The compact approximation of normally solvable operators

THE COMPACT APPROXIMATION OF

SOLVABLE OPERATORS *

L. S. RAKOVSHCHIK

Leningrad

(Receiued 29 July 1970)

NORMALLY

FOR an equation with a normally solvable operator it is established that coincidence of the dimensions of the subspaces of zeros of the original operator and the operators compactly approximating it, is sufficient for the solutions of the approximate equations to converge to the solution of the original equation.

The following scheme for the approximate solution of the equation

Ax = y (0.1)

with a normally solvable operator A is studied in ill. Equation (0.1) is

replaced by the approximate equation

Anxn = P,Y, (0.2)

where \(A, - AlI + 0, and P, is a projector onto the domain of values of the operator A,,. The solution of equation (0.2), orthogonal to the subspace pf zeros of this operator, is taken as an approximate solution of the original equation. It is explained that an essential condition for the convergence of the approximate solutions to a point is concidence of the like defect numbers of the operators A, and A.

In 12-51 a similar scheme for the solution of equation (0.1) with a reversible operator A was studied on the assumption that the operators A, compactly approximate the operator (this condition is weaker than uniform convergence). In our paper the scheme of compact approximation is used in the case of a normally solvable operator with a non-trivial subspace of zeros. It is shown that here also coincidence of the defect numbers of the operators A, and A ensures the convergence of the approximate solutions to the exact solution.

We say that the operators A, compactly approximate the operator A, if:

*Z/Z. uychisl. Mat. mat. Fiz.,u, 5. 1312-1318, 1971.

277

278 L. S. Rakovshchik

(a) the operators A, converge strongly to the operator A, (b) for any bounded sequence x, the sequence A,x, - Ax, is compact.

In the situation considered below this definition differs little from or is

completely identical with the definitions assumed in [3-51.

In everything which follows the operators A, and A are assumed to be bounded linear operators acting from the Hilbert space X into the Hilbert space

Y.

Notation: R, and R are the domains of values of these operators, Z, and 2 are zero subspaces; A,, and A are the orthogonal complements of the zero subspaces. For conjugate operators all the notation is the same, but they are marked by the index “*“.

1. Normal solvability of the approximating operators

In the case of the uniform approximation of a normally solvable operator the approximating operators An must be normally solvable (at least for large n).

In this section it is shown that this fact is true for compact approximation also, provided the zero subspaces of the approximating and approximated operators have the same finite dimension.

Theorem I

Let the following conditions be satisfied:

(1) The sequence of operators A, compactly approximates the normally solvable operator A.;

(2) dimZ,=dimZ<oo for at least sufficiently large n. Then the

sequence of projectors q, on to the subspaces A,, converges strongly to the projector q on the subspace A.

Proof. Let ei(n), i = i,2, . . . , S, be an orthonormal basis in Z,. By condition (1) the sequence

‘(n) A& - Aein).

is compact for a fixed i. Since A,e,(“) = 0, the sequence Aei(n’ is compact. If A-’ is the right inverse operator (from R into A) for the operator A,

(n) e; = A-’ (Aein)) + z,‘l z’i”’ E z.

Since the operator A-’ is bounded, the sequence zi(n) is bounded, and because

The compact approximation of normally solvable operators 279

of the finite dimensionality of the subspace 2, it is compact. Compactness of the sequence Aeifn) and z.(n) entails the compac~ess of the sequence eifn).

By using the diagonal metho; we can select a sequence of indices nk such that

all the sequences eicnk), i = 1,2, . . . , s converge. Let e{ = lim ei(%), The

equation (ei, ej) = lim (e$W, eink)) implies that the elements ei, i = 1, 2, . . . . s

form an orthonormal system.

We consider the projectors

onto the subspaces 2,. For any fixed x the sequence Q,x is obviously compact.

Moreover, as k --f M

Q (x, eJ ei z Qz. i=l

It is obvious that QX is a projector onto an s-dimensional subspace tight on the

vectors ei.

Now let qsk x = 3: - Qnkx be a projector onto the subspace Ank. Then:

(a) qnk x -, z - Qx,

(bl Ank qnkx = Ank x -+ Ax,

(c) A,, (Ink z = Ank [qnkZ - (z - Qx)l + Ank 2 - Ank Qs -a kc - AQs.

It follows from (b) and (~1 that A&x = 0, that is, Q~ E z. Therefore, Qx is a

projector onto an s-dimensional space contained in Z. Since, by condition (2),

dim Z= s, also, Qx is a projector onto the subspace Z, and qx E x - Qx is a

projector onto the subspace A.

This implies that (for fixed xf the sequence Qnx has a unique limit point: Using the preceding scheme, from any convergent subsequence Q, j~ we can

select a new subsequence converging to Qx. Since the sequence Q,Q, is compact, it is convergent, but in this case the sequence

also converges. Thereby its limit is a projector onto the subspace A. The

theorem is proved.


Theorem 2

Let the following conditions be satisfied :

(1) the operators A,., compactly approximate the normally solvable operator

A; (2) dim& = dim2 < 00 at least for large n.

Then for sufficiently large n:

(a) the operators A, are normally solvable: (b) their right inverse operators A;’ (from R, into A,) are bounded in the

aggregate.

Proof. It is sufficient to show that under the conditions of the theorem

lim ( inf liA,z[I}=6>0. 0.1) - XEA=, Ilxlbl

Indeed, if (1.1) holds,

for sufficiency large fixed rr and any z E A, . This inequality shows that the operator A, maps A, on&o-one and continuously both ways, which implies the closedness of R,, equivalent to the normal solvability of the operator A,. Moreover, for z = A,-‘y, y E R, obtain

which is equivalent to the second statement of the theorem.

To prove (1.1) we assume that it is untrue. In this case a sequence x,,~ E Ank, Ilr,,ll = 1, can be found for which

It follows from the condition of compact approximation that the sequence Ax,~. is compact. By the same reasoning as in Theorem 1 we establish that the sequence xGk is compact also. In order not to complicate the notation we will assume that the sequence xnk converges. Let

We also have

s=lim x “k .


The first term on the right side tends to zero because of the strong convergence

of the operators A, to the operator A, the second tends to the same limit

because of the boundedness of the norms /A,[\ and the convergence of xnk to X. The last term tends to zero by hypothesis. Therefore,Ax = 0 and x E 2.

On the other hand,

2 "k = %k%k = t%,kx,,~ - qnk4 + tQ,kz - PI + 4=.

Here qnk and q are projectors onto the subspaces A,, and A. Since

1 qnk2n k -~~k+a%kII II~,k-~ll~II~,k-~ll--r~

and by Theorem 1,

Pnkx --, q=*

we have z=lim 5

“k =qx~A.

The last relation together with the previously obtained inclusion x E z shows

that x = 0. But by construction

ll4l = lim II%J = 1.

This contradiction verifies the initial supposition and thereby proves the

theorem.

In the particular case where dim 2 = 0, condition (2) follows from the

remaining conditions of the theorem.

In fact, in the opposite case a sequence znk, 11 x,.,~ I= 1, could be found for

which Ankxmk = 0. As before this sequence is compact and any limit point x of

it satisfies the conditions Ax = 0 and 11x I( = 1, which is impossible, since

Z = (01. In this case Theorem 2 assumes the following form (compare [2, 4, 51).

Theorem 2

If the sequence A, compactly approximates the normally solvable operator A and dim Z = 0:

(a) the operators A, are normally solvable for sufficiently large n;


(b) the norms of the inverse operators A,” (from R, into X) are bounded in the aggregate.

We notice that if dim 2 ,4 0, condition (2) of Theorem 2 does not follow from the remaining conditions of the theorem and if it is not satisfied the statements of the theorem may be invalid (an example confirming this is given in [ll, section 1).

2. Convergence of the approximate solutions

E-1 uation (0.1) is solvable only for y E R. Hence we replace it by the equation

Ax = Py, (2.1)

where P is a projector onto R. Equation (2.1) is identical with (0.1) for y E R

and is always solvable. Its solution may be regarded as a generalized solution

of equation (0.1).

Together with (2.1) we consider the approximate equation

where y,, E R,.

As,, = un, (2.2)

Theorem 3

Let

(11 the operators An compactly approximate the normally solvable operator

A: (2) dim&, = dim.2 < 00 for sufficiently large n;

(3) lim y, = Py, yn E R,.

Then: (1) the sequence of solutions xn E An of equations (2.2) converges

to the solution Z* E A of equation (2.1) ; (21 the estimate

ClllAnrn - A.x*II G Ilzr, - z*ll (2.3)

G llqnr* - @II + Cz{llyn - PYII + IlAs’ - An~*ll),

holds (for large nl, where C, = sup llAn-*ll, CZ = inf IIAnll-i.

Proof. We note immediately that the required selection of the elements yn is possible. Indeed, if .z is the solution of equation (2.1), we have R,s y, =

A,z -+ AZ = PY, and condition (3) is satisfied. We also note that all the


conditions of Theorem 2 are satisfied.

NOW let x, and x* be the solutions of equations (2.2) and (2.1) indicated above. We consider the difference

l

xn-qflx ,

where q, is a projector onto An. Since this difference belongs to the subspace A, the following equation is valid;

n:-q>,x= = An-‘[An& - &q&].

By Theorem 2 the operators A,-’ are uniformly bounded. Therefore

II&l - q#II G CzllA,J, - Anq#ll f (2.41

G c~{llA,~, - Az’ll + IIAs’ - Ad’II + II&z’ - Anw’ll}.

Since A *a

X, = yn and Ax* = Py, we have Anxn - Ax* + 0. The second term on the right si e of the inequality (2.4) tends to zero because of the strong convergence

of the operators A, to the operator A. The last term equals zero, since the difference 5’ - q,l* E z,. Consequently,

lhn - qnz*ll < Cz{llyn -&/II + ItAs” - Anx*lI) -to.

By Theorem 2 the projectors q, converge strongly to the projector q. Hence

11% - XYI = IIGI - qz’ll G bn - qn5’ll + llqn5* - qz’ll + 0.

The convergence of the approximations is proved. The right side of the

inequality (2.3) has also been established in passing. Its left side follows from the obvious inequalities

II&G --Anz*II < IIAnlI 11~ - s’ll G {sup llAn~~}~~z,, - Z*II.

The theorem is proved.

In the case of uniform convergence it was shown in [ll that as the elements

y, we may take elements P,y, where the P, are projectors onto the subspace R,. In the case of the compact approximation we are considering here, this is not generally true. Indeed, let ei, i = 1, 2, . . . . be an orthonormal basis and AZ =z - (I, ei)el; A,z = Az -j- (5, en)ei. It is obvious that A,x + Ax and that for any bounded sequence x, the set {A,z~-Az,J = {(z,,, e,)eJ is compact. Also, 2 = Z, le,). The projectors onto the subspaces R and R, have the form

PY = Al - (Y, dei,

P*Y = y --(y,el)'l-(y,e.)e,+'y~CL2+ en) (ei+e2).

It is easy to show that IIP,Y - Pyll = ( (u, ei) I* / 2 # 0, provided that y e R.

284 L. S. Rakoushchik

In this case the elements yn = P,y do not satisfy the conditions of Theorem 3.

We note that there is here no convergence of the solutions of equations (2.2)

from A,

xn = c

(v, ei)ei + (v, ei + enI

en 2

f#.l.n

to the solution of equation (2.1) from

5= E

(Il, ei) Ci

wi

We note several cases in which (if the conditions of Theorem 3 are satisfied)

the choice y, = P,y is permissible.

Casel. R,r,R and y E R. Here Pny = Py.

Case 2. The operators A,* compactly approximate the operator A* and dim 2,’ = dim 2’ < 00.

Theorem 1 shows that the projectors onto the subspaces A,’ = Y 0 2,’

converge to the projector onto the subspace A’ = Y 8 Z*. It remains to note that

R, = A,+ and R = A*.

Case 3. dim 2,’ = dim Z* < 00 and for any bounded sequence A,x, a sequence z, can be found such that

weakly

‘&ax, - AZ, + 0.

The sufficiency of condition (3) is less obvious and is implied by the

following lemma.

Lemma

If condition 3 is satisfied the sequences of projectors rn and P, = I- r,, onto the subspaces 2 * and H, = Y e Z,*

onto the subspace .Z*nand R = Y 9 2.

respectively, converge to projectors

Proof. We first show that the sequences of operators rn and P, are weakly compact. This statement follows from Theorem 1, section 34 of [61. but we

here derive a proof adapted to the situation considered, so as to obtain equation

(2.5) which we require later and which does not follow from the general theorem cited.


Under the conditions of the lemma

k

rnx = c (5, fi'",fi'f

c-1

where f<(n), i = 1. 2, . . . , k, k = dim Z* c 00, is an orthononnal basis in Z,*. For any fixed i the sequence f:’ is weakly compact. Hence (see Theorem 11 we can select a sequence of indices nl for which all the sequences

i = 1, 2, . . . , k.

weakly converge weakly. If jp’ + fi, it is obvious that for any x

n;-!5 r 2 (Xv fi) fjGrX. (2.5) i=l

Since the construction described is applicable to any subsequence of the sequence rn, the weak compactness of the latter is established. This also implies the weak compactness of the sequence P,.

We denote by TX the operator on the right side of (2.5), and consequently we write Px 3 x - rx.

Since P,x E IL, elements xn can be found for which A&, = P,z. By condition 3 we can find values of z, such that

weakly 4xn - AZ, -+ o.

It is obvious that

Since the domain of values of the operator A is closed and convex, it is weakly closed, and consequently, PZ E R. The equation

x = r+ + P+

shows that 5 = rz + Ps. (2.6)

Let x h 0 be orthogonal to R. Since pz E R, we obtain from this /X11* = (rz, 4 G Ilrzll 11~11. Hence MI =z II~ZII. On the other hand, since rnFE!$.,

Ilrzll <lim IIrn141 < 11211.

Consequently, for elements x: orthogonal to R,

Ilr4l = 1141.

286 L. S. Rakoushchik

In the case considered x is orthogonal to PX E R. Consequently,

lld~ = 115 - PsllZ = llsll* + llPzll2

and necessarily PX = 0.

Therefore, if x is orthogonal to R, we have Pr = 0 and rx = x, that is,

Z+= YBRcrY.

Since dimZ” 2 dim rY (see (2.5)), we have Z* = rY. Consequently, for any

y the element ry E Z* and by the preceding,

drYI = ry, i.e. rz = r.

The selfconjugacy of the operator r is obvious from (2.5). Hence (see section 36 of [61) the operator r is a projector onto Z*, and P = I-r is a projector onto K = Y 0 Z*. It follows from this reasoning that any weakly convergent sub-

sequence of the sequence r,,x converges weakly to the projection rx of the element x onto the subspace Z*. This means that a weakly compact sequence

rnx has a unique limit point rx, and consequently, converges weakly to it. By Theorem 2, section 38 of [61, the sequence of projectors rR converges strongly to the projector r. Similarly, the sequence Pn converges strongly to the projector P. The lemma is proved. The sufficiency of condition 3 for the convergence of the approximate solutions follows directly from this lemma.

Translated by J. Berry

REFERENCES

1. RAKOVSHCHIK, L. S. Approximate solution of equations with normally resolvable operators. Zh. u~chisl. Mat. mat. Fiz., 6, 1, 3-11, 1966.

2. ANSELONE, P. M. and MOORE, R. H. Approximate solutions of integral and operational equations. J. Math. Anal. and Appl., 9, 2, 268-277, 1964.

3. ATKINSON, K. E. The numerical solution of the eigenvalue problem for compact integral equations. Trans. Amer. Math. Sot., 129, 3, 458.1967.

4. VAINIKKO, G. M. Compact approximation of linear completely continuous operators in factor spaces. Tr. Tartuskogo un-ta, 220, 190-204, 1968.

5. VAINIKKO, G. M. The principle of compact approximation in the theory of approximate methods. Zh. v@zhisl. Mat. mat. Fiz., 9, 4. 739-761. 1969.

6. AKHIEZER, N. I. and GLAZMAN, I. M. Theory of Linear Operators in Hilbert Space (Teoriya lineinykh operatorov v gil’bertovom prostranstve), Fizmatgiz, Moscow, 1966.

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The compact approximation of normally solvable operators