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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(R. Dzhabarzadeh)

(A. Jabrailova)

To cite this article:

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

Definition3 ([4], [5])

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

Definition5 ([6], [7])

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

1. Atkinson F.V. Multiparameter spectral theory. Bull.Amer.Math.Soc.1968, 74, 1-27.
2. Browne P.J. Multiparameter spectral theory.  Indiana Univ. Math. J,24, 3, 1974
3. Sleeman B.D. Multiparameter spectral theory in Hilbert space. Pitnam Press, London, 1978, p.118.
4. Dzhabarzadeh R.M. Spectral theory of two parameter system in finite-dimensional space. Transactions of NAS Azerbaijan, v. 3-4 1998, p.12-18
5. Dzhabarzadeh R.M. Spectral theory of multiparameter system of operators in Hilbert space, Transactions of NAS of Azerbaijan, 1-2, 1999, 33-41
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