The compact approximation principle in the theory of approximation methods

THE COMPACT APPROXIMATIONPRINCIPLEIN THE THEORY OF APPROXIMATIONMETHODS* G. M. VAINIKKO Tartu

(Received

8 January 1968)

THE topics dealt with in the present paper are closely related to Kantorovich’s general theory of approximation methods (see 11, 21, and also 1341). The theorems on the convergence of approximation methods in linear and non-linear problems which are proved here are more general that those in the papers mentioned, and at the same time are convenient for applications. The closeness of the exact and approximation problems is characterized in a new way, using the concept of the compact approximation of operators. The large number of theoretical topics discussed has compelled applications to be relegated to separate papers. The most important applications are to finitedifference methods for solving boundary value problems. 1. Compact

Approximation

of Linear

Operators

Let E be a (real or complex) Banach space and {EnIIm a sequence of closed subspaces of it. Then,

Em= (where

as

{z~E:p(z,E,)-tO

n-too}

p(z, E,,I) = inf llz - znll is the distance from z to En), is also a closed z,~En

subspace of E. Definition 1.1. We shall say that the sequence of linear continuous operators the continuous operator T: E + E if

T,: E, + E, compactly approximates

(a) TE c E,; *Zh. vy’chisl. Mat. mat. Fiz. 9, 4, 739-761,

1

1969.

G. M. Vainikko

(b) given any n, < n, < . . . < nk < . . . , the convergence znk-+ z (z,~ E E, J implies the convergence T, kz,,k-+ TZ (the arrow denotes convergence in the norm); (cl whatever the bounded sequence .znE E, (n = 1,2, . . .) Tnzn + Tz, is compact in E.

the sequence

Let i yk 1 be a sequence in E. Let x 1yk 1 be the exact lower bound of the 6 > 0 for which a finite c-mesh exists in E for 1yk 1. The numberx ( yk 1 character izes the measure of non-compactness of 1yk 1(we recall that, in 171,the concept of measure of non-compactness is used to derive a fixed point principle). Clearly, xi yk 1 > 0 When and Ody when j yk 1 is not Compact. Definition 1.2. Let X be a number(real or complex according as E is real or complex). We shall say that a sequence of linear continuous operators T,: En -) En A-quasicompactlyapproximatesthe linear continuous operator 2’ if conditions (a) and (b) of definition 1.1 are satisfied, together with tc’ ) ~{Tnzn -

Tzn} < x{hzn - Tz,}

whatever the non-compact bounded

sequence {z,}, zn E E, (n = 1,2, . . .) . If, whatever the h from some set A, the sequence T,: En --fEn approximates T: E -f E h-quasicompactly, we shall speak of a Aquasicompact approximation. Notice that, if inf liar - Tzll> 0, aXzE, Ilrll=i

(1.1)

then the compact approximationof 2’ by the sequence Tn will imply that the approximationis Aquasicompact. For, if the sequence ( znl is not compact, it follows from (1.1) that the sequence i AZ, - Tz, 1 must also be non-compact, so that, from (c),

x{Tnzn- Tz,,} =

80<

x{kz, - Tz,},

i.e. condition (c’ ) is satisfied. While (1.1) is satisfied whatever the X from the resolvent set of the operator T, it may also be satisfied when h belongs to the residual spectrumof T.

The compact approximation

3

principle

Let p(T), o(T), a,(T), o,(T), o,(T) and R(3,; T) = (AZ-T)-’ denote throughout the resolvent set, spec,trum, point spectrum, continuous spectrum, residual spectrum and resolvent of the operator T, respectively [81. The notation p(T,), o(T,), etc. will have a similar meaning. We write T, for the contraction of the operator T on to the subspace E,. Notes.1. p(T)

It follows from (a) that

P(T) c p(T,),

the non-zero points of

and p (T,) being identical.

For, if 0 f jl E p(T, ) , it follows at once that the operator (I / h) [I + R (A; T,) T is the two-sided inverse of AI - T in E, so that h.E p (7’). Conversely, if the equation px = TX + f has a unique solution whatever the O#h=p(% f E E; from (a), the solution belongs to E, when f E Em, so that h E p(~,). If0 Ep(T),thenTE=E and (a) implies that E, = E, T, = T, 0~ p(T,). It is easy to give an example in which 0 E p(T,), 2.

but 0 E o(T).

If

E, = U E,, 7,

(1.2)

the statement of condition (b) can be simplified, by requiring that the convergence z, -+ z(z, E En) must imply the convergence Tnzn -, Tz. For, let z ,,k-+~(~,K E Enk). We see from (1.2) that z E E,. Hence the sequence 2 can be complemented up to the sequence z n + z (Z ,, E Ed ): The nk

convergence Tnzn -, Tz implies in particular the convergence Tmdzn h-+ Tz, condition (b) follows from the simplified statement of (b) just mentioned. If (1.2) is not satisfied, adequate for our purposes.

i.e.

the simplified version of condition (b) is not

Notice that (1.2) holds when and only when d(En, En+,) 2, . . . ), where

3. If T is completely continuous, equivalent to

-+0

as p + 00(n = 1,

conditions (c) and (c’ ) are respectively

G. M. Vainikko

(d) whatever the bounded sequence zn E En, the sequence T,z,

is compact

in E; (d’ 1 ‘x{Z’,zn1 -=cIhlx{zll)

whatever the non-compact bounded sequence z, E E,.

It follows from (al, (b) and (d) that the operator T,: E, + E, is completely continuous. This does not follow from (a), (bl and (cl. For instance, the unit operators of the subspaces En compactly approximate the unit operator I of space E provided that E, = E. 4. It follows both from (c) and from (c’ ) that IlT,II G c = const (n = 1, 2,. . .). 5. Let n, < n2 < . . . < nk < . . . . It follows from (c’ 1 that x {TnRznK- % J < This whatever the bounded non-compact sequence znk E _F,,,. X OJnK - %,I can be proved by complementing the sequence z nk by zeros up to zn E E, (n = 1, 2 , . . .I then applying (c’ 1. A similar remark applies to condition (~1. The following conclusion may be drawn from the above: if the operator sequence T,: E, + E, compactly (h-quasicompactly) approximates the operator will also compactly (X-quasicompactly) T, any subsequence T,k:Enk + En, approximate T. Notice that the limiting subspace Em may be widened on passing to the subsequence E No widening will occur if (1.2) holds. *k’

6. Let (1.2) hold. Then, the En h = 1, 2, . . -1 are closed subspaces of E,, and a sequence T,,: E, -+E,, compactly (Aquasicompactly) approximating the operator T: E + E, will also compactly (hquasicompactlyl approximate the operator T,: E, --fE,. 7. If the sequence S,: E, -, E, compactly approximates S: E + E, while T,: En -*E, compactly approximates T: E + E, then S, + T, and S,T, will respectively approximate compactly S + T and ST. 8. Let linear projectors Pn exist, projecting E on to the corresponding E, h = 1, 2, . . . ). In 14-61, the closeness of operators T and T, was characterized by the conditions IIT - I’,, 1’11 E + 0,

these conditions are satisfied

IIT, --P,TIIF,-+o

as

n--+00;

141, in particular, when Kantorovich’s

(1.3)

conditions are

The

satisfied.

compact

approximation

principle

5

It follows obviously from (1.3) that the operator T compactly approximak

the sequence T,.

The converse is false:

the compact approximation of T by the

sequence Tn does not imply the existence

of projectors (or even linear operators)

Pn: E + En, such that (1.3) holds. For instance, let E be a separable Hilbert space with a complete orthonormalized system 1ek )p”; En is the linear envelope It may easily be T,.z = (5, e,,)ei (5 EE,,,). of e,, . ., e,; 7~ = 0 (5 GE), seen that, in this example, T is compactly approximated by the sequence Tn. We have sup x~En,llxll-

IIT& - Tsll= 1

(n =

1, 2,

as

n-k&.

. . .),

1

whereas it would follow from (1.3) that sup IIT,s - T2II-t 0 ~~E,,ll~ll=i

2.

The Linear Inhomogeneous

Equation

Consider the equation hx =

TX +

f,

(2.1)

in space E, where T: E + E is a linear continuous operator, and h is a parameter (it will be regarded as fixed, and possibly zero). Consider, together with (2.1), the “approximating” equation AX =

Tnx f

fw

(2.2)

where T,,: En + E,, is a linear continuous operator in the closed subspace . .). We shall regard the solutions of (2.2) as 6, c E; fn E En (n = I,?. approximate solutions of (2.1). Theorem 2.1 Let the following conditions be satisfied: (1) fn+)asn-+c4; (2) the sequence of operators Tn: En + En kquasicompactly the operator T: E + E;

approximates

G. M. Vainikko

(3) h E p(T); (4) h e o,(T,)

when n is large.

Then, when n is large, (2.2) has a unique solution x, and the sequence of xn tends in the normto the solution x, = R(h, T)f of (2.1). We have Cl&

<

II&

-

Pn4l

<

(2.3)

C2%

where c, and c, = const > 0,

En =

II(Pnf - fn) + (~J-ZXJ - Tn~nGa) II,

(2.4)

and Pn: E + En is a linear operator with a domain of values in E,. Proof. We shall show that (2.5)

lima,>O, R-XG where an =

inf llhz, - T,z,II. ZnGun,iiznIi=1

Otherwise, a sequence zn E E,, llznII = 1, hz, -

Tnzn+O

as

can be found, such that n-+-m.

(2.6)

(We are somewhatsimplifying here: violation of (2.5) in fact implies that the convergence (2.6) only holds for some subsequence z ). We rewrite (2.6) as nk

(AZ, -

Tz,) + (Tz, -

Tnzn) --f

We conclude from this that x{hzn - 2’~~) = x{Tz,

0 as n--t *. -

T,z,}‘.

From condition

(cl ) of definition 1.2, the sequence zn is compact in E. Let z, -f z, as K + CS. Then, T, kZn k -+ TZO by condition (b). Passing to the limit & (2.6) over the sequence nkt we get hzo - Tzi, = Q

(lIzoIl = i),

(2.7)

The.compact

approximation

principle

which contradicts condition (3) of the Theorem. This proves (2.5). It follows from Z, > 0 that h @ op (T,,) and he condition (4), we find that h E p (T,). From (2.5),

oC(Tn) ; recalling

lim’llR(h; Tn) II<‘m.

(2.8)

n-X0

In short, given sufficiently large n, (2.2) has a unique solution zn = R(h; Tn)fn; the sequence x,, is bounded by virtue of (2.8). We rewrite

ax,,f Tnxn -I- fn

(2.9)

as (2.10)

(h&l - TX,,) + (TX, - Tnxn) = fn. From condition (l), XI f,] = 0, and we find from (2.10) that x{&

- TG} =

X{TG - TUG}. By condition (cl ) of definition 1.2, the sequence t xn 1 is compact. Passing to the limit in (2.91, the compact sequence x, is seen to have a unique limit point x ,=R(h,T)f. Hencex,+x,asn+-. The bounds (2.3) follow from (ti - Tn) (xn - Pnxoo) = f,, - Pnf - P,Tx,

+ T,,P,xm

and the uniformboundedness with respect to n of the norms llZ?(h; Tn) Theorem 2.1 is proved. IW - Trill.

II and

Notes. 1. The theoremremains true if conditions (3) and (4) are replaced by the weaker conditions (3’)

llhz- Tzll > 0 inf ‘EBJ&,ll+ll1

(4’ 1 fn E w-

T,)E*

and f E (AZ- T)E;

for large n.

2. The more detailed study of the spectra of operators T and T, in the next Section (see Theorem3.2) will enable cases.to be indicated in which condition (4) of Theorem 2.1 is a consequence of conditions (2) and (3). An example will

G. M. Vainikko

8

also be found, to show that condition (4) is not always a consequence (3). 3. Let

E,’ = (U-

T,)E,

and

E,‘=

{z~E:p(z,

E,‘)+o

of (2) and

asn+m.

It

follows from conditions (2) and (3) of Theorem 2.1 that E,’ = E,. For, let z E E,, hz -

Tz,

z, -f z, z, E E,.

i.e. kz - Tz E E,‘.

we have h G p(T,)

Then,

AZ,,

Hence, (W - T)E,

and (W- T)E,

-

T,z,

c E,‘.

E E,’

and lz, - T,z,

But, since h E p(T),

This shows that E, c E,‘.

= E,.

converse inclusion follows from the inclusions

E,’ c E,

+

The

(n = 1, 2, . . .).

While not as general as Theorem 2.1, the next theorem is more convenient. Theorem 2.2 Let the following conditions be satisfied: (1) f, --tf as n + m, (2) the sequence of operators T,: E, + En compactly approximates the operator T: E + E, (3) X E p(T), (4) the subspaces En are finite- dimensional. Then, for large n, (2.2) has a unique solution x,,, the sequence of xn is convergent in the norm to the solution x_ of (2.1), and (2.3) holds. 3. Spectra of the Approximating

and Approximated

Operators

The space E will be assumed complex in this Section. Lemma 3.1 Let A be compact in the complex number field, and let the sequence of linear approximate the linear continuous operators T, : E, --f En Aquasicompactly continuous operator T. Then, if

The

compact

approximation

inf IIhza-GE, llxll=i

principle

TJEI] > 0,

whateverthe h E A. we have man

> 0,

?Z-

where an =

inf II&z - T&zll. hC%A, ZnEE,’ llZnII=-1

Proof. Suppose that sequences A, EZA and zn E E, such that hnzn -

Tnzn -+

0 as

(Ilz,ll = I), exist,

V.* 00.

(3.3)

Since A is compact, we can assume without loss of generality that the sequence An is convergent, A, + h E A as n --f00. Now, (3.3) may be written in the form (2.6), and, by arguing as in the proof of Theorem 2.1, we arrive at (2.7), which con~adic~ (3.1). The Lemmais proved. We recall [8] that the spectral set of an operator ‘I’ is defined as the subset of the spectrumO(T), which can be surroundedby a small neighbourhoodwhich does not overlap the remainingpart of o(T). In particular, every isolated point of o(T) forms a (one-point) spectral set. Theorem 3.1 Let the sequence of linear continuous operators T, : E, -+ E, ~p~x~a~

compactly

the linear continuous operator T: E + E.

Then we have:

(2) given a spectral set 2 of the operator T_, a sequence hn E o ( T%), exists, such that p(kn, Z) -+ 0 as n -f 0. (We recall, see note 1 to definitions 1.1 and 1.2, that o(~,) c a(~), where those spectra may be identical apart from

10

G. M. Vainikko

the point h = 0). Proof. The sequence of II‘I’, 11is bounded. Let r]TIj, ]IT~J/< c (n = 1,2, Then, ~(7’) and o(T,l are contained in the circle 1x 1< c. Given an arbitraryQ> 0, let AE be the compactumobtained from the circle 1A 1.< t by removingo(T) together with an cneighbourhood. Inequality (3.1) holds for h E A,, and by Lemma3.1, an n, exists such that, when n &n, , . . .).

inf

II?% -

%A e’ZnEE,, llLnII=1

~?lGzII> 0.

This means that, when n + n, , A, contains no points of v~(T,) and @,(T,l, i.e. is located in the E-neighbourhoodof &7. Since r > 0 o#*) u oe(Tn) is arbitrary,this proves (I). To prove (21, we note that inf Ilk‘2- Tzll= 0. hfE% IczEm’ liZll==i

(3.42 If

For, let h be a boundarypoint of 2, iL= lim A,, A, EZp (2%). inf Ilk - Tzll = ?J> 0, &Eoo,iI2ii=: we have, when n is large,

This contradiction proves (3.4). From (3.41, we can see that inf

hfx, ZnGEn’ llxn1i=i

II?& - Td&ll-+ 0

as

n+

00.

(3.5)

The

compact

approximation

11

principle

llzll =

Ilhz -

Then Tz = lim Tnzn

Tzll < 6 / 2.

Let z =

lim zn, zn E E,,.

1.

such that

For, given any 6 > 0, we take h E 2 and 2 E E,,

by

condition (b) of definition 1.1, so that, for large n,

/Ih-,;I, Tn- IIG*. ,,::,I

Since 6 > 0 is arbitrary, this proves (3.5). Suppose we are given any small E> 0, such that the closure ZE of the o (T) \, 2. Let rs be the boundary

e-neighbourhood of I: does not intersect

of XCE. There are no points of a(T) on Ie, and by Lemma 3.1, for fairly large n, llhzll inf be-,, znEEn. Ilznll=i If Z, n a(T,)

is empty, then

Z, c

IIR(h; Tn) II< EE l-l a(T,)

Since

p(T,),

1

(a = const) .

and h E re.

for

is empty, this last inequality can be extended to all h E Z,,

which is incompatible with (3.5).

[21by

TnznII > a > 0

(The possibility

of this extension may be seen

applying the maximum modulus principle to the function

cp(a) = z* (AZ-

analytic in XE, where z E En and a* E E*n are arbitrary elements and

Tn)-iz, functionals).

Hence, for fairly large n, the union X8 17o(Tn)

is not empty,

and since 6 > 0 is arbitrary, this is equivalent to statement (21. The Theorem is proved. Statement (2) of Theorem 3.1 still holds if instead of assuming the

iVotes.1.

compact approximation of T by the sequence Tn, we assume a CC,\ Z)-quasicompac approximation. 2. With the assumptions of Theorem 3.1 it is not essential that a(T)) -to

as n + -.

For instance, let E be a separable Hilbert space with a

complete orthonormal system m and {ek}k=_n;

T( -jj k=-cc

sup ~(h, hEOr

Eke!+)=

{ek}$T_m; while En is a subspace with the basis

jj

Ekeh+l

(x=

k=--or, T,x

=

5 R=-L-0

TX

(5 E E,)

EhekEE);

G. M. Vainikko

12

Conditions (a), (b) and (c) of

(Tn is correctly defined, since TE, c En). definition 1.1 are satisfied. (A ( = 1, while o(T,) while or(T,)

is the circle

In this example, a(T) = a,(T)

is the circle 1h I< 1; uc(Tn) is the circumference 1X / = 1,

is the circle ( h 1 < 1.

3. With the conditions of Theorem 3.1, it is not essential for every point h E o(T) to be a limit point of o(T,). For example, let E be a separable Hilbert space with a complete orthonormal system {ek}kE,i ; let En be the linear envelope ofe 17 . . . , en, and T

(z&k k=l

)=i

(X=

Ehek_1

k=i I,x

= TX

r,&k+; k=i

(x E En).

Conditions (a), (b) and (c) of definition 1.1 are satisfied.

In this example, o(T)

is the circle ( X I< 1 (while o,(T)

is the circle ( X j < 1 and O,(T) the circumference Hence, 0 is the sole point of O(T) lh / = l), and o(T,) = {o) (n == 1, 2. . ..). which is a limit point of a(T,). As is clear from the example in Note 2, in the case of compact approximation, points of o(T) may in general be points of o,(T,) no matter how large n (but not points of up (T,) and uc (T,), in view of Theorem 3.1).

Theorem 3.2 Let A c p(T) be a connected compactum, and let the sequence of linear Aquasicompactly approximate the linear continuous operators T, : E, + E, continuous operator T: E + E. Then: (1) when n is large, either A c

p (T,,),

or A c

or (Tn) ;

(2) if A is capable of connected extension to the compactum A’ C p(T), having a point X for which / h 1 > 11T /I, then A c p (T,,) when n is large enough. Proof. By Lemma 3.1, an n, exists, such that, when n > n,, inf lk, .%A, ZnEE,, llr,ll=i

-

T,z,II 2

a >

0.

(3.6)

The compactum Hence, when n hn,, every point h E A belongs to p(T,) or u,(T,). A cannot contain points of both these sets, since then it would contain points

The

compact

approximation

13

principle

A’E PG%)

and h”e%(T,), as close as desired to one another, with the result that the resolvent R (h, T,) would be able to take values with inde~ni~l~ large norms,

which is incompatible with (3.6). Statement (1) is proved. To prove (2), we continue A’ in turn up to the connected compactum A” C: p (T) , having a Then, the inclusion A” c at(T,) point h such that 1A 1 > sup IlTJ. is possible, and by Segment large. The theorem is proved. 4.

when II is

(11, we have A c A” CI p(T%)

Root Spaces of Approximating and Approximated Operators. Rate of Convergence

Let A be a compactum in the complex number field. Let C,(A) be the Banach space of vector-functions z(X), continuous on A, with the values in E /Z(h)

iic,

(A) =

E

lb@) IlE.

Lemma 4.1 Let the operator sequence T, : E, -+ E, compactly approximate the operator 2’: E -, E. Let the compactum A c p(T) t and let A have an empty inte~ection with O&T,) when n is large. Then, when n is large,

A C p(Tn)

and the following statements

(1) whatever the ni < n2 < . . . < nk < . . . the convergence @Zk E En,)

implies the convergence

R (a; T, J z,~ -+ R (h; T)

hold: z,~-+ z with respect

to the norm of C,(A), (2) whatever the bounded sequence zn E En, the sequence (h; T)z,

R (h; Tn)zn - R

is compact in C,(A),

Proof. The inclusion lemma, we have

A C ~(7~)

follows tiom Lemma 3.1; by the same

14

G. M. Vainikko

sup III?@; Tn) II< c = const X6A

(n=rz@o+1;...).

(4.1)

To prove statement (l), notice that, given a fixed X E A, the convergence -+ z (z, E E, J implies the convergence 271 k

IIR(h;

Tn

For, the sequence of operators

&znk

-

R(3L;

T)zll~-+ 0.

T,, k : En, + E, k (k = 1,2, . . ,) compactly

approximates the operator T, and we obtain the convergence Theorem 2.1. Suppose that the convergence Let Xnk E A be such that IIR(hn, ; TnJznk-R(hnh;

(4.2)

(4.2) by applying

(4.2) is non-uniform with respect to h E A.

T)zIIE 2 27 > 0

(nbd.

(4.3)

We can assume without loss of generality that the sequence h is convergent, nk A,, k -+ h E A as k + 0~. From Hilbert’s identity fi(h’; Tn) -R(h”;

T,) =

(A”-A’)R(h’;

T,)R(h”;

(A’, h” E p (Tn) ) (4.4)>

Tn)

and (4.11, we get IIR(link;Tnk)-R(h;Tnk)ll~c21hnk--h1~0

as

k+-co.

Similarly,

llR(L k;T)-R(A;T)Il-+O

8s

k+m

and it follows from (4.3) that, given large enough n,

IIR(J.;Tn ,Jznk -R(h;

T)zll~ 2 q > 0.

This is incompatible with (4.2). Hence the convergence (4.2) is uniform with respect to h E A. Statement (1) is proved.

The compact approximation

To Prove (2), let z, E En be a bounded sequence. sequence

We have to show that the

R(h; Tn)zn--R(jL; T)z, = R(?L; T) (T, - T)R(h; is compact in C,(A).

15

principle

T,,)z,

(,A E A)

The resolvent R (A, T) maps sequences convergent in C,(A)

into sequences convergent in C,(A),

so that we only need to show that the

sequence

y,(k) = is compact in C,(A),

(Tn-T)R(k

Tn)zn

0

E

A)

i.e. it is sufficient to show [81 that:

ta) the sequence

IIy ,I(A) (IcE (A) is bounded,

(PI the functions y,(X) are equicontinuous with respect to h E A, (Y) given a fixed h EA,

the sequence Y,(X) is always compact in E.

Proposition (a) follows from (4.1) and the fact that the sequence . .);

AZ,.

is bounded;

iiT,lj

to prove Cp), we have to use (4.4) in addition.

(n =

Finally,

(y) follows from condition (c) of definition 1.1. The lemma is proved. Note. Statement (1) of Lemma 4.1 still holds if, instead of assuming the compact approximation of T by the sequence ‘I’,, we assume Aquasicompact approximation.

Theorem 4.1 Let the following conditions be satisfied: (1) the sequence of linear continuous operators T, : E, tE, approximates the linear continuous operator T: E + E.

compactly

(2) the operator T, : E, --f E, has an isolated eigenvalue A_, which corresponds to the finite-dimensional root subspace X,; X, is unique in the circle

i it -

IA--I+,]


(3) for sufficiently large n, at least one point of the circumference h, 1 = 6 does not belong to o,(T,).

G. M. Vainikko

16

Then, for large n, the interaction of CJ(T,) with the circle 1A - h, [ G 6 but contains only a finite number of points, each of which is an eigenvalue of ‘I’, of finite root multiplicity. We have the convergence (n -t WI is no~mpty,

P (2m, X,)-t sup XrnEXoc,Il”coll=~

P (&I, Xco) + 0, sup XnExn’ llrn ll=i

0,

(4.5)

where X, is the linear envelope of the root subspaces of the operator T,, corresponding to the eigenvalues lying in the circle 1il. - h, 1 G 6 Notice that condition (3) is satisfied provided that we can connect by a step line, remaining within p(T), a point of the circumference 13,- L, 1 = 6 and a point h(j h 1 > /f 7’ /If (see Theorem 3.2). Proof.

The circle A = {A: [h - A,/ = S} lies in p(T). By Theorem 3.2 and condition (3), for large n (say n 2 n,), the circle also lies in p(T,). The operator

Q.w=

(4.6)

is [S] a projector, projecting E on to the root subspace X,. define the operator (projector) 1

Qn = &

When n +n,,

we can

P

)

R(h; r,)da,

(4.7)

jh-A,\=8

Projecting En on to the subspace x’,, c (PnEr)L, invariant with respect to T, and corresponding to the spectral set Z,, =

(h: h E o(lTn),

i.i, -A,\

(by Theorem 3.1, xn is not empty when n is large).

< 6)

(4.8)

Now [8]

Zn = o(QnTn) *

(4.9)

Using Lemma 4.1, we obtain fkom (4.6) and (4.7): (a) for any rzl < nz < . . . <

nk <

...

the convergence

zn k-+z

(z,~ E

The compact approximation

E”J

principle

17

implies the convergence (& kznk+ Qmz;

(@ whatever the bounded sequence z,, E E, the sequence QnZn _ Qmz, is compact in E; since Q, is finite-dimensional, the sequence Qnzn is also compact in E. We shall show that the first of (4.5) holds for the subspaces X, c E, and Using reductio ad absurdum,suppose that, given some sequence X,cE, x,, E X, (11x,11 = 1) we have

Since x, = Qnxn, the sequence xn is compact by virtue of (p). Let xnk * 50~ as k + 00. Passing to the limit in the equation 5nk = Qn, “nk and recalling(a), Thus the convergence x,,k 3 xm contrawe get xm = QExoD,i.e. xm E X,. dicts (4.10), so that the first of (4.5) is proved. To prove the second of (4.5), assume that, for some xc0 E X,

p(xm,Xn) 2

rl

>o

(n b no).

Since X, c E,, a sequence zn E E,, zn -+ xoo xmby virtue of (a), and

P(Xmy Xn) < IIXCO - QnznII+

exists. Then Q,,z,, --f QooxoD =

0 as

72-t 00,

in spite of the condition. This proves the second of (4.5). In view of (4.5), dim Xn = dim X, when n is large. Hence the operator Qn is finite-dimensional, and we find from (4.9) that Zn consists of a finite numberof eigenvalues of finite root multiplicity, while Xn is the linear envelope of the corresponding root subspaces. Theorem 4.1 is proved. Notes. 1. By Theorem 3.1, the eigenvalues of the operator T, lying in the circle IA--k( < 6, tend to A, as n + m. 2. Propositions (a) and (p) stated during the proof of Theorem 4.1, in conjunction with the inclusion QmE = X, c E, imply that the sequence of projectors Qn compactly approximatesthe projector Q,. Similarpropositions may be obtained for any functions of the operator 181.

G. M. Vainikko

18

Theorem4.2 Let the conditions of Theorem 4.1 be satisfied. Then,

where c = con&; h, E a( Tn) ( 1A,, - A, 1 < 6) ; value A, E o (T,) ; 4.1;

P,:X,+E,

2 is the rank of the eigen-

while Xn and X, have the same meanings as in Theorem

are linear operators such that Pnx, --f xoc

(n=l,&...)

as n + m whateverthe xm E Xm. The proof is similar to the proof of Theorem 1 of [6]. We only have to modify the argumentsregardingthe proof of inequality (9’ 1 of [61. Let Fn cEn*

be the linear envelope of the root subspaces of the operator

T,’ : En” --f En’, conjugate to Tn, which correspond to the eigenvalues of T*, lying in the circle IA,- h, I < 6 _ Let fn E I;,, Ilfnll= 1. We select a sequence yk E En, Ilydl = 1, such that 1= Since

fn=

Qn*fn= fnQn,

llfnll = Km If71(3%) I we

have

l;m f,, (Q,&

= ~1.

Since Qn is finite-dimensional, some subsequence Qnyk is convergent, Qnyki -+ 1 &, E X,. Obviously, fn(Xn)

while, since ]]+,,I]< I]Q,Jl, we have

=

17

The compact approximatioff

SincefnE F, (IlfJl = 1)

principle

19

is arbitrary,we finally obtain

the uniformboundedness of the 1102111s IIQn /I follows from the fact that the (911 of 161follows in an sequence Qn compactly ~proxima~s Q,. fatality obvious way from (4.13). Notes. 1. It is easily seen that cn + 0 as n + m, where Cnare the quantities defined by (4.12). The proof is based on the compact ~~~~ion of T by the sequence T n’ 2. Generally speaking, the bounds (4.11) cannot be improved(even if -t ~0). For instance, let E = E, = RI, P, = I, E,=EandI)T,-T\(+Oasn while T and Tn are given by i2-matrices, and T = (t
b;“=&

1

‘I’

( > yi

(i = 1, . . . , S),

n

where yi are I-tuple roots of unity (Iy, I = 1, y,” = 1).

We have

On the other hand, in our example,

so that

nl” -k,l=

1

e:”

(i=

I, . . . . I),

G. M. Vainikko

20

and the first of (4.11) is unimprovable.

It can easily be shown similarly that the

second of (4.11) cannot be improved. Obviously, similar examples can be adduced in infinite-dimensional

5.

The Problem

of Eigenvalues

for an Operator

spaces.

Equation

Let us consider the eigenvalue problem in complex Banach space E for the equation

TX = pSx,

(5.1)

where T; E -f E and S: E + E are linear continuous operators, and ,Xis a complex parameter, to be defined so that (5.1) has a norrzero solution. we consider the “approximate” problem

T,x = p&,x,' where T,: E,+

E,,

and

S,: E,, -+ E,

closed subspace E, c E (n = 1,2,

. . .).

(5.2)

are linear continuous operators in the We assume that:

(1) the operators S: E --fE and S,: E, -+ E, (2)

Along with (5.1),

are completely continuous,

0 E p (T) ;

(3) 0 @ or (Tn) for large

n;

(4) the operator sequences S respectively.

‘I’, and S,, compactly approximate operators T and

Using Lemma 3.1, we find from (2)-(4) that 0 E p (Tn) when n is large. Lemma 4.1 (with A = ( 1 I ), it is easily shown that the sequence of operators

Tn = T,-‘S,:

Usin;

E, -+ E,

compactly approximates the operator

.9- = T-‘S: E+-E. The operators Fn

and F

of o(Fn) and o(F) accumulation

are completely continuous, so that the non-zero points

are eigenvalues of finite root multiplicity; the unique’

The compact approximation

point of o (Fn)

and o (F)

principle

21

may be the zero of the complex plane.

Equations (5.1) and (5.2) are respectively equivalent to hx = 9-x and 3Lx= Fnx, where A = l/p. Applying Theorems 3.1, 4.1 and 4.2 to these last equations, we arrive at: Theorem 5.1 Let conditions (So)

be satisfied. Then,

(1) any eigenvalue pooof equation (5.1) is the limit of eigenvalues pn of equations (5.21, and conversely, every limit point of a sequence of eigenvalues of (5.2) is an eigenvalue of (5.11,

where Xoa and X, are the exact and approximateroot subspaces of the operator corresponding to the eigenvalue hm = l/p, (by an approximateroot subspace X, of the operator F we mean the linear envelope of the root subspaces of Fn, which correspond to eigenvalues convergent to X,), (3) we have

where 2 is the rank of the eigenvalue X, = l/p,

c and cl are constants, and p,: X, -+ E, such that Pnxoo + xw

of the operator F,

(n = I,&

. . .) are linear operators

as n -) 00for every xo5E X,.

The inequality (5.5) follows from (5.4) and the easily proved equation

G. M. Vainikko

22

P,Y-xcz, - .T,P,x, where 52 = 9-x,

=

T,-i [ (T,P,f,

E X,;

- PJL)

+

(P&G., -

SJ’nxm)],

the norms jIT;1 11are bounded in aggregate.

Notice that we have assumed the complete continuity of the operators S and S, merely in order to simplify the statement of the theorem. 6.

Compact Approximation

in Non-Linear

Problems

Let $2be an open bounded region in Banach space E, while P is the boundary is the closure of a. As usual, on the basis of the given and c= slur’ sequence of closed subspaces E, c E (n =I, 2, . . .) we construct the limiting subspace E, = {zEE:

p(z,E,)+O

as

n-too}.

We shall assume that the set 151, = 52,fl E, is non-empty. Then, for large n, will also be non-empty. The set an is open in En; the sets Q, = Q 1(7En. let Pn be its boundary in E,, and a,, = Q, U lTn its closure in En. Clearly, I’, c l?n E, (but we do not necessarily have I?, = I’ fl En). Definition 6.1. We shall say that the sequence of (non-linear) continuous operators Tnic, --f E, compactly approximates the (non-linear) continuous operator ‘7: Q + E, if (a) T?i c Em; (b) whatever the ni < n2 < . . . < nk < . . . the convergence (2, k E %,I implies the convergence T, k znk + Tz; (c) whatever the sequence

zn E?&,

the sequence

Tnzn -

Pz,

zn k + z

is compact

in E. The sequence T, will hquasicompactly replaced by the condition k’ 1 X{TnZn -

Tzn} < ~{hz, -

Tzn}

approximate the operator 2’ if(c) is

whatever the non-compact sequence

z7zE K. We turn now to the most important topological

characteristics

of non-linear

The

compact

approximation

principle

23

operators, namely, the rotation of vector fields [9]. The space E will be assumed real. We recall [9] that the lotion y (1 - ‘I’; l?f of the vector field x - TX is defined on the boundary P if the operator T is completely continuous and has no fixed points on r‘, i.e. if I? 8j-l X, is empty, where X, = (2, E 2: z = Y’s} is the set of fixed points of T. Let X,”

=

(2

E

is p(2, X,)

denote the ~neighbourhood of the set X,;

< E}

the set Xcmis empty if X, is empty.

temnaa6.1 Let the operators T,: ?& *En

(n = i,2,

. . .)

and T:% --f E

be

completely continuous, and let the sequence Tn compactly approximatethe operator T. Then, (1) if PI is the boundaryof a subregion 52’ c 52 and T has no fixed points on P1 , then, for large n, the rotation y (I - T,; I?,‘) is defined (i.e. T n has no fixc?dpoints on T’X) and y(Z - T,; I’,‘) = y(Z where PI is the boundaryof the set * n

T; I”),

Sz,’ = Q2’n E,

in En,

for any c > 0, an nf can be found such that, whatever the boundary P’ c (5 \ Xm8) is defined for n 3 n( and (6.1) is satisfied. (2)

Proof. Proposition (1) follows from (21, since the fact that the union I” n X, is empty implies the inclusion I” c (Sz \ Xoo”), where 6 > 0 is small. For, we should otherwise have ok’ - xf*+ 0 as k -f = for certain sequences Xkl E 1’ The set X, = TX:, and xk E X, is compact and closed, so that we can assume that xk -+ xm E X,. But then, we also have xk’ - xm, snd xc0 E r’ since PI is closed. Consequently, zoo E I?’ fl X,, in spite of our ass~~ion that I” fi X, is empty. *If

r(Z-

still holds

T; r’)

=

0,

here provided

the set SPn may be empty no matter how large n. we agree to assume

that y(Z -

T,;

M)

=

0.

The theorem

G. M. Vainikko

24

Let us prove (2). We write ac = a \ .A?,”

and

Since cc is closed, while 7’ is completely continuous and has no fixed points in EC, we have cL> 0 (this is easily proved by reductio ad absurdum,see ES]). Conside of the compact set 2’5; in view of the inclusion the s/4-mesh {$$, . . . , yzj 7’n c Em, we can assume that y 2) +z& E, (i = 1, . . . , r). Let ~2’ = lim Y?, y~kE,(i=1,...,

For large n (say R >,n,J, each of\he r; n=l, 2,. . .)is an c/2-mesh of the set Tc. We introduce the Schauder

sets (y’,“, ‘. . , y$) projection operators (see 193, p. 113)

(z

)A:) (2) =

E

Tsi;

n=%-J,n*+1,...,

yX’II,

a/2 - lb -

II2 - # ,II < a/2; I/z - &’ II 2 a/2,

0,

{

oo),

i=

1,...,

r;

n=no,*..,

00.

The operators Qn are continuous in T& SUPIIQJ’x

XEH

- T+II< f

(n = no, no + 1, . . ., m),

QnTikEn

(n=nt,,no+1,

sup~~Q,Tz-XEH

Q,Tx]j--+O

. . . . m), as n-+w.

(6.2) (6.3) (6.4)

Let I” ~3,. Then, /IX- Fsll > a for x E P , and it follows from (6.2) that the operator QnT has no fixed points on P , while y

(I - T; I?‘) = y(I-

QJ’;

I”)

(n > no).

From (6.3) y(I - Qnl; I”) = y(I - QnT; I’,‘)

The compact

approximation

principle

25

(the I - QnT on the right-hand side of this equation is regarded as an operator in En). Comparing the last two equations, we get (6.5) Let us show that, given large n, the vector fields tQ,Tz

0, (t; z) = z -

do not vanish when 6 < t < assume that sequences

1, z E-&

t, and z,,

Zn =

-

fl En.

(1 - t) T,z Using reductio ad absurdum,

(0 < tn < 1; zn ~3~

$ (1 -

tnQ,Jzn

n E,,),

exist, such tha

tn) Tnzn.

(6.6)

Since T is completely continuous, and from condition (c) of definition 6.1, and (6.41, the sequence zn is compact. Let tn

,,-+

Znk-+

f,

2’ as

k-too

(O< t’<

1;

d&ie).

Passing to the limit in (6.6) over the sequence nk, then using condition (b) of definition 6.1, and (6.4), we get z’ = t’QooTd + (1 - t’) Tz’

(2’ E GE) *

From this and (6.2), llz’ -

WI1 < t’IIQmTz’ -

Till < a / 2

(2’ E

E) .

This inequality contradicts the definition of a. Hence an n, exists, such that, when n bnc, the vector field Qn(t, z) has no zeros when 0 < t < 1; z E KE n E, This means that, when n >,ncLtbe vector fields x - Q,Tx and x - T,x are homotopic on rln, provided that r’ c Q~. This means that the rotation y (I - T,; I?,‘) is defined for n >,n,, and y(I - QnT; r,‘)

= y(Z - Tn; I’,‘),

which, in conjunction with (6.3, gives (6.1). The lemma is proved.

G. M. Vui~~kko

26

Note. Lemma6.1 still holds if, in&& of compact approximation,we have lquasicompact approximationof T by the sequence T,. The proof is virtually the same, except that condition (c’j has to be used to prove that sequence (6.6) is compact. A similar remarkapplies to the next theorem. Theorem 6.1 Let the sequence of non-linear completely continuous operators T,:gr, -+ E, compactly approximatethe non-linear completely continuous operator T: iI + E. Let T have no fixed points on the boundaryr and y(I---;I’)

(6.7)

#O.

Then, for large n, the set

X, =

{xnE

a,:xn

=

Tnxn}

of fixed points of T, is nonempty and sup P(~~,X~)-+O XnEX fl

as n-too.

(6-S)

Proof. The set XW of fixed points of 7’ is non-emptyby virtue of (6.7). Using Lemma6.1 and (6.7). we see that

y(Z - T,; I’,) = y(f - T; I’) # 0 when n is large, and Xn is non-emptyfor these n. Suppose that, for some sequence xn E X, p (Gl, X00) b rl > 0.

(6.9)

Since x, = Tnxn = TX, + (TAG - TXn) f the sequence x, must be compact in accordance with condition (c) of definition 6.1. Let xn h--+ xm. Then, T, kxn k+ TG=, and passage to the limit in the equation xn &= T, Ken k

The

compact

approximation

principle

!27

gives ga, = Tz,, i.e. xoo E X,. Hence the convergence %nk -+x(6.9). Thus (6.8) must hold. The theoremis thus proved.

contradicts

Corollary

Let the sequence of non-linear completely continuous operators T,: 2, -+ E, compactly approximatethe non-linear completely continuous operator 2’: f2 -+ A’& and let ‘I’ have an isolated fixed point x, E fi of non-zero index [9], unique in Q. Then, for large n, the operator T, has -at least one fixed point xn in En and every sequence x, of fixed points of T, in C12, tends in the normto x, as n + 00. We tnrn to the question of the eigenvalues and eigenvectors of non-linear operators. We introduce the space A’ = RI X E, the elements of which are the pairs??= (A, x), where h E Rt, z E E; lIZI = fh/ -I- bll. Let

where 6 > 0 is a given number,r’ is the boundaryof a given subregion ,Q’ c Q

rnf

is the boundaryof the region lsl,’ = Q‘ n E, in En. We recall (see [9], p. 189) that _G,,(a; r’) is non-empty,provided that y (61- T; I”) # 0 and O@fi’ or ~(61 - T; I?) f: 1 and 0 E $31; here A, > 6. Theorem6.2 Let the sequence of non-linear completely continuous operators T,: 2, A-E, compactly approximatethe non-linear completely continuous operator T; z -, E. Let the operator 61- T have no zeros on I? and let either of the following two conditions be satisfied:

(1) y(6LTT;Y)

Then, H, (6; I’,‘)

#OuOcf:iP,

is non-emptywhen n is large and

p f&, _Fm(S; sup &EQ% m’, Proof. Notice first that 0 @ @

ry 3 0 as n -+ 00.

implies 0 @%,’

(6.10)

while 0 E Ql implies

G. M. Vainikko

28

0 E 9,’

By Lemma 6.1, for large n,

(n = 1,2, . * .). Y (61-

Tn; I’,‘) =

my(61-

T;

r’) ,

and the fact that X, (6; I?,‘) is non-empty follows from the statement just made [9]. The convergence (6.10) is proved in the same way as (6.8). The theorem is thus proved. Notes.

1. Let o -c hi -c 6.

Under the condition of Theorem 6.2, we have

2. Theorem 6.2 still holds if, instead of compact approximation, we assume &compact approximation of T by the sequence Tn. Under the conditions of the next theorem, the eigenvectore of ‘I’form a continuous branch [91. The theorem tells us that this continuous branch is approximated by the continuous branches of the eigenvectors of the operators T,. Theorem 6.3 Let the sequence of non-linear completely continuous operators T,: & compactly approximate the non-linear completely continuous operator T: c+ Let T have a positive eigenvalue A, (if A, is negative we simply replace T zoo E 52 of non-zero index, which corresponds to the isolated eigenvector inE. Let O@%. Then the following propositions

-+ En E. by -T), unique

hold:

(1) if r’ is the boundary of a subregion Q’ C 9, containing x,, then 9, (A,; I’,‘) is non-empty when n is large, and the convergence (6.10) holds whatever 6,O < 6 < h,; (2) for any c > 0, an nE exists such that x, (A,; I’,‘) is non-empty when n pc whatever the subregion ,SYc 52, which contains x, and the boundary r’ of which is at a distance >/E from x,, (3) the convergence for c > 0,

(6.10) is uniform with respect to Q’ in the sense that,

The

compact

approximation

principle

29

where

Proof.

Proposition (1) follows at once from Theorem 6.2, Note 1 on this

theorem, and the inequality

y&J-T;r’)

=ym#O,

where y, is the index of the eigenvector X, and I’ is the boundary of any region 52’ c 52, such that xa, E Q’. To prove (21, we also have to recall the second statement of Lemma 6.1. Finally, (3) is easily proved by reductio ad absurdum. The theorem is thus proved. Note.

Theorem 6.3 still holds if, instead of compact approximation, we assume

h,quasicompact

approximation of T by the sequence Tn.

Finally, let us dwell very briefly on the question of the branching points [9] of non-linear completely continuous operators T: E + E and T, : E, -+ E,. Let TO = 0 and

T,O= 0,let T and T, beFnSchetdifferentiable at zero, and let the sequence of linear completely continuous operators T,'(0) :En * En compactly approximate the linear completely continuous operator T'(O) :E -+E. If the completely continuous operator T’ (0) and T’,(O) are considered in complex extensio of spaces E and En, we have, in accordance with the results of Section 3, p (o(Tn’ (0)),

‘o(T’(0)))

--f 0

of odd root multiplicity.

as n -$m. Suppose that

T'(0)has

a real eigenvalue p,

Then, at least one of the eigenvalues of T’,(O), which

tend to /.&V will also be real and of odd n>ot multiplicity. This implies that the branch values of T,which correspond to odd-dimensional root subspaces of the operator T’ (0). are the limits of the branch values of the operators Tn. This result may easily be extended to the case when two or more of the rotations [9] r(~Loo),Y(CLm-O),y(CLOO+O) 7.

are defined and distinct for the branch value p-00’

Case of Differentiable

Consider the equations Ax = TX and AX = T,x,

Operators where T and T, are non-linear

G. M. Vainikko

30

operators in space E and the subspace E, c E (n = 1,2, . . .) respectively. We shall not require complete continuity of tk operators T and T, in this section; the case X = 0 is not excluded. The Banach space E may be real or complex. Theorem 7.1 Let the following conditions be satisfied: (1) the equation Xx = TX has a solution zoo E E,; (2) T is F&ret-differentiable at the point x,, the linear operator ?,J - T’ (z,,)

being continuously invertible, i.e. h E p (T’ (5,) ) ; (3) the operators T,

are Frkhetdifferentiable when

(n = 1, 2, . . .)

2 E E,, llltr- zooll < 6 (6 = const > O), and 6,, (0 < 6n <. 6)) such that IIT++)

-

for any v > 0, there is an n,.,

T,t’(zn2) 11< q

when n > n,, .zni E J%, II&i - 5mll< 6?j (i = 1,2); (4) if z,, +zoo (5) if z,, +xT,,‘(z,,) : E, --f&x T’(xm): E+E;

(z, E En),

then Tnzn +- TX,;

(zn E En), the sequence of linear continuous operators compactly approximatesthe linear continuous operator

(6) if zn--+xca (zn EE,,),

then h @ or (Tn’(zn) ) when n is large,

Then, not and 6, > 0 exist such that, when n bn,, the equation Ax = T,x has a solution xn and this solution is unique in the sphere 11x- XJ < 6,. The convergence xn * xoo, holds, and

Cl&n < llxn - Pnxooll G 4&l,

(7.4)

where c, and c, = const > 0, En

and P,,: E-E,

(n=

=

I,%...)

IjPnTxc,, -

TnPnxcd,

ace any operators such that Pnxoo + x-

(7.2) as

The compact ~proxim~ion

n

3

principle

31

by.

Proof. Let znO-+ xm, where zno = Pnzm E E,. Using Lemma3.1, we see from (2), (5) and (6) that, for large n, the operators j,J - T,,‘(z,,O) are invertible in E,, the normsof the inverses being bounded in aggregate: 11[hi -

We take some Q (0 < q <

(7.3)

T,‘(zno) ]-*I\ < x = conk 1) and put q = q ,/ 1~.

From condition (3), we

choose n, and 6, > 0 such that, when n 2 n,,

From conditions (4) and (l), we have hzno -

T,,zn”+

?a, -

Ix,

= 0,

(7.5)

so that, when n is large (see (7.2) )

En =

II?i+s”- Tnzn”ll<

60(1- Q) x

(7.6)

*

Let n be so large that (?.3), (7.4) and (7.6) are satisfied sim~~ously. It follows from these inequalities (see 143,p. 729) that the equation AX= T,x has a unique solution X, in the sphere II- - znOllz$ 8~ where

CiEnG II&l - .&loll4

(7.7)

C2ht,

c, and c, being positive constants independentof R (they depend on q, x and 1~’ = sup IIW - T,z’(znO)1). We see from (7.7) and (7.5) that, as n + m, n x,+xa? ~ Hence the solution x, of the equation hx = T,x will be unique in the sphere 11~ - smll < 6, of small radius 6_ > 0 when n 3~~. The bounds (‘7.1) are the same as (7.7). The theoremis proved. Notes. 1. Theorem 7.1. still holds if, instead of (5), we require A-quasicompt approximationof T’(s,) by the sequence T,‘(zn), where zn +z,, zn EE,. 2. Let the conditions of Theorem 7.1 be satisfied, and let E, c

E,,+,

(n =

1,

G. M. Vainikko

32

2 , . . .) .

Then, if the initial n = R, is sufficiently large, and the initial approxima-

tion y,@ is s~ficien~y

close to the solution x,~ of the equation

~CZ = T,,x,

the

iterative scheme Yn+i=

Yn

-w--

will be realizable and

T7a’(yn)l-i(Ry, - T,y,)

y, --f 5,

(n = no, no + i, . . J)

as n -+ m.

This proposition is proved in the same way as Theorem 6 of [4]. Translated by D. E. Brows

1.

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Part I, General theory,