On Existence of Eigen Values of Several Operator Bundles with Two Parameters
Makhmudova Malaka Gasan^{1}, Sultanova Elnara Bayram^{2}
^{1}Department of Higher Mathematics, Baku State University, Baku, Azerbaijan
^{2}Department of Applied Mathematics, Baku State University, Baku, Azerbaijan
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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 whenis 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 spacewe have and on all the other elements of the spaceoperator 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 оperatorsrepeated 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 parameterWithout 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 operatorsare 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 ifand 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 elemententers 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 spaceentering 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 parameterThe greatest degree of parameterin 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 spaceand 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 repeatedtimes.
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 valueand 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 andare 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 andFromit 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 spaceand the variables play the role of the parameters, correspondingly. Numbersare the bounded operators in the space.
The operators andturn 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 theand.
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