Multiparameter System of Operators with Two Parameters in Finite Dimensional Spaces
Rakhshanda Dzhabarzadeh, Afet Jabrailova
Department of functional analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku
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Rakhshanda Dzhabarzadeh, Afet Jabrailova. Multiparameter System of Operators with Two Parameters in Finite Dimensional Spaces. Pure and Applied Mathematics Journal. Special Issue: Spectral Theory of Multiparameter Operator Pencils and Its Applications. Vol. 4, No. 4-1, 2015, pp. 1-4. doi: 10.11648/j.pamj.s.2015040401.11
Abstract: The authors have proved the existence of the multiple basis on the eigen and associated elements of the two parameter system of operators in finite dimensional spaces. The proof uses the notion of the abstract analog of resultant of two operator pencils, acting, generally speaking, in different Hilbert spaces. In this paper necessary and sufficient conditions of the existence of multiple completeness of the eigen and associated vectors of two parameter system of operators in finite dimensional Hilbert space is given.
Keywords: Multiparameter, Spectrum, Operator, Space, Eigenvector
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. For example, a multivariable problems cause problems in quantum mechanics, diffraction theory, the theory of elastic shells, nuclear reactor calculations, stochastic diffusion processes, Brownian motion, boundary value problems for equations of elliptic-parabolic type, the Cauchy problem for ultraparabolic equations and etc.
Despite the urgency and prescription studies, spectral theory of multiparameter systems was not enough studied. The available results in this area until recently only dealt with systems of self- adjoint multiparameter system of operators, linearly dependent on the spectral parameters.
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 selfadjoint compact operators in infinite-dimensional Hilbert spaces.
In the further Browne, Sleeman, Roch and other mathematicians built the spectral theory of selfadjoint multiparameter system in infinite-dimensional Hilbert spaces[2],[3]
In particular, in this work the existence of multiple basis on eigen and associated vectors of two parameter non-selfadjoint system of operators in finite-dimentional Hilbert spaces is proved. Definitions of the associated vectors, multiple completeness of eigen and associated vectors of two-parameter non-selfadjoint systems, are introduced in [4],[5].
At the proof of these results we essentially use the notion of the analog of an resultant of two polynomial bundles [6],[7].
(1)
when operator (correspondingly, acts in Hilbert space correspondingly, )
is the tensor product of the spaces and .
Let
(2)
be the nonlinear multiparameter system in two parameters. Operators (correspondingly, ) act in the the Euclidean space (correspondingly, ), is a tensor product of spaces and .
For nonlinear algebraic system with two variables sufficient conditions of existence of solutions are given. The proof of these statements are received as a corollary of more common reviewing considered in this paper.
2. Preliminary Definitions and Remarks
Definition 1[1]
is an eigenvalue of the system (2) depending on two spectral parameters if there are such non-zero vectors that the equations
are satisfied. Decomposable element is called an eigen vector of multiparameter system (2)
Definition2 [1]
Operator (correspondingly ) is induced into the space by the operator (correspondingly ), acting in the space (correspondingly, ), if the following conditions satisfy : on decomposable tensor,
and
on other elements of the space -on linearity and continuity
A tensor is called the th associated vector to an eigenvector if the following conditions (3) satisfy.
(3)
are arrangements from set of the whole nonnegative numbers on 2 with possible recurring and zero.
Definition 4
Systems of elements of finite-dimensional space form multiple bases in this space if any elements. of space can be spread out in series with coefficients, not depending on indices of the vectors . If system coincides with the system of eigen and associated vectors of an operator, and systems are constructed, proceeding from sequences on to some rules speak about -fold bases on system of eigen and associated vectors of an operator
Let be two operator pencils depending on the same parameter and acting in, generally speaking, in various Hilbert spaces
Operator is presented by the matrix
(4)
which acts in the - direct sum ofcopies of the space In a matrix (4). number of rows with operators is equal to leading degree of the parameter in pencils and the number of rows with is equal to the leading degree of parameter in Notion of abstract analog of resultant of two operator pencils is considered in the [7] for the case of the same leading degrees of the parameter in both pencils and in the [6], generally speaking, for different degrees of the parameters in the operator pencils.
Theorem 1
Let operators (correspondingly, ) are bounded in corresponding Hilbert spaces, one of operators or has bounded inverse Then operator pencils and have a common point of spectra if and only if
(5)
Remark1. If the Hilbert spaces and are the finite dimensional spaces then a common points of spectra of operator pencils and are their common eigenvalues. (see [6], [7].)
3. Nonlinear Nonselfadjoint Multiparameter System in Two Parameters
Consider the system (2). Operators act in the finite dimensional Hilbert spaces and , correspondingly . For study of the spectral properties of the (2) we use the notion of abstract analog of resultant of and .
Fix the one of the parameters in (2). Let it is the parameter and Then we have two operator pencils in one parameter
Arrange the pencils on increasing of the degree of the parameter and denote the operator coefficients of the parameter in the degree in the operator pencil though and in the pencil operator coefficient of the parameter in degree we denote though .
These operators, induced into the space , we denote and , correspondingly
Introduce the notations
,
Construct the resultant of operator pencils and (the parameter is fixed).
(6)
The number of rows with the operators is equal to the leading degree of the parameter in the operator pencil, that is; number of rows with the operators is equal to the leading degree of the parameterin the pencil , that is
Let realizes at and also and where . Let the highest degree of in the operator coefficient at in the operator pencil be By analogy the highest degree of in the operator coefficients at in the operator pencil be . So the parameter is fixed arbitrarily, further in the system we miss the index 0 of the parameter ^{。}
4. Spectral Properties of Two Parameter System
Theorem2. Let all operators (correspondingly, act in finite dimensional Hilbert space (correspondingly)
and one of the following conditions:
a),
b) >, operator has inverse and selfadjoint
c) , operator
has inverse and selfadjoint are fulfilled. Then the eigen and associated vectors of the system (2) form multiple basis in the tensor product of the spaces .
Proof of the Theorem 2
Using the results of the theorem1 we have that under the conditions of the theorem2 operator pencils in (1) have the common point of spectra (in the finite dimensional Hilbert space the common point of spectra is the common eigenvalues of operators). It is not difficult to see, that under the conditions of the theorem 2 the. In fact, decomposition of the (6) is the operator pencil in the parameter The spacehas a finite dimension and in each case operator coefficient of the leading degree of the parameterin the obtained in the result of the decomposition of the resultant is selfadjoint operator. Consequently, eigen and associated vectors of the obtained pencil form the corresponding to leading degree of the parameter fold basis in the tensor product space . Earlier in the [5],[6] is proved that the system of eigen and associated vectors of obtained pencil, depending on parameter,coincides with the system of eigen and associated vectors of the system (2).
Remark1.Besides of these eigen and associated vectors are the first components of the elements of resultant of operator pencils and , when the parameter is the first component of the eigenvalue of the system (2).
Remark 2. If the is infinite dimensional Hilbert space then the system of eigen and associated vectors of the system (2) coincides with the system of eigen and associated vectors of the operator pencil obtaining as result of decomposition of the resultant of operators and ,when the parameter is fixed. Thus, the systems of eigen and associated and vectors of (2) and the resulting expansion of the resultant simultaneously fold complete or fold up basis in the space at the same time.
5. The Nonlinear Algebraic System in Two Variables
The following result for the algebraic system of equations
(7)
Proving in the work [8]is the corollary of the theorem2.
Really. instead of the spaces and we adopt the space , parameters play the role of variables
Let and one of the following conditions
a) ,
b),
c) ,
are satisfied then the algebraic system (7) has not less than the solutions.
6. Conclusion
In this paper the conditions of existence of multiple basis of eigen and associated vectors of the system (2) in the finite dimensional tensor product space are proved.
References