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 Operator1. 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 spaces2 1
,
H
H
, correspondingly.))
(
),
(
(
Re
s
A
λ
B
λ
is the resultant of two operator pencils( )
A
λ
andB
( )
λ
. It is presented by the matrices (2) andacts in the space
(
H
1⊗
H
2)
n m+ - direct sum ofn 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 parameterλ
inpencilB( )λ , that is
m
, the number of rows in matrix 1 2Re ( ( ), ( ))s A λ A λ with operators
B
i coincides with the leading degree of parameterλ
in pencilA( )λ
, that isn
.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
iandB
i are bounded and one of them has bounded inverse.b) Spectra of operator pencils
A
( )
λ
andB
( )
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 meaningsm
,
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
−
−
−
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 parameterλ
=
+
+
+
=
n
i
B
B
B
B
k k ii 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 spacei
H
, correspondingly. Without loss of generality we supposen
k
k
k
1≥
2≥
...
≥
.Definition1. [1,3,11]
B
s+,i -the operator induced by operatorB
s,i, acting from the spaceH
iinto tensor productspace
H
=
H
1⊗
...
⊗
H
n and is defined the following way: on decomposable tensorx
=
x
1⊗
...
⊗
x
n:n i
i i s i i
s
x
x
x
B
x
x
x
B
+,=
1⊗
...
⊗
−1⊗
,⊗
+1⊗
...
⊗
, onother 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 space2 1 k
k
H
+ (the direct sum of2
1
k
k
+
copies of tensor productn
H
H
H
=
1⊗
...
⊗
of spacesH
1,
H
2,...,
H
n) by meansof 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
i−1the number of lines with operators1 1
,
,
s
0
1,
,...,
k
B
s+=
is equal tok
2and the number of lines with operatorsB
s+,i,
s
=
0
,
1
,...,
k
iis equal tok
1.
R
i−1are the operators in the spaceH
k1+k2representing by the matrices in (6). (3.13)We shall designate
σ
p(
B
i(
λ
)
)
the set of eigenvalues ofan 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- valueλ
0. For everyonei
there are such elementsx
i∈
H
i , that.
,...,
2
,1
,
0
...
)
(
0x
1x
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 2in the kernel of an operator
R
i for eachi
=
1
,
2
,...,
n
−
1
,then 1
1
. n
i i
X −KerR
=
∈
1
Sufficiency. Let 1
{ }
1n i i
KerR
θ
−
= ≠
1
and an element1 1 2
1
( , ,..., k k1 2) n i, i .
i
X = x x x + ∈ −KerR x H∈ =
1
Then elementX
inthe kernels of operators
R
1,
R
2,...,
R
n−1 , i. e.1
,...,
2
,1
;
)
~
,...,
~
,
~
(
x
1x
2x
1+2=
s
=
n
−
R
s k kθ
. 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 theequalities 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 ofResultant of operators
B
1(
λ
)
andB
++(
λ
,
α
2,...,
α
n)
=
).
(
...
)
(
)
(
3 32
2
λ
α
λ
α
λ
α
+++
+++
+
++n n
B
( 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 spacen
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 ofk
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 numberλ
being the common point of spectra of operatorsB
1(
λ
)
andB
(
λ
,
α
2,...,
α
n)
+ +
at all values
α
i . The last means, thatλ
there is a common eigenvalue of all pencilsB
i(
λ
)
,i
=
1
,
2
,...,
n
. Theorem2is 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 spaceH
iDefinition 2. [1,2,11]
λ
=
(
λ
1,
λ
,...,
λ
n)
∈
C
n is aneigenvalue 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 tensorx
=
x
1⊗
x
2⊗
...
⊗
x
nis called the eigenvector corresponding to an eigenvaluen
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
1λ
2λ
nλ
=
; then
m m m
x
1, 2,...., ism
1,
m
2,...,
m
n - the associated vector (see[5]) to an eigenvectorx
0,...,0of thesystem (8) if there is a set of vectors
n i
i
i
H
H
x
n)
⊂
⊗
...
⊗
(
1,2,..., 1 , satisfying to conditions0
...
)
(
, ,..., 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
, whens
i<
0
,
,...,
1
;
,...,
1
,
n i
i
i
H
H
x
n)
⊂
⊗
...
⊗
(
1,2,..., 1 there are variousarrangements from set of integers on
n
with0
≤
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 operatorsi
∆
are defined with the help the matrices=
∆
∑
α
i ix
⊗
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
, , 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 nx
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) whereα
0,
α
1,...,
α
n are arbitrary complex numbers. Thedecomposition 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∆
kx
, wheren
x
x
x
x
=
1⊗
2⊗
...
⊗
. On all the other elements of thespace
H
operators∆
iare defined on linearity and continuity.)
,...,
2
,1
(
s
n
E
s=
is the identity operator of spaceH
s.Suppose that for all
x
∈
H
,x
≠
0
,0
,
0
)
,
(
)
,
(
∆
0x
x
≥
δ
x
x
>
δ
>
, and allB
i,k areselfadjoint operators in the space
H
i. Inner product [.,.] isdefined as follows; if
x
=
x
1⊗
x
2⊗
...
⊗
x
m and my
y
y
y
=
1⊗
2⊗
...
⊗
are decomposable tensors, then),
,
(
]
,
[
x
y
=
∆
0x
y
where(
x
i,
y
i)
is the inner product inthe space
H
i. On all the other elements of the spaceH
theinner product is defined on linearity and continuity. In space
H with such metric all operators
Γ
i=
∆
−1∆
i0 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. Lethighest degree of parameter
λ
in (5) ism
. In (5) we make the transformationsm
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 operatorB
i(
λ
1,...,
λ
m)
, that acts in spaceH
i.The equation
B
i+(
λ
1,...,
λ
m)
~
x
1=
0
i
=
1
,...,
k
we consider together with the following equations0
)
(
..
...
...
...
...
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
0t
,
=
0
0
0
1
1t
, = 1 0 0 0 2
t (14)
In the space
R
2the equations (13) on its eigenvectors realize the connections between the parametersm
λ
λ
λ
1,
2,...,
according to the requirements of (5).We build
k
linear multiparameter systems0
~
)
,...,
(
1 1=
+
x
B
iλ
λ
m0
)
(
..
...
...
...
...
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 11 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 tensorproduct
H
factorR
2 repeatsm
−
1
time. Constructthe 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 mi i i
t
t
t
t
t
t
B
B
B
B
α
α
α
α
α
(16) WhenB
j,i=
0
ifj
>
k
i. All operatorsB
i+,+j,
t
s+,+kin the expression (16) are induced in space2 2
1
...
H
R
...
R
H
H
=
⊗
⊗
k⊗
⊗
⊗
operators
B
i+,j,
t
s,k, correspondingly.If operators
B
i,kare selfadjoint in spaceH
kand aboveaccepted 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 aneigenvalue 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
andx
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
1β
1+
λ
2α
1=
0
,β
1+
λ
1α
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
thenthe equalities
λ
2β
2+
λ
3α
2=
0
,
λ
1β
2+
λ
2α
1=
0
and2 2 3 1
λ
λ
λ
=
.Earlier we proved 21 2
λ
λ
=
, therefore 31 3
λ
λ
=
.By analogy for other equations: if
(
λ
1,
λ
2,...,
λ
m)
is an eigenvalue of the system (15) then 41 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) foreach of linear multiparameter system (15). If kernels of operators
∆
0,1,...,
∆
0,k.
are zero then (see [1],[3],[5]) theparameter
λ
i in (15) are separated. The formulae k s x x xx 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
1m
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 thenm
i
k
s
x
x
i ii 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 operatorsB
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 ,
0s∆isxi = ixi xi =x i= m s= k
∆−
λ
or;
,...,
1
;
,..,
1
;
, 1
,
0s
∆
isx
=
ix
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
itfollows 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
)
,...,
,
(
)
,...,
,
(
22
1
λ
λ
mλ
λ
λ
mλ
=
.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+,+iΓ
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 givenx
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
2λ
m is a eigenvalue of all systems in (15). Soλ
is the eigenvalue of (15), then there is eigenvector
x
x
x
x
m
=
=
=
=
2...
(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 haven
polynomials
=
+
+
+
=
n
i
x
b
x
b
b
x
b
ii k i k i
i i
,...,
2
,1
...
)
(
0, 1, ,(23)
when the variable
x
is the parameterλ
in (5). Then the polynomials (23) have the common solution then and onlythen when
{ }
.
1
θ
≠
= i
n
i
KerR
1
R
i are constructed by theformula (6), in which the operators
B
s,iare replaced by thenumbers
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).