Multiparameter Operator Systems with Three Parameters

Rakhshanda Dzhabarzadeh¹, Kamilla Alimardanova²

¹Department of functional analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku

²Department of functional analysis of mathematics and Mechanics of NAS of Azerbaijan, Baku

Email address:

(R. Dzhabarzadeh)

(K. Alimardanova)

To cite this article:

Rakhshanda Dzhabarzadeh, Kamilla Alimardanova. Multiparameter Operator Systems with Three Parameters. Pure and Applied Mathematics Journal. Special Issue: Spectral Theory of Multiparameter Operator Pencils and Its Applications. Vol. 4, No. 4-1, 2015, pp. 5-10. doi: 10.11648/j.pamj.s.2015040401.12

Abstract: For the multiparameter system of operators in three parameters the conditions of the existence of multiple basis of eigen and associated vectors in finite dimensional space is proved. The proof of this fact uses essentially the notion of the Resultant of two operator pencils, acting in, generally speaking, in different Hilbert spaces and the criterion of existence of common eigenvalues of several operator pencils, acting in Hilbert spaces.

Keywords: Eigen and Associated Vectors, Finite Dimensional Space, Multiparameter System of Operators, Nonlinear Algebraic System of Equations, Resultant-Operator of Two Pencils

1. Introduction

The founder of researches of spectral problems of the multiparameter selfadjoint systems was F.V. Atkinson [1]. Studied the outcomes which are available for multiparameter symmetrical differential systems, Atkinson has constructed the spectral theory of multiparameter systems in finite dimensional spaces. Further, by means of passage to the limit Atkinson has generalized the received outcomes on a case of multiparameter systems with the selfadjoint compact operators in infinite-dimensional Hilbert spaces. In the further, a design introduced by Atkinson for study of multiparameter systems in finite-dimensional spaces, it has appeared possible to build and in infinite-dimensional spaces that has allowed to construct the spectral theory of multiparameter systems in Hilbert spaces [2],[3] ,etc.

But, unfortunately, the technique of research in these works demands all operators in the system to be selfadjoint. For non- selfadjoint multiparameter systems the investigated technique does not allow to solve the simplest problems of the spectral theory.

The paper is devoted to the study of nonlinear multiparameter system of operators in finite dimensional Hilbert spaces. Previously, for nonlinear algebraic of system of equations was built analog of determinant of Cramer and it is given a necessary and sufficient condition for the existence of solutions of nonlinear algebraic systems with a complex dependence on the variables [4]. For two-parameter systems in the abstract case, it is obtained the results to determine the conditions for finding the number of solutions [6].

In this paper describes the method of determining the conditions on the coefficients of the multiparameter system to establish the existence of its eigenvalues.

Consider a multiparameter system of operators in three parameter of the form.

(1)

when operators (correspondingly,  and ) act in Hilbert space  (correspondingly,  and ).

2. Preliminary definitions and Remarks

Let's reduce a series of known positions from the spectral theory of multiparameter systems

Definition1. [1,2,3]. An operator  (accordingly, ), where  (accordingly,  and ) is identical operators in  (accordingly,  and ), is called the operator, induced in  by the operators  (accordingly,  and ).

Definition 2. [1], [2], [3]  is an eigen value of the system(1) if there are such nonzero  elements   that equations (1) and also the equations

are satisfied, then the decomposable tensor is called the eigenvector of the multiparameter system (1).

Definition 3. [9],[10] A tensor  is called

the-associated vector to an eigenvector  if there is the set of elements  such that the following conditions (2) are satisfied

(2)

 is arrangement from set of the whole nonnegative numbers on 3 with possible recurring and zero.

Remark 1. We gave here the definition of the associated vector for the case when the number of parameters in the system is equal to 3 (in [10] the definition is given for the common case)

Definition 4. [11],[13]

Let

(3)

be two operator pencils depending on the same parameter  and acting, generally speaking, in different Hilbert spaces, correspondingly. Resultant of two operator pencils and  is the operator, presented by the determinant (4) and acting in the space  - direct sum of copies of tensor product  of spaces

(4)

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 matrix  with operators  coincides with the leading degree of parameter  in the operator pencil , that is . One major application of the matrix theory is calculation of determinants. It turns out that a mapping is invertible if and only if the determinant of this matrix is not zero.

This definition is the generalization of the notion of the definition of resultant for two polynomials

Resultant of these polynomials is the operator acting in the space  or  (probably, in some expansion of a field).

In a matrix  the number of rows with coefficients  to equally leading degree of unknown of a polynomial, that is , the number of rows of a matrices with numbers  coincides with the leading degree of unknown  of a polynomial , that is. Continuant of this Resultant is a polynomial from coefficients and equal to zero then and only then when polynomials also have a common roots (probably, in some expansion of a field).

The study of multiparameter system (1) of equations is spent with help of following result from [5]:

3. On common Eigenvalues of Several Operator Bundles

Let the  bundles depending on the same parameter

- operator bundles acting in Hilbert spacecorrespondingly. Suppose that. In the space  (the direct sum of  tensor product  of spaces ) are introduced the operatorswith the help of operational matrices (4).

Let  be the operational bundles acting in a finite dimensional Hilbert space, correspondingly.

(4)

The number of rows with operators  in the matrix  is equal to  and the number of rows with operators  is equal to We designate  the set of eigenvalues of an operator.From [5] we have the result:

Theorem 1.  if and only if

In particular, in [6] the two-parameter system operators

(5)

in tensor product  of finite-dimensional spaces  and is studied.

Dimension of space is the product of dimensions of spaces  and. In (5) linear operators  act in finite-dimensional space; and linear operators  act in finite-dimensional space.

If  and  then inner product of these elements in space  is defined by means of the formulae

(6)

This definition of inner product is spread to other elements of tensor product space  on linearity.

If  is the Hilbert space then the inner product (6) is spread to other elements of on linearity and continuity.

By means of the approach stated in [5,6], we can establish completeness, multiple completeness of system of eigen and associated vectors, a possibility of multiple decompositions on system of eigen and associated vectors of multiparameter system (5).

Theorem 2. [6] Let operators also act in finite-dimensional spaces  and , accordingly, and one of three following conditions is fulfilled:

a) , ;   are self-adjoint operators everyone in their space

b) ,

- self-adjoint operators, acting in the corresponding spaces

c) , , are the self-adjoint operators, acting everyone in finite-dimensional space, correspondingly.

Then the -fold basis of system of eigen and associated vectors of (5) takes place.

At the study of the system (1) we will be used essentially the results of [5,6]:

4. Three Parameter System of Operators in Finite Dimensional Hilbert Spaces

In (1) we fix parameters . Let be . Then the system (1) contains three operator bundles depending on one parameter . Using the result of the theorem 2, we build operators  и .They are the Resultants of operator bundles  and , and also operator bundles  and , correspondingly.

Introduce the notations:

(6)

Let  For the polynomials  and  Resultant has a form

(7)

In the matrices of the operator  the number of rows with  equal toand the number of rows with  is equal to1.

By analogy operator  in the space  is determined with the help of matrices

(8)

In the matrix of the operator  the number of rows with is equal toand the number of rows with is equal to1

Operators  and act in the same space - direct sum of the  copies of tensor product space

Continuants of Resultants and are equal to zero if and only if matrices of operators  and have nonzero kernels.

So at the decompositions of continuants of Resultants and  we obtain very bulky forms, in obtained equations we will operate with the leading degrees of parameters in and. Further we will work only with the members having the leading degrees. We can write the decompositions of continuants of Resultants and in the forms, correspondingly:

(9)

(10)

Expression (9) is the decomposition of the continuant of the Resultant of the operator bundles  and  from (1), when  and. Expression (10) is the decomposition of the continuant of the Resultant of operator bundles  and . Let the couple (, ) is the eigen value of the (9), then from the definition of Resultant follows that at these meanings of parameters andwe have that operator pencils  and  have a common point of spectra . By analogy if the couple (, ) is the eigen value of the (10), then from the definition of Resultant follows that at these meanings of parameters and  we have that operator pencils and  have a common point  of spectra of these operator bundles.

Now we consider the system of equations

(11)

If the  is the eigenvalue of the system (11) and is the corresponding eigen vector of (11) then  So the [8], [9] are decompositions of Resultants and, correspondingly, and (11) is satisfied then the element  is the first component of the element of  and also the eigen vector of the system (1) or its associated vector in definition of which no the differential on and.All the associated vectors of the system (11) are also the associated vectors of the system (1).

Operator pencils (9) and (10) form the nonlinear two parameter system of the kind of the system (5).

We apply the results of the theorem 2 to the system (9), (10).

Theorem 3. Let operators ,  and also act in finite-dimensional spaces ,  and , accordingly, and one of three following conditions is fulfilled:

a), ;  are self-adjoint operator in the space

b) ,

, are the self-adjoint operator, acting in finite-dimensional space , correspondingly.

c)   is selfadjoint operator in the space , Then the eigen and associated vectors of the three parameter system of operators in finite dimensional space form the  multiple basis in .

Proof of the theorem3 uses the approach applying of the proof of the theorem2.We fix one of the parameters of the system (9),(10). For the definite let it is the parameter . Then we have two operator bundles depending on the parameter Construct the resultant of (9) and (10), when parameter  is fixed. It is clear that the decomposition of this resultant is the operator bundle. In fact, the leading degree of the parameter  depends on values of numbersConditions a), b) and c) are only some variants of possibility. If in each of the cases a), b) and c) operator coefficient at the parameter with the greatest degree  is selfadjoint operator with the zero kernel then the system of eigen and associated vectors of this bundle forms the multiple basis in the tensor product space. The multiple of the system of eigen and associated vectors coincides with the leading degree of the parameter . In works [8] and [9] it is proved that the system of eigen and associated vectors of decomposition of resultant coincides with the system of eigen and associated vectors of two parameter system (9),(10).

So the (9) and (10) are also decompositions of corresponding of resultants, we obtain that the system of eigen and associated vectors of (9) and (10) coincides with the system of eigen and associated vectors of the three parameter system (1).

5. Conclusion

In this paper the conditions of existence of multiple basis on eigen and associated vectors of three parameter operator system in finite dimensional spaces are proved.

References

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