On Some Properties of Systems of Linear Inequalities for Operators

Journal of Science and Technology

3 (2), 2009, 161 - 164

©BEYKENT UNIVERSITY

On Some Properties of Systems of Linear

Inequalities for Operators

Afgan ASLANOV

Department of Mathematics and Computing Beykent University, Istanbul, Turkey

afganaslanov@beykent.edu. tr

Accepted: 11.11.2009

Abstract

In this paper, the existence of the solution of linear system of (operator) inequalities is considered using the theory of tensor products of linear spaces, linear operators and some properties of tensor determinantal operators.

Keywords: tensor determinants; systems of equations; multi-parameter

spectral theory

Özet

Operatörlerin Lineer Eşitsizlik Sistemlerinin Bazı Özellikleri

Bu makalede, lineer uzayların, lineer operatörlerin tensör çarpımından ve tensör determinantlarının bazı özelliklerinden yararlanarak operatörlerin lineer eşitsizlik sistemlerinin çözümünün varlığı ele alındı.

Anahtar Kelimeler: tensör determinantları; denklem sistemleri; çok

parametreli spectral teori

INTRODUCTION

The tensor determinants are important for the solutions of problems related with multi-parameter spectral problems (MPSP). The system of linear equations

b 1 + b 21+ . . . + K K = a, i = U . . . n (1)

has a unique solution if det(bi k) ^ 0, where bin, ai are fixed numbers. In this paper, we try to find some analogue of this fact in the case, when instead of bin, ai we have operators in finite dimensional spaces, then we show some application to the MPSP.

Let Bin, Ai be self-adjoint linear operators (that is the corresponding matrix is equal to its own conjugate transpose if they act in Cn) in finite dimensional inner-product space Hi, Bin, A î L(Hi ) . The spectrum (or eigenvalues)

of MPSP is the set of

(Àl,À2,...,Àn )î Cn

such that

( A ~lB

n

-A

2

B

i 2

- . . . - 1

n

B

m

)Xi = 0, (2)

for some

0 ^ xi î Hi, i = 1,2,...,n.

The tensor vector

x1 ®... ® xn î H1 ®... ® Hn is called an eigenfunction of the problem (2). For the definition and properties of tensor products of spaces, operators and for the multiparameter eigenvalue problems, see [1,2].

We denote by

A:

B1 1 - B1 n

B ... B„

® det(B

ik

), i, k = 1,2,..., n,

for example if n=2, we have A = Bu ® B22 — Bu ® B21. Main Result

The next theorem is an interesting improvement of the theorem on the solvability of the system of linear equations.

Theorem 1. If

dim Hi < i = 1,2,...,

n, and

A >

0, then for any set of self-adjoint operators Ai e L(Hi) and for any choice of

£i = ± 1 , i = 1 , 2 , . . . , n , there exist a unique ( 11, 12, . . . , 1n) e Rn, such that

e ( A — 1B

a

—lB

r

2 —... — 1

n

B

m

) > 0, and (3)

Afgan ASLANOV

that A — IB > 0 and Ker ( A — IB) ^ {0} . Let the statement of the theorem be true for k = n — 1. We shall show that it holds for k = n.

Since A » 0 , we can replace Bjk by their linear combinations such that these new operators satisfy the condition

... B,„

22 2n

B

2₂

... B

» 0.

(5)

Indeed, for a fixed 0 ^ x e H we obtain

( B1 1 X i , X1 ) . . . ( B 1 nx i )

B,

B.

2n

B

1₁

... B

» 0.

At least one of

(B

1k

x

1

, x

1

) ^ 0,

say

(B

11

x

1

, x

1

) ^ 0.

To eliminate the elements

(B

1k

x

1

,x

1

), k ^ 1,

add

—(B

1k

x

1

, x

1

)/(B

11

x

1

, x

1

)

times column1 to column k: ( B1 1 xı, x1) 0

B.

₂₁

B

( B1 2 X1, X1) ( B1 1 ^ X1)

B

B

B

( B1 2X1, X ) B ( B1 1 X1, X1)

Thus the tensor determinant

B

( B1 n X1, X1) ( B1 1 X1, X1)

B

B _ ( B1 nX1, X1) B nn f T-i \ n ( B1 1X1, X1)

» 0.

0

( Bl nXı, Xi) B _ ( Bi 2Xi , xi) B B b22 / r > x B2 1

•••

B2n ( B1 1 X i , xi) (

B

n

Xi,

x

i

) -Br,

B.

( Bi 2 ^ Xi) ( Bi iX1, Xi)

B

••• B

( B1 n X1, X1) ( B1 1 X1)

B

is strongly positive or negative, say positive. Taking

(B

x x

)

B'k = Bjk —

T7T~I 1xL)Bj1,

j = 1,2,...,n, k = 1,2,...,n

we obtain j j ( B1 1 ^ x1)

D =

B1 1 B'i2 ••• B1n Bn1 B'n2 ••• B L

» 0

and

® det(B'

k

) » 0, j , k = 2,..., n.

That is without changing the notations for Bjk we suppose that (5) holds.

Now by the induction, for any fixed \ e R, there exists a unique ( 1 ( 1 ),..., 1n ( 1 ) ) e R"~1 such that

Ai

— I B

— 1 B I 2

—... —

1„Bi„

> 0, i = 2,3,... and

(A — l Bi 1 — 12Bi2 —...— kBnn)X = 0 fo r some 0 * Xi e H, i = ., n. It follows from f A2 11B21 - 12B2 2 " •• -1nB2n B2 3 • •• B2 , A

(X2 &••• ÄXn),(X2 &••• ÄXn)

An - \ B m -12Bn2 ~ ••• - 1nBn n Bn 3 " Bn ( ( A2 - 11B2 1 - 12B2 2 - ••• -1nB2n ) X2 , X2 ) ( B2 3X2X2 ) ••• ( B2 nX2 , X2 )

0

Afgan ASLANOV A2 B2 3 " B2 n

A B

_{n n3 n} 3...

B

(X2

® ... ®

Xn ),(X2

® ... ®

Xn

)

1

B B

. . .

B

B2 1 B2 3 . . . B2 n - ^ 2

B B

_{n1 n3 nn} . . .

B

B B

. . .

B

B2 2 B2 3 . . . B2 n

B B ... B

_{n 2 n3 n} (X2

® ... ®

Xn ),(X2

® ... ®

Xn

)

(X2

® ... ®

Xn ),(X2

® ... ®

Xn

) = 0

or ( (G2 - 11D1 2 + 4 A l ) X' X) = °

where A1k is the algebraic cofactor of B1k in the determinant A, Tk is the algebraic cofactor of B1k in the determinant

r =

A1 B1 2 . . . B 1n

A B

,...₂

B

and X = X ® . . . ® X .. In like manner we have ₂

((G —

I1A13 +I3A11)

X, X ) = 0,....

((G—1A„ + i A

n

) X, X ) = 0,

( G2 X X ) + 3 ( D1 2 X X ) i2( 1 )

=

^ r c ' + 1 ^ ^

(D

11

X, X) (D

11

X, X)

3 ( 3 )

= - G „ X X ) +

3 ( Aı nX , X ) 1 ( 1 )

/A

+ 1

/A ~ •

(A

11

X, X) (A

11

X, X)

Consider the expression

((Ai - I 1 B 1 1

-1(1)B12 - •••

- 1 ( 1 ) B i n

)

Xi,

X )

= ((

A -1

1

B

11) X

,

X ) ^2

(1

1

)

( ^ X )-•••

- 1

n

( 1 )

(

B

1n

X

1

,

X) =((

A

1

- 1 B

11) X

,

X )

1 (

Bi2 Xi, Xi

) ( A ^ ) + (

Bi2 Xi, Xi

)

^

^

(A

11

X, X) (A

11

X, X)

( A1 n X X )

,

X

„

^ ( Gn X X ) X1, 1 ( B i n X i , Xi + ( B i n X i , Xi )

(A

11

X, X) (A

11

X, X)

1 r( A1X1, X1 ) ( A1 1 X X ) + (B1 2X1, X1 ) ( G2 X , X ) + ••• + (B1 nX1, X1 ) ( Gn X , X ) ( A1 1 X, X )

1

( A1 1 X X) ( Bi X X1 ) ( A1 1 X , X ) + (B1 2 X1, X1 ) ( A1 2 X , X ) + ••• + ( ^ n ^ X1 ) ( A1 nX , X )

= (

[ G —

^1A]( X1 ®

X

)

,

( X1 ®

X

)).

Since A > 0 , there exists a unique 1 e R such that r — 1 A > 0 and Ker ( r — 1 A ) ^ {0}, which means that

Afgan ASLANOV

Thus we have that the statement of the theorem holds for all n.

Now let us show that this theorem is the generalization of the main theorem of linear algebra on the existence and uniqueness of solution of the system (1) when

det(b

kk

) > 0

. If

H

1

=

H2

= ... = H

n

= R

the condition (3) together with (4) implies

(ai — b

i1

1

1

— bi21 —... — b

in

l

n

) Xt = 0

for some

X e

R,

that is for all

X e

R, in fact. Therefore the condition

det(b

;k

) > 0

implies that the system of equations (1) has a unique solution. If

det(b

;k

) <

0 ,

multiplying one of rows by -1 again we have that the system (1) has a solution. Let us note that the Theorem 1 can be applied to solve some problems of MPST and make some new observations. Atkinson [2] established that if the tensor determinant A is positive then for any choice of

e = ± 1 , i = 1,2,..., n there exists ( ( 1 , 12, . . . , ln ) e Rn such that

e ( 1 B + 1 Bi 2 +... + 1

n

B

m

) » 0, i = 1,2,..., n .

Now using the Theorem 1 we can prove the following

Theorem 2. If

dim Hi < ¥, i = 1,2,...,

n, and

A >

0, then for any choice

of e =±1, i = 1,2,...,n and fixed i0 e{1,2,...,n} there exists

( 1 , 1 , . . . , 1 ) e Rn such that

1) e

(1Bi1 +12Bi 2

+... + 1

n

B

in

)> 0, i = 1,2,..., n and

2) e ( M . 1 + 1 B ,

2

+... + 1

n

B

i o n

)>0

3) Ker (1B

n

+ 1 $

a

+... + 1nBin ) * {0} for i * v

Proof. For simplicity we consider the case i0 = 1 a n d £i = 1, i = 1,2,..., n

As in case of the proof of Theorem we may suppose that (5) holds and then using the fact that there exists

(1

1

,...,1

n

)e Rn

1 with

Bi1 —12B

t

2 — 3 —... — 1

n

B

in

> 0, and

Ker(Bn —12Bt2 — M3 —... — 1nBm) * {0}, i = 2,3,...

we consider the determinant

D =

B1 1 - 12B1 2 - 13B1 3 " ••• - 1nB1 n ••• B1 n

Bn1 - 1 Bn2 - 13Bn 3 - ••• - 1nBn n ••• Bn

It is not difficult to see that B1 1

-1

2

B

i 2

- 1

3

B

i3

- ••• - 1

n

B

in » Indeed, if

((B11 - I 2 B 1 2 - I 3 B13 - ••• - 1 n B m ) X , X ) £ 0,

for some

x

1

e H

1

.

taking

X e Ker (Bi - ^Bi 2 - • • •- ¿nB,n), i = 2 , 3 ,• • •, n we ha v e

(A(x ® x

2

®...®X

n

),(x ® x

2

®...®x„))

= ((

Bn - l B12 - l B13 - . . . - l R n ) Xl, X ) (A„ ( X2 ® ... ® X„ ) , ( X2 ® ... ® X„ )) +((B21-I2B22 -I3B23 -... - l B n ) X2, X) ( A i (Xl ®X ®...®X„),(Xl ®X, ®...®X))+.... < 0, where Ai k is the algebraic cofactor of Bk in the determinant A , which

contradicts the positivity of the A .

REFERENCES

[1] Atkinson, FV. Multiparameter spectral theory. Bull. Amer. Math. Soc., 74: 1-27, 1968

[2] Atkinson, FV. Multiparameter eiganvalue problems, v.l, matrices and compact operators, Acad Press N-Y, 1972.