Research Methods of Multiparameter System in Hilbert Spaces
Rakhshanda Dzhabarzadeh
Department of Functional Analysis of Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan
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To cite this article:
Rakhshanda Dzhabarzadeh. Research Methods of Multiparameter System in Hilbert Spaces. Pure and Applied Mathematics Journal. Special Issue: Spectral Theory of Multiparameter Operator Pencils and Its Applications. Vol. 4, No. 4-1, 2015, pp. 38-44. doi: 10.11648/j.pamj.s.2015040401.18
Abstract: The work is devoted to the presentation of the methods, available in the literature, of the study of multiparameter spectral problems in Hilbert space. In particular, the method of Atkinson and his followers for a purely self-adjoint multiparameter systems and methods proposed by the author for the study, in general, non- selfadjoint multiparameter system in Hilbert space. These approaches solve questions of completeness, multiple completeness, the basis and a multiple basis property of eigen and associated vectors of multiparameter systems with a complex dependence on the parameters.
Keywords: Method, Atkinson, Multiparameter Systems, Basis, Complete
1. Introduction
Spectral theory of operators is one of the important directions of functional analysis. The physical sciences open more and more challenges for mathematical researches. In particular, the resolution of the problems associated with the physical processes and, consequently, the study of partial differential equations and mathematical physics equations required a new approach. 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 of a solution of a system with 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 ultra-parabolic equations, etc.
Despite the urgency and prescription studies, spectral theory of multiparameter systems was not enough investigated? The available results in this area until recently only dealt with systems of selfadjoint multiparameter operator, linearly depending on the spectral parameters.
F.V. Atkinson is the founder of the research of the spectral multiparameter system of operators. Atkinson [1] studied the results available for multiparameter symmetric differential systems, built multiparameter spectral theory in Euclidean spaces. Further, by taking the limit Atkinson summarized these results on the case of multiparameter systems with self-adjoint compact operators in infinite-dimensional Hilbert spaces.
Later, method of investigation, introduced by Atkinson to study multiparameter systems in finite-dimensional spaces is used by Browne, Sleeman and others for constructing the spectral theory of selfadjoint multiparameter system in Hilbert spaces [2], [3], etc.
2. Some Aspects of the Method of Separation of Variables: Abstract Analog of a Separation of Variables for the Operational Equations
In the works of Morse, Feshbach [4], Roche [5], Sleeman [6] are paid the great attention to detail shown in the essence of the emergence and development of a multiparameter system as a result of applying of the method of separation of variables in the partial differential equations and equations of mathematical physics.
The method of separation of variables is applied to various boundary value problems, when the initial conditions are considered as a special case of the border
[4], in particular, contain a series of coordinate systems in which the equations admit separation of variables. Equations which may be separated have the special form .and they can be written as:
where is the tensor product of the spaces. are unity operators of.
It is interesting that all solutions of partial differential equations can be obtained by linear combinations of family members separated solutions.
Let be linear operators acting in Hilbert space, and let be the tensor product of spaces. Suppose that an equation is given in the form
(1)
Theorem 1. [7] For every decomposable tensor , that is solution of (1), there exist complex numbers such, that
For the proof of the Theorem1 we use the famous property of the elements of tensor product space. It is known that the representation of the element in tensor product space is not unique. For each element of tensor product space there is the number coinciding with the minimal number of decomposable tensors, necessary for the representation of this element. This number is named the rank of element. If the number of decomposable tensors in the representation of element are more than the rank of element then one-nominal components of decomposable tensors in the representation of the given element are linear dependent
3. Selfadjoint Multiparameter Systems
Research method in spectral theory of self adjoint multiparameter systems.
The first works on spectral theory of multiparameter system of operators were the investigations of Faierman[8] concerning to the system of differential equations. Faierman [8] considered the system
(2)
when , are continuous ,real valued and differentiable on the interval of real axis.
System (2) with the common boundary conditions
(3)
is the parameter Sturm-Luiville system.
If condition forin the systems (2) and (3) are satisfied then their eigen values belong to and do not have a finite limit point.
Let be the set of non-negative integers, then there is exactly one eigenvalue of the problem (2) and (3), for which the function has exactlyzeroes in the interval of .
Assuming and are differentiable functions Faerman [8] also proves that the eigenfunctions of the problem (2) and (3) form a complete orthogonal system with respect to the weight function, where the Cartesian product of intervals.
Later Browne established this result without conditions of differentiable of functions
A brief proof of the fundamental theorem of Browne gives research methods for study of selfadjoint multiparameter systems
For presenting of the method of investigation of multiparameter systems we present some results in this direction.
Famous results in the spectral theory of selfadjoint multiparameter systems:
Let
(4)
be -parameter system
Operators are bounded and selfadjoint in the space . For any each set of elements determinant , where (.,.) is the inner product in.
Operators are defined as follows: let be a decomposable tensor inandbe an arbitrary complex numbers. Then are determined by equation
(5)
where the determinant can be extended to the whole space with the help of the tensor product.
is determined on the decomposable tensor of the spacewhen in the (5) with help of(5) and on all other elements of the space is defined on linearity and continuity.
The inner productis given by the expression.The norms, induced by these inner products are equivalent, and thus topological concepts as continuity of the operators and the convergence of sequences of elements are equal with respect to these standards. Further, we will denote the operator.
Theorem 2. [1,2]. Suppose, then .
Theorem3. ([1],[2]).Suppose, the inner productis given by the expression,then = are selfadjont operators.
Operators are projection operators, as operators are mutual commute. Thus, we adopt the as the spectral measure on the Borel subsets of the space, which carrier is set, and for each pair of elements has a function with complex valued Borel measure, turns to zero out The type of measures are nonnegative, essentially finite Borel measures, vanishing outside
The spectrum of the system is defined in ([1].[2]) as a vehicle operator-valued measure. Then there is a compact subset of measures and indeed fade out If there is a Borel function defined on, and we can define the operator as follows:
and to have for an arbitrary
If is a bounded function, then . If is unbounded and has a dense domain, the details of these results can be found in [17] E. Prugoveĉku.
Definition 1. Operator is named by operator, induced to space by and is constructed by following:: on decomposable tensor of and on other elements ofoperatoris defined on linearity and continuity.
Theorem 4 ([1], [2]) For each there are elements such that, the operators; are mutual commute.
4. Non-Selfadjoint Multiparameter System of Operators
We research the multiparameter system
(6)
In (6) operators act in separable Hilbert spaceand bounded.
Definition 2. ([1]], [2])is eigen value of multiparameter system (6) (see [1,2]), if there are such nonzero elements that equalities (6) are fulfilled. Vector named eigenvector of the system (6), corresponding to eigen value.
Definition 3. ([8],[9]).Let be an eigenvector of the system (6), corresponding to its eigen value; the is - th associated vector to an eigenvector of the system (6) if there is a set of vectors , satisfying to conditions
.
Indices are the various arrangements of set of integers on with . ; .
Under canonical system e.a. elements we understand system
(7)
having the following properties: elements form base of an 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 for everyone fixed value of number .
Elements from (7) form a chain of e.a. elements at everyone fixed value .The sum - we name a multiplicity of an eigen value .
We introduce the necessary definitions and notions for the statement of the main results of the spectral theory of nonselfadjoint multiparameter system in Hilbert space.
Some positions play the essential role in the investigation of multiparameter system of operators
Definition 4. ([10],[11]). Let be two polynomial bundles
(8)
depending on the same parameters and acting, generally speaking, in various spaces, accordingly. The operator is given by matrix
(9)
In [10],[11] operator is named by abstract analog of a Resultant for polynomial bundles (8).
In definition of a Resultant (9) of bundles (8) the rows with operators are repeated times, and rows with оperators are repeated exactly times. , there are the highest degrees of parameter in bundles of and , accordingly. Thus, the Resultant (9) is an operator, acting in space that is a direct sum of copies of tensor product spaces Value of Resultant is equal to its formal expansion when each term of this expansion is tensor product of operators. Let all operators (correspondingly, are bounded in the Hilbert space (correspondingly,) and operator or is invertible.
By [10],[11] it 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 then a common point of spectra of these operators and is their eigenvalue.
Let now we have the bundles depending on the same parameter
(10)
are operational bundles with the discrete spectrum, acting in a Hilbert space, accordingly. Without loss of generality we shall suppose, that. We shall introduce operators which act in space (the direct sum of copies of tensor-product of spaces and these operators are defined by means of operational matrixes
(11)
We shall designate the set of eigen values of an operator.
Theorem 5 [12]. Let the operator has inverse and operatorshave discrete spectrums. Then in the only case if
A decisive result in the spectral theory of nonselfadjoint multiparameter systems began with the following Theorem 5. If the number of parameters in (6) is equal 2 with the help of the abstract resultant of two operator bundles in one parameter (the other parameter is fixed) we obtain one equation in one parameter in the tensor product of the Hilbert spaces. The obtained equation is studied. It is proved that the eigen subspace of this pencil coincides with the subspace spanned by eigen and associated vectors of kindor(index 0 stands on the place of the fixed parameter). Associated vectors of the pencil are also the associated vectors of the system (6). The author proves that the system (6) and the obtained the pencil have the common system of eigen and associated vectors.
Theorem 6 [9]. Let in (6). Operators and , eigenvectors and associated vectors of an operator at any fix meanings of parameter form basis in Hilbert space; operators are bounded in corresponding spaces. Then the eigen and associated vectors on the direction is a linear combination of the elements of an aspect
where (accordingly, there is a restricted chain of e.a. vectors of an operator (accordingly, corresponding to some common eigen value of both operators and
Let's designate through a subspace spanned by eigenvectors and associated vectors of system (6), corresponding to an eigenvalue.
Linearly-independent elements from the set form a chain of eigenvector and associated (e.a) vectors
Theorem 7. [8],[9].. Let following conditions satisfies: operatorsare bounded for all meanings and in space, the operator exists and bounded. Then system of e.a. vectors (6) coincides with e.a. vectors of each of operators.
The proof .when in (6). We fix parameters. Then we have the operational bundles linearly depending on parameter. From Theorem 5 it follows, that bundles have a common eigenvalue if and only if under conditions, that one of operators (at fixed n) exists and bounded . Existence (not limiting a generality, we suppose, that ) follows from existence and bounded of an operator . As the result of use the Theorem 5 we obtain equation with the parameters. The main operator for the obtained exists and bounded. Continue this process, at last we have the one equation in one parameter in the tensor product space Further, it is proved that all eigenvalues and the systems of eigen and associated vectors of original , intermediate and last multiparameter systems, presenting in the process of proof theorem, coincide
Theorem 7. [18] Let the eigen and associated vectors of operatorin (6),when any of parameters are fixed, form a basis in space,. If …,is the chain of e.a. vectors of an operator on parameter,corresponding to its eigen value, then associated vectors on direction of (6) is the linear span of the sum of various combinations of decomposable tensors , for which ,
5. Some Approaches to the Investigation of the Multiparameter Systems of Operators Complicated Depending on Parameters in the Hilbert Spaces
Below we give two approaches by which we study multiparameter systems with complex dependence on the parameters.
For the example we consider
(12)
when bounded operators in a separable Hilbert space.
Definition 5. Elements
are named the associated vectors to the eigen vector of the system (12) if the following conditions
are satisfied. If for some meanings , then element. Linear independent elements of form the chain of eigen and associated vectors of multiparameter system (12), corresponding to the eigenvalue .
One of methods of investigation of the system(12) is the method of converting into a linear multiparameter system of type (6)
Introduce the notations
and add system (12) by following equation (13)
(13)
where operators are set by means of matrixes
(14)
If the is the eigenvalue of the system [(12),(13)] and is the corresponding eigenvector, rhen on the eigenvalues and the corresponding eigenvectors of the system (12) and (13) the equations (13) are realized connections between parameters according to requirements of system (12).
((12), (13)), considered together, form the multiparameter system consisting of the equations and containing parameters.
To obtained system all procedures on investigations under certain conditions by the method of Atkinson and by the method offered by author are possible. The main goal of this approach is to reduce the investigation of complicated system (12) to the investigation of linear multiparameter system. On the eigenvalues of linear system we have, …,, … ,, If the, all operators , forming the system selfadjoind, then under the additional conditions Browne ‘s theorem states the existence and a reality of a spectrum of a multiparameter system [(12),(13)]. If the operators in (12) are bounded but the system [(12),(13)] is not selfadjoint, we can apply the results of the work [9], or [8].
Such approach allows solving more complex multiparameter systems containing products of parameters. In this case the additional equations contain non- selfadjoint operator giving with the help of matrixes. And it is possible the applying only the results of [8],[9].
Second approach of investigation of n-multiparameter system is to use the criterion of existence of common point of spectra of several polynomial pencils, acting, generally speaking, in different Hilbert spaces. Fixed all parameters in the system besides one parameter we obtain several operator pencils, depending on one parameter. With help of criterion (Theorem5) we come the multiparameter system in which number of parameters is equal to . Continue this process we at last obtain the one equation in the tensor product of spaces with the one parameter. Further we proved the coincidence of the system of eigen and associated vectors considering multiparameter system and the last equation.
This method allows solving the problems when the number of equations in the multiparameter system is greater than the number of parameters.
The special case of the multiparameter system is the nonlinear algebraic systems. In the case the complicated nonlinear algebraic systems with many variables with help of these two approaches we find the number of solutions and prove the reality of solutions [13], [15],,ets.
6. Conclusion
The methods of investigations of non-self- adjoint multiparameter system are stated.
References