Spectral Problems of Two-Parameter System of Operators
Rakhshanda Dzhabarzadeh
Department of functional analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan
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To cite this article:
Rakhshanda Dzhabarzadeh. Spectral Problems of Two-Parameter System of Operators. Pure and Applied Mathematics Journal. Special Issue: Spectral Theory of Multiparameter Operator Pencils and Its Applications. Vol. 4, No. 4-1, 2015, pp. 33-37. doi: 10.11648/j.pamj.s.2015040401.17
Abstract: The author has proved the existence of the multiple basis of the eigen and associated vectors of the two parameter system of operators in Hilbert spaces. The proof essentially uses the theorem of the existence of multiple basis of operator bundles and the notion of the abstract analog of resultant of two operator pencils, acting, generally speaking, in different Hilbert spaces. Considerable non-selfadjoint two parameter systems depend on both parameters in a complicated manner.
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.
F.V. Atkinson [1] studied the fragmentary results 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 compact selfadjoint operators in Hilbert spaces.
Later Browne, Sleeman, Roch and other mathematicians built the spectral theory of selfadjoint multiparameter system in infinite-dimensional Hilbert spaces [2],[3].
In this work the existence of multiple basis on eigen and associated vectors of two parameter non-selfadjoint system of operators in 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],[6].
2. Preliminary Definitions and Remarks
Let
(1)
be the non-linear multiparameter system in two parameters.
Operators(correspondingly,) act in the Hilbert space (correspondingly, ),is a tensor product of spacesand.
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.
Definition 1.
is an eigenvalue of the system (1) depending on two spectral parameters if there are such non-zero pair vectors that the equations (2)
(2)
are satisfied .Decomposable element is called an eigen vector of multiparameter system (1).
Definition 2 [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
Definition 3 ([4], [5])
A tensor is named th the associated vector to an eigenvector , if the following conditions (3) are satisfied
(3)
- arrangements from set of the whole nonnegative numbers on 2 with possible recurring and zero.
Under canonical system of e.a.vectors (this definition is generalization of the definition of canonical system, introduced in [10] in the case polynomial pencils in one parameter) for we understand the system
(4)
possessing the following properties: elements form basis of a eigen subspace ; there is eigenvector which multiplicity reaches a possible maxima; there is eigenvector which is not expressing linearly through which sum of multiplicities reaches a possible maxima Let's designate through a subspace spanned by eigen and associated vectors of the system (1), corresponding to an eigenvalue .
Linearly-independent elements form a chain of a set eigen and associated (e.a) vectors of (1). The multiplicity of eigenvalue designates the greatest number of associated vectors to a plus 1.
The sum is a multiplicity of an eigen value .
Elements in (4) form a chain of eigen and associated vectors for every fixed value .
Definition 4.
Systems of elements of Hilbert space form multiple bases in this space if any elements of space can be spread out in series with coefficients, not dependent on an index of vectors. If system coincides with the system of eigen and associated vectors of an operator, and systems are constructed, proceeding from sequences on according to some rules then speak about -multiple basis on system of eigen and associated vectors of an operator
Definition5( [6],[7])
Let be two operator pencils depending on the same parameter and acting in, generally speaking, various Hilbert spaces
Operator is presented by the matrix
(5)
which acts in the- direct sum ofcopies of the space. In a matrix (5) number of rows with operators is equal to leading degree of the parameter in pencilsand 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 [8] for the case of the same leading degree of the parameter in both pencils and in the [7] for, generally speaking, different leading degrees of the parameters in the operator pencils.
Theorem 1.
Let all the operators be 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
(6)
Remark 1. If the Hilbert spacesandare the finite dimensional spaces then a common points of spectra of operator pencilsandare their common eigenvalues.(see [6], [7].)
3. Multiple Basis of Eigen and Associated Vectors of Multiparameter System with Two Parameters
Consider the system (2). Operators act in the Hilbert spacesand ,correspondingly . For study of the spectral properties of the system (1) we use the notion of abstract analog of resultant ofand. Fix the one of the parameters in (1). Let it is the parameter and Then we have two operator pencils in one parameter
(7)
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 pencilthough and in the pencil operator coefficient of the parameter in degree we denote though. Operators, induced into the space, we denoteand, correspondingly
Introduce the notations
= , =
Construct the resultant of operator pencils and (the parameter is fixed).
(8)
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 tthe pencil ,that is
Let realizes at where . Let the greatest degree of in the operator coefficient of in the operator pencil be By analogy the greatest degree of in the operator coefficients at in the operator pencilbe. So the parameteris fixed arbitrarily, further, in the system we miss the index 0 of the parameter.
Let denotes the term of the expansion of the resultant (7) free from parameter and let be a selfadjoint operator with the.
It is known that the self-adjoint completely continuous operator has a discrete spectrum, i.e. the spectrum of the operator contains only eigenvalues of finite multiplicity. If the series of modules of eigenvalues of completely continuous operator in some positive degree converges then this operator belongs to class
Arrange the bundle produced by the decomposition of the resultant (7), to increase the power of the parameter and denote the operator coefficients of though .
Let's remind, that under an eigen value of polynomial bundle is understood such non-zero vector,that equalityis satisfied.. -th associated vector to an eigenvector is the vector, satisfying to conditions
Letbe system of eigen and associated vectors of a bundle.
On system of eigen and associated vectors of polynomial bundle derivative systems of vectors are constructed by rules
,
4. Spectral Properties of Two Parameter System
For the study of the spectral problems of two parameter system (1) we use the following result
In a separable Hilbert space we consider an operator pencil
when the operators, are completely continuous in the space
Let ()be the normal operator, characteristic values of which lie on a finite number of rays, emanating from the origin. Denote an increasing sequence of modules characteristic numbers of the operator, taking into account their multiplicity.
Theorem 2.[5] Suppose that one of the following two conditions:
а), operators are bounded
b) completely continuous operators are satisfied.
Then different eigenvalues ofcan be arranged in a sequence such that for some increasing sequence of positive integers, and for all elements of the spacesatisfying the conditions, the expansions
where the number of different eigenvectors of multiplicity corresponding to eigenvalue with eigenvector
.
-canonical system, corresponding eigen values are satisfied. .
Arrange the bundle produced by the decomposition of the resultant (8), to increase the power of the parameter and denote the operator coefficients of though and the modules of characteristic numbers of operatorsthough, correspondingly. are defined below.
Theorem3. Let be
is selfadjoint completely continuous opertator and one of following conditions are fulfilled
operators, are bounded,
operators are the completely continuous
Then characteristic valuesofcan be arranged in a sequence such that for some increasing sequence of positive integers, and for all elements of the spacesatisfying the conditions, we have the expansions
where the number of different eigenvectors of multiplicity corresponding to eigenvalue .
Proof of the Theorem3.
If any of the conditions a), b) is satisfied then conditions of Theorem 2 are fulfilled, therefore, multiple basis of eigen and associated vectors of the operator bundleexists. Studies conducted in the papers [4],[5] and [6] show that the eigenvectors of the bundle (8)(decompositions of resultant) are either eigenvectors or associated vectors of the system (1) in the definition of which there is no differentiation on. The associated vectors of the bundle (8) (decomposition of the resultant (7)) are also eigen- vectors of the system (1). The results of the theorem1 mean if the conditions a), b) of the theorem 2 hold 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 ) . Really, each condition a) , b) of the theorem 2 means the. Consider the decomposition of the resultant (8) which is the operator pencil in the parameter Consequently, eigen and associated vectors of this operator pencil form the multiple basis in the tensor product space and the multiplicity of this basis coincides with the greatest degree of the parameter in the operator pencil (the parameter is fixed) . Earlier in the [4],[5],[6] it is proved that the system of eigen and associated vectors of obtained pencil, depending on parameterand coincides with the system of eigen and associated vectors of the system (1).Therefore, the eigen and associated vectors of the two parameter system (1) form the multiple basis in. Thus, the systems of eigen and associated vectors of (1) and the resulting expansion of the resultant simultaneously multiple basis with brackets.
Theorem4
Let be >, operator has inverse, selfadjoint completely continuous and one of following conditions
are bounded operators
are completely complete operators
is satisfied. Then the assertions of theorem 3 is fulfilled.
Theorem 5. Let be=, operator has inverse, selfadjoint completely continuous and one of following conditions
are bounded
completely continuous operators is satisfied. Then the statements of the theorem 3 is fulfilled.
The next theorem is the special case of the theorem2
Theorem6. Let all operators(correspondingly,act in finite dimensional Hilbert space (correspondingly, ), and one of following conditions:
a) ,
b) >, operator has inverse and selfadjoint
c) =,operator + has inverse and selfadjoint
is satisfied. Then the eigen and associated vectors of the system (2) form multiple basis in the tensor product of the space .
Remark1. Eigen and associated vectors are the first components of the elements of the kernel of the resultant of operator pencilsand
Remark2. If the is infinite dimensional Hilbert space then the system of eigen and associated vectors of the system (1) coincides with the system of eigen and associated vectors of the operator pencil obtaining as result of decomposition of the resultant of operators
Theorems 4,5,6 are proved by analogy with the proof of the Theorem 3.
5. Conclusion
In this paper it is given the conditions of multiple basis of eigen and associated vectors of two parameter system of operators in Hilbert space.
References