Criterion of Existence of Eigen Values of Linear Multiparameter Systems
Rakhshanda Dzhabarzadeh, Elnara Sultanova
^{1}Department of functional analysis of Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku
^{2}Department of Applied Mathematics of Baku State Universiteties, Azerbaijan, Baku
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Rakhshanda Dzhabarzadeh, Elnara Sultanova. Criterion of Existence of Eigen Values of Linear Multiparameter Systems. Pure and Applied Mathematics Journal. Special Issue: Spectral Theory of Multiparameter Operator Pencils and Its Applications. Vol. 4, No. 4-1, 2015, pp. 11-15. doi: 10.11648/j.pamj.s.2015040401.13
Abstract: It is considered the linear multiparameter system of operators when the number of equations may be more than the number of parameters. For such multiparameter systems the authors have proved the criterion of existence of eigen values. Under certain conditions, the authors a have proved that all components of the eigen values of the considered multiparameter systems are real numbers.
Keywords: Operator, Parameter, Eigenvalue, System, Multiparameter
1. Introduction
Spectral theory of operators is one of the important directions of functional analysis. The method of separation of variables in many cases turned out to be the only acceptable, since it reduces finding a solution of a complex equation with many variables to finding a solution of a system of ordinary differential equations, which are much easier to study ..F.V. Atkinson [1] studied the results available for multiparameter symmetric differential systems, built multiparameter spectral theory of selfadjoint systems in finite-dimensional Euclidean spaces. Further, by taking the limit Atkinson generalized the results obtained for the multiparameter systems with self-adjoint operators in finite dimensional space on the case of the multiparameter system with self-adjoint compact operators in infinite-dimensional Hilbert spaces. These multiparameter systems were studied of many mathematicians in the case when the number of parameters is the system is equal to the number of equations. Until recently time in [1],[2],[3] the main requisition to the operators, forming the multiparameter system were to be self-adjoint and bounded. Browne received the spectral expansion of Parseval- Atkinson when operators in (1) may be symmetric unbounded operator.
2. Preliminary Definitions and Remarks
Let be the linear multiparameter system in the form:
(1)
when operators act in the Hilbert space
Definition 1. [1,2,3] is an eigenvalue of the system (1) if there are such non-zero elements s that (1) is satisfied, and decomposable tensor is called the eigenvector corresponding to eigenvalue.
Definition 2.[1,2,3]
The operator is induced into the space by an operator , acting in the space,, if on each decomposable tensor of tensor product spacewe have, on all the other elements of the operator is defined on linearity and continuity.
Definition 3. For the system (1) in [1,2,3] 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 operators are defined with help of the matrices.
(2)
where are arbitrary complex numbers, under the expansion of the determinant means its formal expansion, when the element is the tensor products of elements If, then right side of (2) is equal to , where On all the other elements of the space operators are defined by linearity and continuity. is the identity operator of space. Suppose that for all, and all are selfadjoint bounded operators in the space . Inner product [.,.] is defined as follows; if and are decomposable tensors, then where is the inner product in the space. On all the other elements of the space the inner product is defined on linearity and continuity. In spacewith such a metric all operators are self adjoint.
Spectrum of the system(1)[see[1],[2]) coincides with the joint spectrum of the operators so in this metric operators are self adjoint operators. Moreover, the joint spectrum of operatorscontains only real numbers. In all these works number of parameters is equal to the number of equations.
Consider the multiparameter system of operators when the number of operators in it is more than the number of parameters.
Let be
(3)
And are the self-adjoint bounded operators acting in the Hilbert space,
System (4) differs from system (1) so the number of equations is more than the number of parameters
Let be the operators, induced into the spaceby the operators, correspondingly. So we have the system
(4)
We form of the equations of the system (1) various multiparameter systems consisting of equations and n parameters. If several equations have been left out of groups, supplement them with any of the equations of the system (1). Each group represents a multiparameter system previously studied, since the number of parameters is identical to the number of variables
Let be the groups . Denote the operators entering in -th multiparameter system though .We have the systems
(5)
We will consider each group separately. For each group the following results hold. From ]1],[2] follows that if the operators all operators are self adjoint operator in corresponding spaces then the parameters of each multiparameter systems are separated and we have the following equalities
In fact, that all components of eigenvalues in each multiparameter system are real numbers, so in this metric of the spaceoperators for all are the selfadjoint operators. Consequently, for all meanings of I we have
For all fixed we have n equations on one parameter. Further we use the results of the work [11]. We give some notions, necessary for understanding of
3. The Notion of Resultant’s Analog of Two Operator Pencils
Let
be two operator pencils depending on the same parameters and acting, generally speaking, in various Hilbert spaces, correspondingly.[4,12]
Resultant of two operator pencilsand is the operator, presented by the determinant (2) and acting in the space - direct sum of copies of tensor product of spacesand
(6)
In a matrix the number of rows with operators is equal to leading degree of parameter in the operator pencil, that is, the number of rows in matrices with operators coincides with the leading degree of parameter of the pencil, that is. One major application of the matrix theory is calculation of determinants, a central concept in linear algebra. It turns out that a mapping is invertible if and only if the determinant is non-zero.
The concept of abstract analog of resultant for two operator pencils when they have the identical degrees concerning parameter, has been given in work of Khayniq [9], for operator pencils, generally speaking, with the different degrees of parameter the abstract analog of a Resultant is studied by Balinskii [4].
Let all operators (correspondingly, are bounded in the Hilbert space (correspondingly,) and operator or is invertible.
By [2] follows that the existence non-zero kernel of the operator is the necessary and sufficient conditions for the existing the common point of spectra of operators and, if the spectrum of each operatorand contains only eigen values. In this case a common point of spectra of these operators and is their eigenvalue.
4. Necessary and Sufficient Conditions of Existence of Eigen Values of Several Operator Polynomials
(7)
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 tensor product space .
Introduce the operatorswith help of the operator matrices (5)
(8)
Rows with the operators are repeated time and rows with the operators are repeated time. The operator is induced by an operator acting in the space into the space by the formulae
Denote the set of eigen values of operator.
Theorem 2. [7],[11]. 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
(9)
This result is obtained in [7], [11].
Each group we study separately. Results for any group will be same. Without loss of generality of proof we consider the first group. Further, we assume that all operators are induced in space.
In order not to complicate the entry of a further induced in the space of operators leave designation.Denote though operators, constructing for th multiparameter system on the rules (3). All operators act in the space.
Theorem3. Let all operators in the system are self adjoint and
Then for all multiparameter systems with parameters the following equalities
(10)
Or
(11)
hold.
Proof of the Theorem3
It is clear that the conditions of the theorem3 mean the fulfilling of the conditions of the theorem from the work[1],[2]. In the adopting metric of the tensor space operators are selfadjoint operators in the spaceand, moreover, the separation of the parameters in the form(10) takes place.
Let the inner product [.,.] is defined as follows; if and are decomposable tensors, then where is the inner product in the space On all the other elements of the space the inner product is defined on linearity and continuity In this metric of the spaceoperators are not self adjoint, but all operators are self-adjoint and the formulas (11) take place.
Remark All are real numbers.
Theorem4. Let the conditions of the theorem3 is fulfilled.. Operators
have inverse, spectrum of the bundle (6) contains only eigen values, then the multiparameter system (4) have a common real eigenvalue if and only if
Proof of the Theorem4.
We have systems of operator equations
, or (12)
, or
For any multiparameter system from (5) the separation of parameters holds.
,….,
(13)
,….,
Each system in (12) contains operator equations in one parameter. All operator in (12) and (13) act in the space.
The common eigenvalue of the system (12) when is the first component of the eigen value of the system (4).
The proof at the fact is analogous to the proof of the theorem 2. Construct the resultants of operator equations and, . We have operator equations with one parameter. From the results of the theorem 2 we have that By analogy for other systems in (12) at we have the conditions
Are the necessary and sufficient for the existence of the common eigenvalue of each operator groups.
Condition allows to collect the separate common components of the eigen values of the systems and to obtain the common eigenvalue of the multiparameter system (4).
Remark. If is an eigenvalue of the system (4), then all are real numbers.
5. Conclusion
In this work the necessary and sufficient conditions of existence of eigenvalue of the multiparameter system, when the number of operator equations is more than the number of parameters in it.
References