SELF-ADJOINT POLYNOMIAL OPERATOR PENCILS, SPECTRALLY EQUIVALE~

TO SELF-ADJOINT OPERATORS

B. M. Podlevskii UDC 5~7.984

The solving of various problems of mathematical physics by the Fourier method leads naturally to appropriate polynomial operator pencils of the form

L(~) = ~Ao + ~-~A~+. . . + ~A~-~ + A~ (1)

with operator coefficients A~ = A~, i = 0, I ..... n, A~IE~(H), H being a Hilbert space.

One of the possible approaches to the spectral investigation of such pencils consists in their linearization. The latter is especially effective in those cases when the equiva- lent linear problem can be symmetrized by some self-adjoint operator, possessing the definite- ness property. This takes place in the case of the so-called strongly Hilbert operator pen- cils [I, 2].

In the space H of the direct orthogonal sum of n copies of the initial space H ((y.z) =

k ) ((y~y~ . . . . . y~)~,(zt, z~ . . . . . z~/) = ~_~(y~,%) , to the penc i l (~) one a s s o c i a t e s the l i n e a r penc i l

(2) where L (~) = ~ -- ~'

L : L y = z , z~ = Y k + l , It = 1, n - - 1, z n = - - ~ A ~ l A , ~ + t _ ~ y i , (3)

being the i d e n t i t y ope ra to r in ft.

LEMMA 1. The spec t r a of penc i l s (1) and (2) co inc ide .

This assertion follows from the relation S0(XI -- L) = B(X) diag (L(X), Q)C(X), where

So: Soy = i = E k = t, =

zk = - - ~ A . - k - ~ y t + 2 , k = 2, n: [~ (~.) �9 [~ (~) y = z, zk = Yk - - ~Yk+r,

k = 2 , n

A more detailed assertion regarding the relations between the spectra of pencils (I) and (2) can be found in [3].

The operator L is not self-adjoint. However, any of the operators of the form

where p is a real polynomial, is a symmetrizer of L, i.e., (SL)* = SL. If, however, the poly- nomial p(%) = X n + cz1% n-I +... + an with zeros ~i > ~2 >... > ~n is such that the conditions

( - - l)' (L(~Oy, y ) > O Vy(~0)EH, i = l,--n, (4)

Institute for Applied Problems of Mechanics and Mathematics, Academy of Sciences of the Ukrainian SSR, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 36, No. 5, pp. 660-662, September-October, 1984. Original article submitted March 15, 1983; revision sub- mitted February 17, ~984.

498 0041-5995/85/3605-0498508.50 �9 1985 Plenum Publishing Corporation

o r

( - - 1)' (L (a~) y, y) ~ 6, (y, y), 6, > 0 V Y (~= 0) E H, i = l,---n, (5)

are satisfied, then the correspondin$ conditions will be satisfied by the operator S [I]: (S~, ~) > 0 (strictly positive) or (S~, ~) > y(~, ~), u > 0 (uniformly positive).

It should be emphasized that, as a rule, the verification of conditions (4) or (5) for concrete operator pencils is connected with considerable difficulties. Consequently, it is not easy to construct a positive-definite symmetrizer for operator L.

In the present paper we single out a class of pencils for which one can construct effec- tively a symmetrizer of the operator L, possessing the property of strict or uniform positive-definiteness.

L e t g (y, ~) = ~2n-'c o (y) + %2n-2c~ (y) + ... --[- ~C~n-2 (y) + C2n-~(y), i w h e r e c~ (y) : ~ a~ (~) az (y) (n --- l), ~ : ~+l=1

0, 2 n - - 1; k = 0, n; ~ = 0, n - - 1, a j ( y ) = ( A j y , y ) , j = 0 , n .

D e f i n i t i o n . P e n c i l L(X) i s s a i d t o be c o m p l e t e l y h y b e r b o l i c i f t h e c o n d i t i o n

Bez (g (y, L), g' (y, ~)) > 0 is satisfied.

2n--2 I n f o r m u l a ( 6 ) , B e z ( g , g ' ) - - I l b a i ia=o i s t h e B e z o u t m a t r i x ( B e z o u t i a n ) o f two p o l y n o m i a l s , whose e l e m e n t s a r e d e f i n e d f rom t h e e q u a t i o n [4 ]

~a b~a~"~' = (g(Y, L )g ' (be, ~) - - g (y, ~) g' (y, ~,))/(L --F~). ~,]=0

(6)

As it is known, inequalities (6) are the necessary and sufficient conditions in order that all the zeros of each pair of polynomials g(y, X) and g'(y, X) be simple, real and that they should alternate. We denote them by Pk(Y), k = I, 2,...,2n-- I. Since pk(~y) = Pk(Y), y ~ O, i z 0, one can consider the functionals Pk on the unit sphere K of the space H. Ob- viously, Pk(Y) are bounded continuous functionals on K and, therefore, the sets Wk of their values (the spectral zones) are nonempty, connected, bounded subsets of the real line. We arrange these zones in decreasing order. Then the following property holds.

LEMMA 2~ ai N ~j=~ for i ~ j.

Taking into account these properties of the domains ~i and the structure of the poly- nomials g(y, X), we obtain the following assertion.

LEMMA 3. The spectral zones of pencils L(X) and L'(k) do not intersect.

From here it follows that for a completely hyperbolic pencil there exist numbers a i such that conditions (4) or (5) hold and for a i one can take the zeros of the polynomial l ~ (~, y)~-! (L' (~) y, y), y C H.

Thus, we obtain the following result.

THEOREM. If the polynomial operator pencil L(X) is completely hyperbolic, then the sy~a- metrizer S = S0~'(L, y) of its accompanying operator L will be strictly positive-definite for an arbitrary y E H �9 This symmetrizer will be uniformly positive if the spectral zones of the pencil L'(X) do not degenerate into points or if at the degeneration of some zone Wk into a point one has sup ~k+,#inf~k

Remark. The verification of the complete hyperbolicity of a pencil is essentially sim- pler if for the determination of the Bezoutian one makes use of the following representation:

Rez (g (~), g' (M) = S~e' (@), ( 7 ) where

2n--k--[

S g : - S g y = z . z h = ~ c2n_k- , - ,y i+, , k = l , 2 n - - 1 ; G:Gq---=z, i ~ O

'in--|

zh = yk+~, k = 1, 2 n - - 2, z2,,-, = - - j ~ (c~,,-dco)y~.

499

The elements of the matrices occurring in formula (7) are very easily determined taking into account the relations ~g~k = diag (Gk, Gzn-k-l), where

m

G : G ~ = ~, ~ = - E ~ - , - ~ - i y ~ - ~ + ~ _ . m = 1, 2 . . . . . k. k > o; ; ~ I

2~--|--m

G , , - , , - , : G , - , , - , ~ = ~, z~, = E ~ , , - , - , , , -~y~+,~+, , m = k + I , k + 2 . . . . . 2 , , - j , k < 2 ~ , - 1. ~ 0

LITERATURE CITED

I. A.I. Balinskii, "Certain methods of investigation of generalized problems regarding eigenvalues," Author's Abstract of Doctoral Dissertation, Lvov (1972).

2. A.S. Markus, V. I. Matsaev, and G. I. Russu, "On certain generalizations of the theory of strongly damped pencils to the case of pencils of arbitrary order," Acta Sci. Math. (Szeged), 34, 245-271 (1973).

3. A.I. Balinskii and B. M. Podlevskii, "The variational characterization of the eigen- values of certain polynomial pencils of differential operators," in: Mathematical Meth- ods and Physical-Mechanical Fields [in Russian], Naukova Dumka, Kiev (1983), pp. 17-21.

4. M.G. Krein and M. Yu. Neimark, The Method of Symmetric and Hermitian Forms in the Theory of the Separation of the Roots of Algebraic Equations [in Russian], Gosnauchtekhizdat Ukrainy, Kharkov (1936).

IMBEDDING OF KERNELS BY REGULAR TRANSFORMATIONS

A. V. Revenko UDC 517.521.8

Let S and T be sets of real numbers with finite or infinite limit points ~ and B, re- spectively.

By F(S) (F(T)) we denote some linear space of complex-valued functions defined on S(T), and we shall assume that F(S) contains all the constant functions on S. The linear opera- tions in F(S) (F(T)) are defined in the usual manner. Also in a natural manner one can de- fine the operation of the multiplication of functions.

Everywhere below we shall consider linear operators A:F(S) + F(T) such that A(x) is a bounded function on T if x is bounded on S.

The operator A is said to be regular if for each x 6 F (S) such that lim x(s) exists, there exists also lim (A(x))(t) and both limits are equal. ~

Examples of such regular operators are the operators of the form

(A(x))(t)= lira ~ x(s)d~t, t6T, (1) S\U,

where ~t are measures of the bounded variation v(~, S), defined on the algebra F of the sub- sets of the set S; Ur6 P is a neighborhood of the point =, contracting to e as r § ~ or r + 0 (depending on whether e is a finite or infinite point); x is a function bounded on S~Ur vr ands__measurable with respect to any measure ~t [I]; v(~t,S)<H , t6T, v(~,,S\Ur)-~0, t-+~, V r,

Jd~t-+l, t--~. Operators similar to those of the form (I) have been considered in [2]. S

Particular cases of operators of form (I) are the integral transforms, matrix operators, and semicontinuous matrices.

We shall consider that the functions xl, xz6F(S ) are identical if lim(xl(s)mx~(s))=O.

Similarly for 9,, yz6F(T). Then by the norm of the bounded function x6F(S ) (y6F(T)) we mean

Kiev Pedagogical Institute. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 36, No. 5, pp. 662-666, September-October, 1984. Original article submitted March 11, 1983; revision submitted February 6, 1984.

500 0041-5995/85/3605-0500508.50 �9 1985 Plenum Publishing Corporation