D(A) and D(B)

are densely defined; also in the first inclusion D(A + B), and in the second

D(BA) must be dense so that every adjoint can be calculated.

Better results can be obtained if B is assumed to be bounded and

everywhere defined. In that case we have

(A* + B*) _ (A+ B)*, A*B* _ (BA)*, (4.8)

under the sole assumption that A is densely defined (note that D(A + B) =

D(BA) = D(A)). The proofs are again a direct consequence of the definition

of adjoint and are omitted.

4.6 Lemma. Let A be densely defined. Then

p(A*) 2 p(A)

and

R(A; A*) = R(A; A)*.

Proof. Let A E p(A). By virtue of the first equality (4.8), (Al - A)*

= Al - A*; we apply then the second inclusion relation (4.7) obtaining

R(A; A)*(AI - A*) c I, (4.11)

which is (3.3) for A*. On the other hand, making use of the rather evident

fact that

A C B implies B* C A*, (4.12)

we apply the second equality (4.8) to R(A; A)(AI - A) c I obtaining

(AI - A*) R (A; A)* = I. (4.13)

This shows that A E p (A*) and that (4.10) holds.

4.7 Corollary.

Let A be densely defined and closed and E reflexive.

Then

and (4.10) holds.

p(A*) = p(A) (4.14)

Proof. According to (4.6), A** = A. The results then follow from

two applications of (4.9).

4.8 Example.

Let A, B0, B be the operators in Example 3.8. Then A* _

- B, B* = - A, Bo = - B1, Bf B0, where B, is the restriction of A

defined by u(1) = 0.

4.9 Example. Let A E (E), f E JC (A) (see Example 3.12). Then

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

5. Generalized Sequences. Weak Convergence

GENERALIZED SEQUENCES. WEAK CONVERGENCE

A set d = (a, $, ...) with a partial ordering relation < is called a directed set

if given a, $ E 9 there exists y E d with a< y, $ < y. A generalized sequence

or net in E is a function a - ua from a directed set into E. We say that ua

converges to u E E (in symbols, ua -+ u) if, given e > 0 there exists ao e Q

with

Ilu - uall <- E

whenever a >_ ao.

A net (v.; $ E

is a subnet of (ua; a E 6) if and only if there exists a map

j : JB - C@ with the following properties:

ul(s) = vv and for every a e e there

exists $ E 3 such that if y >_ $ in JB, then j(y) >_ a in d (note that, although

we use the same symbols for elements and order relations in the two

directed sets d and 6J3, these sets may be quite different).

A generalized sequence in E is said to converge E*-weakly to u E E if

and only if

(ua, u*) - (u, u*)

for all u* E E*, and a generalized sequence (ua) in E* converges E-weakly

to u* E E* if and only if

(ua, u) - (u*, u)

for all u E E. Clearly both notions are weaker than ordinary convergence;

note that in the space E* we have two different definitions, that of E-weak

convergence and E**-weak convergence, the second one being in general

more demanding than the first since E c E**. Both notions coincide if E is

reflexive, of course.

It is easy to see that if ua - u weakly (in any of the two possible

ways if the net is in E*), then

Ilull < lim inf7luall,

where the liminf of a real-valued generalized sequence is defined in the

same way as for a sequence.

5.1

Theorem of Alaogha Let (ua) be a bounded generalized se-

quence in E*. Then there exists a subnet (vi*) of (u*) that converges E-weakly

to some u* E E*.

A proof can be found in Dunford-Schwartz [1958: 1, p. 424].

A space E is separable if it contains a countable dense set. The

following sequential version of Theorem 5.1 holds:

5.2

Theorem. Let E be separable, and (u*) a bounded sequence in

E*. Then there exists a subsequence (u*(k)) of (u*) that converges weakly to

some u* E E*.

Elements of Functional Analysis

We return for a moment to the matter of adjoints. In case E is not

reflexive, it is important to have some indication that D(A*) is "large

enough." This is provided by the following weak analogue of Theorem 4.4.

5.3 Theorem.

Let A be a densely defined, closed operator in the

Banach space E. Then D(A*) is E-weakly dense in E* (that is, for every

u* E E* there exists a generalized sequence (ua) C D(A*) with ua - u*

E-weakly).

For a proof, see Hille-Phillips [1957: 1, p. 43].

The role of generalized sequences in topology is discussed with more

detail in Dunford-Schwartz [1958: 1, 1.7.11 or in Kelley [1955: 1, Ch. 2].

6. NORMAL, SYMMETRIC, SELF-ADJOINT,

AND UNITARY OPERATORS IN HILBERT SPACE

Let H be a complex Hilbert space. Then, as is well known, if u* E H*, there

exists an element yu* E H such that

(u*, u) = (yu*, u) (u E H). (6.1)

It is immediate that the map y: E* -* E is additive, isometric, and trans-

forms E* onto E; on the other hand, since (u*, u) is linear in u* while

(v, u) is conjugate linear in v, we have

y(Au*)_Xyu* (u*EE*,XEC).

(6.2)

Hitherto we have defined adjoints with respect to the bilinear form

(u*, u); in the present case we can also use the scalar product (u, v) and the

operators thus obtained will be called Hilbert space adjoints (we use the

notation A+ for them). Clearly A+ is an operator in H (instead of H*), and

A* and A+ are related by

A*=y-'A+y

A+=yA*y-'.

(6.3)

Most of the time we will ignore this distinction, however, and write A* also

for A+. This will originate no confusion. A closed, densely defined operator

A is called normal if

A*A = AA*. (6.4)

(This equality of course includes the requirement that D(A*A) = D(AA*).)

A densely defined operator A is symmetric if

(Au, v) = (u, Av)

(u, v E D(A)).

Clearly this is equivalent to

AcA*. (6.5)