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Criteria for the Existence of Common Points of Spectra

of Several Operator Pencils

R.M. Dzhabarzadeh

Institute of Mathematics and Mechanics, National Academy of Science of Azerbaijan, Azerbaijan

Copyright © 2015 Horizon Research Publishing All rights reserved.

Abstract

In this paper we present two criteria for the existence of common eigen values of several operator pencils having discrete spectrum. One of the given criteria is proved by using analogs of resultant for several operator pencils; proof of the other criterion requires the use of the results of the multiparameter spectral theory. In both cases the number of operator pencils is finite, operator pencils act, generally speaking, in different Hilbert spaces.

Keywords

Operator Pencils, Resultant of Two Operator Pencils, Hilbert Space, Common Eigenvalue, Kernel of Operator

1. Introduction

The definition of abstract analogue of resultant of two operator pencils in one parameter is given in [2] and [10] with the help of definitions of tensor product of spaces and tensor product of operators. When the operator pencils have the identical degree concerning parameter concept of resultant has been given in work of Khayniq [9], for operator pencils, generally speaking, with the different orders of parameter , the abstract analog of a resultant is studied by Balinskii [2].

Let be

2

0 1 2

2

0 1 2

( )

...

,

( )

...

n n m

m

A

A

A

A

A

B

B

B

B

B

λ

λ

λ

λ

λ

λ

λ

λ

=

+

+

+ +

=

+

+

+ +

(1)

two operator pencils depending on the same parameters

λ

and acting, generally speaking, in various Hilbert spaces

2 1

,

H

H

, correspondingly.

))

(

),

(

(

Re

s

A

λ

B

λ

is the resultant of two operator pencils

( )

A

λ

and

B

( )

λ

. It is presented by the matrices (2) and

acts in the space

(

H

1

H

2

)

n m+ - direct sum of

n m

+

copies of tensor product of spaceH1⊗H2.

       

 

       

 

⊗ ⊗

⊗ ⊗ ⊗

⊗ ⊗

⊗ ⊗ ⊗

=

m m

n n

B E B E B E

B E B

E B E

E A E A E A

E A E

A E A

B A s

1 1 1 0 1

1 1

1 0 1

2 2

1 2 0

2 2

1 2 0

.... ...

. .

. ... . ...

. .

0 .. ...

... ...

0 0

. ... . ...

. .

0 ... ...

)) ( ), ( ( Re λ λ

(2) In a matrixRe ( ( ), ( ))s A1 λ A2 λ the number of rows with

operators

A

iis equal to leading degree of parameter

λ

in

pencilB( )λ , that is

m

, the number of rows in matrix 1 2

Re ( ( ), ( ))s A λ A λ with operators

B

i coincides with the leading degree of parameter

λ

in pencilA( )

λ

, that is

n

.

In [2], [10] operator Res(A(λ,)B(λ)) is named by abstract analog of a resultant for operator pencils (1).Value of a resultant Res(A(λ,)B(λ)) is equal to its formal decomposition wherein each term of decomposition is tensor product of operators.

Theorem 1. ([2],[10]). Let the following conditions satisfy: a) Operators

A

iand

B

i are bounded and one of them has bounded inverse.

b) Spectra of operator pencils

A

( )

λ

and

B

( )

λ

are discrete set.

Then the pencils (1) have a common eigenvalue if and only if

Ker

Re

s

(

A

(

λ

,)

B

(

λ

))

{ }

θ

.

In the case

m

=

n

in (1) Theorem1 is proved in [10], at arbitrary whole meanings

m

,

n

Theorem1 is proved in [2].

Operator (2) is the generalization of Resultant

Re ( , )s f g =

1 1 0

2 1 0

1 0

1 3 2 1 0

4 3 2 1 0

1 1 0

... 0 ... ... 0 0

0 ... ... ... 0 0

. . ... . . . ... ... . .

. . ... . . . ... ... . .

0 0 ... 0 0 ... ...

... ... 0 0

0 ... ... 0 0

. ... . . ... . .

. ... . . ... . .

0 0 0 0 ...

n n

n

n

m m

m

m m

a a a a

a a a a

a a a

b b b b b b

b b b b b b

b b b b

 

 

 

 

         

 

(2)

constructed for polynomials 1

1 0

1

1 0

( )

...

,

0;

( )

...

,

0;

n n

n n n

m m

m m m

f x

a x

a x

a

a

g x b x

b x

b

b

− −

− −

=

+

+ +

=

+

+ +

(4)

At the proof of the second criterion of existence of common eigenvalue of several operator pencils in Hilbert space we essentially use the results of multiparameter spectral theory [1,3,5,11] and also the notion of abstract analog of Cramer’s determinants.

Criterion for the existence of common eigenvalues of several operator pencils in Hilbert space

1. Let now we have

n

the operator pencils depending on the same parameter

λ



=

+

+

+

=

n

i

B

B

B

B

k k i

i i

i i

i

,...,

2

,1

...

)

(

λ

0,

λ

1,

λ

,

(5) (3.12)

when

B

i

(

λ

)

be the operator pencils acting in Hilbert space

i

H

, correspondingly. Without loss of generality we suppose

n

k

k

k

1

2

...

.

Definition1. [1,3,11]

B

s+,i -the operator induced by operator

B

s,i, acting from the space

H

iinto tensor product

space

H

=

H

1

...

H

n and is defined the following way: on decomposable tensor

x

=

x

1

...

x

n:

n i

i i s i i

s

x

x

x

B

x

x

x

B

+,

=

1

...

−1

,

+1

...

, on

other elements of space

H

=

H

1

...

H

n operator

+ i s

B

, is defined on linearity and continuity.

Introduce the operators

R

i

(

i

=

1

,...,

n

1

)

in space

2 1 k

k

H

+ (the direct sum of

2

1

k

k

+

copies of tensor product

n

H

H

H

=

1

...

of spaces

H

1

,

H

2

,...,

H

n) by means

of the operational matrices

0,1 1,1 ,1

0,1 1,1 1.1 ,1

0,1 1,1 ,1

1

0, 1, ,

0, 1, ,

0, 1, ,

0 1

0 1 1 0

0 0 1

0 0

0 0

0 0

k

k k

k i

i i k i

i i k i

i i k i

B B B

B B B B

B B B

R

B B B i

B B B i

B B B i

+ + +

+ + + + −

+ + +

− + + +

+ + +

+ +

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅

⋅⋅ ⋅⋅

⋅ ⋅ ⋅⋅⋅⋅⋅ ⋅ ⋅⋅⋅⋅⋅⋅ ⋅

⋅⋅ ⋅⋅⋅⋅⋅⋅

=

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅

⋅⋅⋅ ⋅ ⋅⋅

⋅ ⋅ ⋅⋅⋅⋅⋅⋅ ⋅ ⋅⋅⋅⋅⋅⋅ ⋅

⋅⋅⋅ ⋅⋅⋅⋅⋅⋅

, i 2,..., .n

+

 

 

 

 

 

 

  =

 

 

 

 

 

 

 

 

(6)

In the matrix

R

i1the number of lines with operators

1 1

,

,

s

0

1,

,...,

k

B

s+

=

is equal to

k

2and the number of lines with operators

B

s+,i

,

s

=

0

,

1

,...,

k

iis equal to

k

1

.

R

i−1are the operators in the space

H

k1+k2representing by the matrices in (6). (3.13)

We shall designate

σ

p

(

B

i

(

λ

)

)

the set of eigenvalues of

an operator

B

i

(

λ

).

Theorem 2. [4] Let the operator

B

k1 has inverse.

{ }

1 ( ( ))

n

p i

i1=σ B λ ≠ θ iff 1

{ }

.

n i i

KerR ≠ θ =

1

Proof of Theorem 2. Necessity. We suppose, that pencils

)

(

λ

i

B

have a common eigen- value

λ

0. For everyone

i

there are such elements

x

i

H

i , that

.

,...,

2

,1

,

0

...

)

(

0

x

1

x

i

n

B

i+

λ

n

=

=

It is easy to see, if the element

( )

(

n

)

k k n

n

x

x

x

x

x

x

X

=

...

,

...

,...,

+ − 1

...

1 0 1

0

1

λ

λ

1 2

in the kernel of an operator

R

i for each

i

=

1

,

2

,...,

n

1

,

then 1

1

. n

i i

XKerR

=

1

Sufficiency. Let 1

{ }

1

n i i

KerR

θ

= ≠

1

and an element

1 1 2

1

( , ,..., k k1 2) n i, i .

i

X = x x x + ∈ −KerR x H∈ =

  

1

 Then element

X

in

the kernels of operators

R

1

,

R

2

,...,

R

n1 , i. e.

1

,...,

2

,1

;

)

~

,...,

~

,

~

(

x

1

x

2

x

1+2

=

s

=

n

R

s k k

θ

. Expression

{ }

1

1

n i i

KerR

θ

≠ =

1

means there is such nonzero element

( ) ( ) ( )

1, 2, , 1

1 1

1 2

1 2

... ( ... )

k k s

k k s s s

i i n i n

i s

x x x H H

+

+

⊗ ⊗ ⊗ ⊗

 

=

 = that the

equalities satisfy. Then an element

( ) ( ) ( )

1, 2, ,

1 1

1 2 ...

k k

s s s

i i n i

i s

x x x

+ ∞

⊗ ⊗

 

=

 = enters the kernel of

Resultant of operators

B

1

(

λ

)

and

B

++

(

λ

,

α

2

,...,

α

n

)

=

).

(

...

)

(

)

(

3 3

2

2

λ

α

λ

α

λ

α

++

+

++

+

+

++

n n

B

(3)

( 1) ( 1) (1) (1) (1)

0,1 1, 2, , ,1 1, ,

1 1

( ) ( )

0,1 1, 2,

1

1

1

1

1

1

...

...

1

...

0

...

2

2

2

...

s s

k k

i i n i k i n i

i i

s

k k

i i

i

k

B

x

x

x

B

x

x

k

B

x

x

+ +

+ +

= =

+

=

+

⊗ ⊗

+ +

⊗ ⊗

=

( ) ( ) ( )

, ,1 1, ,

1

( 1) ( 1)

(1) (1) (1)

0, 1, 2, , , 1, ,

1 1

( ) ( ) ( )

0, 1, , , 1,

1

1 2

2

...

1 2

...

1 2

0

1

1

1

2

...

...

...

0

1 1

...

1

...

1 2

s

k k k k k

n i k i n i

i s

s

k k

i i i n i k i i n i

i i

s

k k k k

i i n i k i i

i i

k k

x

B

x

x

k

i

i

B

x

x

x

B

i

x

x

k

B

x

x

B

i

x

+ +

+

=

+ +

+ +

= =

+

+ +

=

+

+ +

⊗ ⊗

=

+

⊗ ⊗

+ +

⊗ ⊗

=

⊗ ⊗

+ +

( )

, 1

1 2

1 2

...

0

2,...

s

k k n i

k k

x

i

n

+

=

+

⊗ ⊗

=

=

(7)

) ,..., ,

( 2 n

B++

λ

α

α

-the operator induced to the space

n

H

H2⊗...⊗ by an operator

B

i

(

λ

)

and αi

(

i=2,3,...,n

)

-

arbitrary complex numbers. The Resultant of pencils

B

1

(

λ

)

and B++(

λ

,

α

2,...,

α

n)acts in the direct sum of

k

1

+

k

2 copies of tensor product H1⊗...⊗Hnof spacesH1,...,Hn.

Further we use the known property of the elements of tensor product space. It is known that the representation of the element in tensor product space is not unique. For each element of the tensor product space there is the number coinciding with the minimal number of decomposable tensors, necessary for the representation of this element. This number is named the rank of element. If the sum decomposable tensors in the representation of element are more than the rank of this element then one-nominal components of the given element are linear dependent. Having transferred to each line of equalities (7) one decomposable tensor from left side in the right side of the this eqalities, we get, that the series standing at the left side in each equation have a rank 1 as they are equal to a decomposable tensor, standing in the right side of this equality. Thus, between one-nominal components of all terms entering into expression (7) there is a linear dependence and an element standing at the left in all equalities in (7) is a decomposable tensor.

We have that

n s

i n s

i s

i s

i

x

x

H

H

x

=

...

...

( ) 1

, )

( , 2 1

) (

, 1

is decomposable vector for all values of number

s

.

Using [2] we prove there is a number

λ

being the common point of spectra of operators

B

1

(

λ

)

and

B

(

λ

,

α

2

,...,

α

n

)

+ +

at all values

α

i . The last means, that

λ

there is a common eigenvalue of all pencils

B

i

(

λ

)

,

i

=

1

,

2

,...,

n

. Theorem2

is proved.

2. Give some definitions and concepts of the theory of multiparameter operator systems in the case when the number of parameters is equal to the number of equations.

Let the linear multiparameter system in the form be:

0

)

(

1 ,

,

0

=

+

=

= k

n

k k ik

k k

k

x

B

B

x

B

λ

λ

k=1,2,...n

(8) when operators

B

k,i act in the Hilbert space

H

i

Definition 2. [1,2,11]

λ

=

(

λ

1

,

λ

,...,

λ

n

)

C

n is an

eigenvalue of the system (8) if there are non-zero elements

i

x

H

i

(

i

=

1

,

2

,...,

n

)

such that equalities in (8) are true , then a decomposable tensor

x

=

x

1

x

2

...

x

nis called the eigenvector corresponding to an eigenvalue

n

n

C

=

(

λ

1

,

λ

,...,

λ

)

λ

.

Definition 3 ([5], [6]).

Let

x

0,...,0

=

x

1

x

2

...

x

nbe an eigenvector of the system (8), corresponding to its eigenvalue

)

,...,

,

(

λ

1

λ

2

λ

n

λ

=

; the

n

m m m

x

1, 2,...., is

m

1

,

m

2

,...,

m

n - the associated vector (see[5]) to an eigenvector

x

0,...,0of the

system (8) if there is a set of vectors

n i

i

i

H

H

x

n

)

...

(

1,2,..., 1 , satisfying to conditions

0

...

)

(

, ,..., 1, 1, ,..., , , ,..., 1 ,

0 ,1, 2

+

1 2

+

+

1 2 −

=

+ −

+ +

n n

n i s s s ki s s s

s s s

i

x

B

x

B

x

B

λ

.

x

i1,i2,...,in

=

0

, when

s

i

<

0

,

,...,

1

;

,...,

1

,

(4)

n i

i

i

H

H

x

n

)

...

(

1,2,..., 1 there are various

arrangements from set of integers on

n

with

0

s

r

m

r

,

n

r

=

1

,

2

,...,

.

Definition 4. In [1,3,11] for the system (8) analogue of the Cramer’s determinants, when the number of equations is equal to the number of variables, is defined as follows: on decomposable tensor

x

=

x

1

x

2

...

x

n operators

i

are defined with the help the matrices

=

α

i i

x

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

, , 2 , 1 , 0 3 3 , 3 3 , 2 3 3 , 1 3 3 , 0 2 2 , 2 2 , 2 2 2 , 1 2 2 , 0 1 1 , 1 1 , 2 1 1 , 1 1 1 , 0 2 1 0 n n n n n n n n n n n n n

x

B

x

B

x

B

x

B

x

B

x

B

x

B

x

B

x

B

x

B

x

B

x

B

x

B

x

B

x

B

x

B

α

α

α

α

(10) where

α

0

,

α

1

,...,

α

n are arbitrary complex numbers. The

decomposition of the determinant (10) is its formal decomposition, when all terms of decomposition are tensor products of elements. If

α

k

=

,1

α

i

=

0

,

i

k

,then right side of (10) is equal to

k

x

, where

n

x

x

x

x

=

1

2

...

. On all the other elements of the

space

H

operators

iare defined on linearity and continuity.

)

,...,

2

,1

(

s

n

E

s

=

is the identity operator of space

H

s.

Suppose that for all

x

H

,

x

0

,

0

,

0

)

,

(

)

,

(

0

x

x

δ

x

x

>

δ

>

, and all

B

i,k are

selfadjoint operators in the space

H

i. Inner product [.,.] is

defined as follows; if

x

=

x

1

x

2

...

x

m and m

y

y

y

y

=

1

2

...

are decomposable tensors, then

),

,

(

]

,

[

x

y

=

0

x

y

where

(

x

i

,

y

i

)

is the inner product in

the space

H

i. On all the other elements of the space

H

the

inner product is defined on linearity and continuity. In space

H with such metric all operators

Γ

i

=

−1

i

0 are selfadjoint [1,3,11].

We give a new criterion for the existence of common points of spectra of all operator pencils (5). Let us assume that each operator pencil in (5) has a discrete spectrum. Conditions on the operators

B

j,k remain the same. Let

highest degree of parameter

λ

in (5) is

m

. In (5) we make the transformations

m

i

i

i

,...,

1

,

2

,...,

,...,

1

=

=

=

λ

λ

λ

λ

. (11)

then the operator pencils (5) are written in the form

k

i

x

B

i+

(

λ

1

,...,

λ

m

)

~

1

=

0

=

1

,...,

(12) by the operator

B

i

(

λ

1

,...,

λ

m

)

, that acts in space

H

i.

The equation

B

i+

(

λ

1

,...,

λ

m

)

~

x

1

=

0

i

=

1

,...,

k

we consider together with the following equations

0

)

(

..

...

...

...

...

0

)

(

0

)

(

1 0 1 2 2 3 1 3 0 2 2 1 2 1 2 0 1 2

=

+

+

=

+

+

=

+

+

m m m

m

t

t

t

x

x

t

t

t

x

t

t

t

λ

λ

λ

λ

λ

λ

λ

λ

(13)

where the operators

t

0

,

t

1

,

t

2are defined with help of the matrices





=

0

1

1

0

0

t

,





=

0

0

0

1

1

t

, 

     = 1 0 0 0 2

t (14)

In the space

R

2the equations (13) on its eigenvectors realize the connections between the parameters

m

λ

λ

λ

1

,

2

,...,

according to the requirements of (5).

We build

k

linear multiparameter systems

0

~

)

,...,

(

1 1

=

+

x

B

i

λ

λ

m

0

)

(

..

...

...

...

...

0

)

(

0

)

(

1 0 1 2 2 3 1 3 0 2 2 1 2 1 2 0 1 2

=

+

+

=

+

+

=

+

+

m m m

m

t

t

t

x

x

t

t

t

x

t

t

t

λ

λ

λ

λ

λ

λ

λ

λ

. ,..., 1 ; 1 ,..., 2 , , ... ~ 2 1

1 H H x R s m i k

x ∈ ⊗ ⊗ k s∈ = − =

(15) and introduce the space

2 2

1

...

H

R

...

R

H

H

=

k

. In the tensor

product

H

factor

R

2 repeats

m

1

time. Construct

the analog determinants of Cramer for the linear multiparameter system (15) of the formulae

=

+ + + + + + + + + + + + + + + + + + + + =

.

.

.

0

0

0

.

.

.

.

.

.

0

.

.

0

0

.

.

.

.

.

.

, 2 3 , 0 3 , 1 2 , 2 2 , 0 2 , 1 , , 2 , 1 , 0 2 1 0 0 m i m i i i m m

i i i

t

t

t

t

t

t

B

B

B

B

α

α

α

α

α

(16) When

B

j,i

=

0

if

j

>

k

i. All operators

B

i+,+j

,

t

s+,+kin the expression (16) are induced in space

2 2

1

...

H

R

...

R

H

H

=

k

(5)

operators

B

i+,j

,

t

s,k, correspondingly.

If operators

B

i,kare selfadjoint in space

H

kand above

accepted conditions are satisfied, then all the eigenvalues of each operator

.)

,...,

1

;

,...,

1

(

, 1

, 0

,s s is

s

k

i

m

i

=

=

=

Γ

, are real numbers [1,3,11]. Suppose now that

(

λ

1

,...,

λ

m

)

is an

eigenvalue of the system (15). Take two equations of the system, namely the equations

0

)

(

0

)

(

3 1 3 0 2 2 1

2 1 2 0 1 2

=

+

+

=

+

+

x

t

t

t

x

t

t

t

λ

λ

λ

λ

λ

(17)

For eigenvector

k k

m

m

x

R

x

R

x

x

H

H

x

x

...

,

,...,

2

,

(

1

...

)

1

...

1 2 1

1 of

the system (15) we have the following:

If

λ

1

0

and

x

2

=

(

α

1

,

β

1

)

is eigenvector of the first equation of (16) then









+





+





0

0

0

1

0

1

1

0

1

0

0

0

2

1

λ

λ

(

α

1

,

β

1

)

=

0

,

or

λ

1

β

1

+

λ

2

α

1

=

0

,

β

1

+

λ

1

α

1

=

0

,

2 1 2 2

0

;

λ

λ

λ

=

.

Further if

λ

1

0

,

λ

2

0

,

x

n+2

=

(

α

2

,

β

2

)

0

then

the equalities

λ

2

β

2

+

λ

3

α

2

=

0

,

λ

1

β

2

+

λ

2

α

1

=

0

and

2 2 3 1

λ

λ

λ

=

.Earlier we proved 2

1 2

λ

λ

=

, therefore 3

1 3

λ

λ

=

.

By analogy for other equations: if

(

λ

1

,

λ

2

,...,

λ

m

)

is an eigenvalue of the system (15) then 4

1 4

λ

λ

=

…,

1 1

,

λ

λ

λ

λ

=

m

=

m are valid.

In the space 2 2

1

...

H

R

...

R

H

H

=

k

we build operators

0,1

,...,

0,k(accord to the definition 4) for

each of linear multiparameter system (15). If kernels of operators

0,1

,...,

0,k

.

are zero then (see [1],[3],[5]) the

parameter

λ

i in (15) are separated. The formulae k s x x x

x s ms m m m

s

s 1, 1 1 1 , ...., 01, , ; 1,..., 1

,

0 ∆ = ∆ ∆ = =

∆−  λ  −  λ

2 2

1

...

H

R

...

R

H

H

x

=

k

are true. So on eigenvectors of the system 4

1 4 3 1 3 2 1

2

λ

,

λ

λ

,

λ

λ

λ

=

=

=

,…,

λ

=

λ

1m

,

λ

=

λ

1

m

and

m m m k m k i

i

x

x

x

x

1 ,

1 , 0 1

1 1 1 , 1

1 ,

0

=

λ

,

....,

=

λ

− −

Besides of

λ

=

λ

1 then

m

i

k

s

x

x

i i

i s i

s ,

,

1

,

2

,...,

;

1

,

2

,...,

1

,

0

=

=

=

λ

(18)

Theorem3. Let the spectra of the operator pencils

B

i

(

λ

)

contain only eigenvalues , operators

−01,i

(

i

=

1

,

2

,...,

k

)

exist and bounded. Then

λ

is a common eigenvalue of all the operators

B

i

(

λ

)

in (5) iff

}

{

)

ker(

...

)

(

1,1

λ

0,1

1,k

λ

0,k

θ

Ker

1

1

(19)

Proof of Theorem3. Suppose that conditions of the Theorem 3are fulfilled. From [ 5] it follows: system of eigen and associated vectors of all multiparameter systems (15) and system of eigen and associated vectors of operators

)

,...,

2

,1

(

....,

,

, 01, , 1

, 1

1 , 0 1

, i ik k ik

i

m

i

=

Γ

=

=

Γ

− −

(20) coincide for the each fixed meanings of number

i

.

In virtue of the connections between the components of eigenvalues of the system(15) we have

; ,..., 1 ; ,.., 1 ; ;

, 1 ,

0sisxi = ixi xi =x i= m s= k

∆− 

λ

   or

;

,...,

1

;

,..,

1

;

, 1

,

0s

is

x

=

i

x

i

=

m

s

=

k

λ

(21) Let

λ

is the first component of eigenvalue of all systems (15). From condition

}

{

)

ker(

...

)

(

1,1

λ

0,1

1,m

λ

0,m

θ

Ker

1

1

it

follows that there is the element

k

m

x

H

x

H

H

x

x

x

...

=

;

~

...

~

1 1 2

1

, being the common eigenvector of all systems (15), corresponding to the common eigenvalue

)

,...,

,

(

)

,...,

,

(

2

2

1

λ

λ

m

λ

λ

λ

m

λ

=

.

We use the known formulae of multiparameter spectral theory [1,3,11]

k

i

B

B

B

0+,+i

+

1+,i+

Γ

1,i

+

...

+

m+,+i

Γ

m,i

=

0

,

=

1

.

2

,...,

0

..

...

...

...

...

0

0

(

, 1 , 1 , 1 1 , 0 , 2 1 , 2

, 3 3 ,, 1 , 2 3 , 0 , 1 3 , 2

, 2 2 , 1 , 1 2 , 0 2 , 2

=

Γ

+

Γ

+

Γ

=

Γ

+

Γ

+

Γ

=

Γ

+

Γ

+

+ +

− −

+ +

− −

+ +

+ + +

+ +

+

+ + +

+ + +

i m m i m m i m m

i i

i

i i

t

t

t

t

t

t

t

t

t

(22)

. ,...,

1 k

i= Substituting into the first equality in all multiparameter systems (22) values of operators

Γ

s,rfrom (20),(21) and given

x

1

=

x

2

=

...

=

x

m

=

x

we establish that all operator pencils in (5) have a common eigenvalue.

It is true an inverse preposition. Let

λ

there is a common eigenvalue of the operator pencils (5), consequently,

)

,...,

,

(

λ

λ

2

λ

m is a eigenvalue of all systems in (15). So

λ

is the eigenvalue of (15), then there is eigenvector

x

x

x

x

m

=

=

=

=

2

...

(6)

(15).From [5] it follows that the system of eigen and associated vectors of multiparameter system (15) and of each equation

(

1,1

0,1

)

x

=

0

,...,

(

1,k

0,k

.)

x

=

0

λ

λ

coincide. So

x

is an eigenvector of (15), then

}

{

)

ker(

...

)

(

1,1

λ

0,1

1,k

λ

0,k

θ

Ker

1

1

Theorem3 is proved.

Application. In the case when all operators in all operator pencils (5) are the real numbers, Hilbert spaces

).

,...,

2

,1

(

,

i

n

R

H

i

=

=

we have

n

polynomials



=

+

+

+

=

n

i

x

b

x

b

b

x

b

i

i k i k i

i i

,...,

2

,1

...

)

(

0, 1, ,

(23)

when the variable

x

is the parameter

λ

in (5). Then the polynomials (23) have the common solution then and only

then when

{ }

.

1

θ

= i

n

i

KerR

1

R

i are constructed by the

formula (6), in which the operators

B

s,iare replaced by the

numbers

b

s,i

.

REFERENCES

[1] Atkinson F.V. Multiparameter spectral theory.

Bull.Amer.Math.Soc.1968, 74, 1-271.

[2] Balinskii A.I Generation of notions of Обобщение понятия Bezutiant and Resultant DAN of Ukr. SSR, ser.ph.-math and tech. of sciences,1980,2,pp.3-6 ( in Russian).

[3] Browne P.J. Multiparameter spectral theory. Indiana Univ. Math. J,24, 3, 1974.

[4] Dzhabarzadeh R.M. On existence of common eigen value of some operator-bundles, that depends polynomial on parameter. Baku.International Topology conference, 3-9 оct., 1987, Теz., 2, Baku, 1987, p.93

[5] Dzhabarzadeh R.M. Spectral theory of two parameter system in finite-dimensional space. Transactions of NAS Azerbaijan, v. 3-4 1998, p.12-18

[6] Dzhabarzadeh R.M. Spectral theory of multiparameter system of operators in Hilbert space, Transactions of NAS of Azerbaijan, 1-2, 1999, 33-40.

[7] Dzhabarzadeh R.M. Multiparameter spectral theory . Lambert Academic Publishing, 2012, pp. 184(in Russian)

[8] Dzhabarzadeh R.M .Nonlinear algebraic system. Lambert Academic Publishing, 2013,pp. 101 (in Russian )

[9] Dzhabarzadeh R.M. About solutions of nonlinear algebraic system in two Variables. Pure and Applied Mathematics Journal,vol. 2, No. 1, pp. 32-37, 2013

[10] Khayniq Q. Abstract analog of an eliminant of two polynomial bundles. The Functional analysis and its applications, 1977,2, issue 3, pp.94-95(in Russian).

References

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