On Existence of Eigen Values of Several Operator Bundles with Two Parameters
Makhmudova Malaka Gasan1, Sultanova Elnara Bayram2
1Department of Higher Mathematics, Baku State University, Baku, Azerbaijan
2Department of Applied Mathematics, Baku State University, Baku, Azerbaijan
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To cite this article:
Makhmudova Malaka Gasan, Sultanova Elnara Bayram. On Existence of Eigen Values of Several Operator Bundles with Two Parameters. Pure and Applied Mathematics Journal. Special Issue: Spectral Theory of Multiparameter Operator Pencils and Its Applications. Vol. 4, No. 4-1, 2015, pp. 16-21. doi: 10.11648/j.pamj.s.2015040401.14
Abstract: For the several operator bundles with two parameters when the number of equation is greater than the number of parameters in the Hilbert spaces is given the criterion of existence of the common point of spectra. In the special case the common point of spectra is the common eigen value. In the proof of the theorem the authors use the results of the spectral theory of multiparameter systems.
Keywords: Operator, Space, Resultant, Criterion, Eigenvector, Bundle
1. Introduction
Spectral theory of multiparameter systems arose as a result of studying questions related with the solutions of partial differential equations and equations of mathematical physics. Founder of the spectral theory of multiparameter systems is F.V Atkinson. Atkinson [1] studied the available at the time fragmentary results for multiparameter symmetric differential systems, built multiparameter spectral theory of multiparameter systems of operators in finite-dimensional Euclidean spaces. Further, by taking the limit Atkinson built the spectral theory of multiparameter systems with self-adjoint compact operators in infinite-dimensional Hilbert spaces.
Later, many of his followers have studied spectral problems of self-adjoint multiparameter systems.
2. Necessary Definitions and Notions
Consider the two parameter system
(1)
in the Hilbert spaces ;
is the tensor product of spaces
We consider the case when the number of operator equations in (1) is greater than the number of parameters. Let the number of equations in (1) is equal to The case when
is considered in the work [9].
In (1) linear operators act in the Hilbert space
.
Let and
be two decomposable tensors .The inner product of these elements in the space
is defined by means of the formulae
This definition is spread on other elements of the tensor product space on linearity and continuity
Let's reduce a series of known positions concerning the spectral theory of multiparameter system.
Definition1. ([1, 3]) is an eigen value of the system (1) if there are non-zero elements
such that (1) is fulfilled. A decomposable tensor
is named an eigenvector of the system (1), corresponding to the eigenvalue
.
Definition2. The operator is induced by an operator
, acting in the space
, into the tensor space
, if on each decomposable tensor
of tensor product space
we have
and on all the other elements of the space
operator
is defined on linearity and continuity.
is an identical operator in
Definition3.[5]. Let be two polynomial bundles
,
(2)
depending on the same parameter and acting, generally speaking, in various Hilbert spaces
, accordingly.
The operator is given with help of matrix
(3)
In [4,11] this operator is called an abstract analog of an Resultant for polynomial bundles (2).
In definition of a Resultant (3) of bundles (2) lines with operators repeated
times, and lines with оperators
repeated exactly
times Numbers
,
are the greatest degrees of parameter
in bundles
and
accordingly. Thus, the Resultant is an operator, acting in space
that is a direct sum
of copies of tensor product spaces
Value of
is equal to its formal expansion when each term of this expansion is tensor product of operators. Let operator
, or
is invertible. If the greatest degrees of parameter
in bundles of
and
coincide (see [11]) or if the greatest degrees of parameter
in bundles of
and
, generally speaking, can not coincide (see [4]) bundles (2) have a common eigen values in the only case when the Resultant of these bundles has non-zero kernel. In fact, that in case of when bundles act in finite-dimensional spaces, common point of spectra of these bundles is a common eigen value.
When in (1) the number the following result is proved in the work [9]:
Theorem1. Let operators also
act in finite-dimensional spaces
and
, accordingly, and one of three following conditions is fulfilled:
a),
;
are selfadjoint operators everyone in their space
b),
operators
are selfadjoint everyone in their spaces
c),
,
are the self-adjoint operators, acting everyone in its finite-dimensional space.
Then the-multiple basis of system of eigen and associated vectors of (2) takes place
This paper is devoted to the investigation of the system (1) when the N>2. The goal of our investigations is the finding the criterion of existing of common point of spectra of the system (1).
For proving we use the notion of criterion of existence of common point of spectra of several bundles with the same one parameter.
3. Necessary and Sufficient Conditions of Existence of Eigen Values of Several Operator Polynomials
(4)
when is an operator bundle with a discrete spectrum, acting in Hilbert space
(
Without loss of generality, we assume that.
.
is the direct sum of
copies of the tensor spaces
.
Introduce the operatorswith help of the operator matrices (5)
(5)
Rows with the operators are repeated
times and rows with the operators
are repeated
times. The operators
are induced by an operator
, acting in the space
, into the space
by the formulae
Denote the set of eigen values of operator
.
Theorem 2.[10]. Let all operators are bounded in the corresponding spaces
, the operator
has an inverse. Spectrum of each operator pencil
contains only eigen values.
Then if and only if
(6)
This result is obtained in [10].
4. Criterion of Existence of Common Eigen Values of Two Polynomial Bundles with Two Parameters
We fix in the all operators (1) the parameter Let be
Then we have
operators, depending on the same parameter
Without loss of generality we suppose
.
Then the system (1) turns out to the system
(7)
So the parameteris fixed arbitrarily we missed the index 0 in the
Denote though
the operator in the space
, induced into
by the operator
acting in the space
Let all operators exist and bounded.
Introduce the notations
Construct the resultants of operatorsand
(8)
The rows in (8) with the operatorsare repeated
time, rows with the operators
are repeated
time for all meanings of
The operators
act in the direct sum of
copies of tensor product space
If
,
,
, all operators
,
are bounded for all meanings of
. Suppose that the eigen and associated vectors of the bundle
for each fixed
form the basis of the space
. Consider the operators
. It is known they act in the direct sum of
copies of tensor product space
From the results of the spectral theory of multiparameter system the first component of the element of the kernel of
is the linear combination of elements of an aspect
(9)
when ((accordingly,
) there is a restricted chain of eigen and associated vectors of an operator
(accordingly,
) , corresponding to some common eigenvalue
of both operators. It is clear if
and all other conditions are fulfilled the element of the type
form the first component of the element entering the kernel of the resultant of operator bundles
and
If the condition is fulfilled and the element
,
then this element
enters the kernel of resultant of operator bundles
and
for the first component of the element
expression (9) are fulfilled. From [12] it follows that element of the space
entering the
is the linear combination of elements of the aspect
(10)
when (accordingly,
) there is a restricted chain of e.a. vectors of an operator
and operator
, correspondingly ,
Naturally, when in (10) decomposable tensor
is the first component of the element of
which has the form
(
). Element
is the eigen vector of the system (1), corresponding to the eigen value
.
Denote though the decompositions of the resultants
. In fact,
are the operator bundles, acting in the space
We have the operator bundles with one parameter
The greatest degree of parameter
in the operator bundle
is equal to
. Moreover, we have the following result:
Let be all operators bounded in the corresponding space
,
then the
operator bundles (7), depending on the parameter
have the common point of spectra if and only if
Remark1. If spectrum of each operator bundle in (7) at each fixed meaning of the parameter contains only eigen values then the common point of spectra of these operator bundles is their common eigenvalue.
Remark 2. If the spaces are finite dimensional spaces, then this common point of spectra of (7) is the common eigenvalue of (7).
So we have the operator bundles
, depending on one parameter
and acting in the space
Without loss of generality we suppose
(11)
Denote the operator coefficients of the parameter in the
though
.
All operatorsact in the space
Thus we have operator bundles, acting in the space
and depending on one same parameter:
(12)
Further, the proof is spent by analogy of the proof the Theorem2. Construct the resultants of the obtained operator bundlesand
,
and
,…,
and
The resultants have the form
In the resultant the rows with the operators
are repeated
times, and the rows with the operators
are repeated
times.
Theorem 3. The system of operator bundles (1) have a common points of spectra if and only if
,
,
, all operators, forming the system (1) and
,
are bounded in corresponding spaces.
Proof of the Theorem 3.
Necessity. Let the system (1) has a common point of spectra. Then for fixed first component of
the spectrum of each operator bundle contains only eigen values . From the Theorem1 it follows that
Last means that the system (12) has the eigen value
and there is the non-zero decomposable tensor of the space
that following equations (13)
(13)
are satisfied.
So (13) is fulfilled then the result of theorem2 demands the fulfilling of the condition.
Sufficiency. Let be
,
, operators
and
are bounded.
Let be. Last means there is non-zero element of the space
(direct sum of the
copies of the space
), entering
. Each operator
is the resultant of the bundles
and
From
it follows the (12) are satisfied. Therefore , each equation in (13) means that the first operator bundle with the any other operator bundle has the common eigen value
and common for all operator bundles eigenvector.
is the common eigen value of the system(1) for this fixed
All operators act in the same Hilbert space- direct sum of
+
copies of the space
The condition means that the results of the theorem2 take place for the system (1). Consequently, the system (1) has a common eigen value.
Theorem3 is proved.
5. Nonlinear Algebraic System of Equations with Two Unknown Variables
Consider the algebraic system with two variables, when the number of equations is greater than 2.
(14)
Instead of the spaceswe adopt the space
and the variables
play the role of the parameters
, correspondingly. Numbers
are the bounded operators in the space
.
The operators and
turn out to the form
when
Denote the coefficients at in decomposition
though
So we obtain
polynomials
(14)
Accord to our notations we construct the operators
Operatoris the resultant of polynomials, obtaining as result of decompositions of the
and
.
Criterion of existence of solution of the system (14) is defined by the
Theorem 4. The system of polynomials (12) has a common solutions if and only if ,
,
.
6. Conclusion
In this paper the necessary and sufficient conditions of existence of the common eigenvalue of the two-parameter system of operators in the Hilbert space in the case when the number of operators is greater than the number of parameters are proved. As corollary of this result for the non-linear algebraic system of equations is solved the problem of existence of solutions.
References