The factorization of operators in L 2(a, b)

10.4.

THE FACTORIZATION

OF OPERATORS IN n2(a, b)~

I. The operators S-, S+, bounded in LZ(a, b) (--~ ~ a < b < ~), are said to be lower and upper triangular operators, respectively, if for all ~ (a ~ ~ ~ b) we have

where P~f = X[a,~]f. We shall say that an operator S, bounded in the space L2(a, b), admits a left factorization if S = S-S+, where S- and S+ are lower and upper triangular operators, bounded together with their inverses. Gokhberg and Krein tion that

[I] have investigated

the problem of factorization

under the assump-

S - I In this case, the factorizing

operators have been sought in the form

S_

+X_,

The factorization methods have played an important role in a series of spectral theory problems. By getting rid of condition (I) and switching to the general form of the triangular operators, one can extend in an essential manner the domain of the applicability of the factorization method. Example.

We consider

[2] the operator

= It is easy to see that the operator

~&t,

-~4

S B (B ~ 0) does not satisfy condition

admits the factorization S B = w~w~, where ~ = is defined by the formula

~ ,

(I).

However,

S~

while the lower triangular operator w a

S In order that the operator S should admit a left factorization following condition be satisfied: (*) For any ~ (a ~ ~ ~ b), the operator

S~ = P~SP~ is invertible

Problem I. Find those classes of operators for the existence of a left factorization of S. In general

this condition

it is necessary

is not sufficient.

that the

in L2(a, ~).

S for which the condition

(*) is sufficient

The operator

(2) O

possesses property (*) but does not admit a factorization ially satisfied if we have (**) The operator An important

~L. A. SAKHNOVICH. 270000, USSR.

S is bounded

[2].

The condition

(*) is triv-

together with its inverse and it is positive.

special case of Problem ] is the following problem.

A. S. Popov Odessa Electrotechnical

Communication

Institute,

Odessa,

2157

Problem 2.

Does every operator,

If S -- I ~

ym, then the answer here is in the affirmative

2.

satisfying the condition

It is interesting to investigate Problems

(**), admit a factorization? [I].

I-2 for operators with a difference kernel

0

Setting M(x) = s(x), N(x) = --s(--x), 0 ~< x ~< ~, we write

[3]

~O

(AoS-S~o)~ :~I ~(t)[M(m)* N(t)]&t, where ~'o~ = ~ I ~ (t)~

If the condition

(*) holds,

(3)

then the following second-order matrix

functions make sense:

[ q' w(g,z):

I-

~"

p,CAo-,# M, ,

_,

.,

-,

5(A) = (5~.,P~M,I)(5~P~_,M,N)] (%p~,~) (s~P~,N) tion.

THEOREM I. Assume that the operator S with a difference kernel admits a left factorizaThen the matrices w(~, z) and B(~) are absolutely continuous relative to ~ and

~-~T:TWCg,z)HcD, HcD =~(~),

(4)

where the elements hij(~) of the matrix H(~) have the form

~q (g) = R~CD 0 i(D,

(5)

and we also have

The functions Ri, ~i can be expressed in terms of the factorizing operators ,-4

-4

*-4--

S-, S+:

-I

o,(~)=s,I, n,(m=s+ N, R,(~)=S_M, %(m) =S_ 4.

(7)

Formulas (5)-(7) associate to each factorizable operator S with a difference kernel some system of differential equations (4). In the theory of the inverse problems, such a procedure has been developed by Krein [4] for the case when the operator S is positive and S -I E E2~ In addition, Theorem I means that the transfer matrix-function [5] w(~, z) admits a multiplicative representation

@

If the o p e r a t o r S i s p o s i t i v e , t h e n i t f o l l o w s from (3) t h a t w(m, z) i s t he c h a r a c t e r i s t i c m a t r i x - f u n c t i o n of t h e o p e r a t o r S-1/zAoS 1/2 Then t h e r e l a t i o n s ( 4 ) , (8) c o i n c i d e w ith t h e known r e s u l t s o f [6, 7] . The e q u a l i t y

is new also in this case. It follows directly from Theorem I that for the possibility of the factorization of an operator with a difference kernel the following conditions are necessary: 2158

(***) The operator S with a difference kernel satisfies condition (*), B(~) is an absolutely continuous matrix-function, and relation (9) holds. We note that in the example (2) all the requirements of the condition (***) hold, except relation (9). Problem 3.

Does every operator, satisfying the condition (***), admit a factorization?

THEOREM 2. If the operator S satisfies the conditions (**) and (***), then it admits a factorization. LITERATURE CITED I 9

2. 3. 4. 5. 6. 7.

I. Ts. Gokhberg (I. C. Gohberg) and M. G. Krein, Theory and Applications of Volterra Operators in Hilbert Space, Amer. Math. Soc., Providence (1970). L. A. Sakhnovich, "The factorization of operators in L2(a, b)," Funkts. Anal. Prilozhen., 13, No. 3, 40-45 (1979). L. A. Sakhnovich, "On an integral equation with a kernel dependent on the difference of the arguments," Mat. Issled., 8, No. 2, 138-146 (1973). M. G. Krein, "The continual analogies of the conjectures on polynomials orthogonal on the unit circumference," Dokl. Akad. Nauk SSSR, 105, No. 4, 637-640 (1955). L. A. Sakhnovich, "On the factorization of an operator-valued transmission function," Dokl. Akad. Nauk SSSR, 226, No. 4, 781-784 (1976). M. S. Livshits, Operators, Oscillations, Waves. Open Systems, Nauka, Moscow (1966). V. P. Potapov, "The multiplicative structure of J-contractive matrix-functions," Tr. Mosk. Mat. Obshch., 4, 125-136 (1955).

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