Pure and Applied Mathematics Journal
Volume 4, Issue 4-1, August 2015, Pages: 27-32

On Multiple Bases of Eigen and Associated Vectors of Operator Pencils in the Hilbert Spaces

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

Rakhshanda Dzhabarzadeh. On Multiple Bases of Eigen and Associated Vectors of Operator Pencils in the 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. 27-32. doi: 10.11648/j.pamj.s.2015040401.16

Abstract: It is proved the theorem about of multiple basis of eigen and associated vectors of the operator pencil, non-linear depending on parameter in the Hilbert space. This work is the generalization of existing results on the multiple completeness of the eigen and associated vectors of polynomial pencils, rationally depending on parameters. At the proof the author uses the methods of spectral theory of operators.

Keywords: Multiple Basis, Eigen And Associated, Residue, Bounded

1. Introduction

Spectral theory of nonselfadjoint operators attracted great attention of mathematicians, physicists, technicians. For a long time the most essential part of this theory remained the study of Hilbert and F.Riss. In theory of nonselfadjoint problems for ordinary differential equations Birkhoff and Tamarkin achieved major successes, based on the study of the analytic properties of the resolvent of the operators. This situation changed only after the publication of the Keldysh’s work.(1951). Keldysh M.V. had proved the fundamental result [1] about the multiple completeness of the system of eigen and associated (e.a.) vectors and properties of the eigen values ​​for a wide class of polynomial pencils, like

(1)

in the Hilbert space. Keldysh in [1] under the conditions that operatorsare completely continuous; is positive completely continuous operator of finite order (series of the norms of eigenvalues in some positive degree converges),  proved the multiple completeness of the eigen and associated vectors of the operator pencil (1) in the Hilbert space. This result allowed studying of boundary problems for partial differential equations and getting strong results for ordinary differential operator systems which can be applied to the study of equations of mathematical physics and differential equations with partial derivatives.

Later, this result was generalized by many mathematicians in different directions. It should be noted of the works of J..E.Allahverdiev, M.G.Gasymov, A.G.Kostyuchenko, G.Radzievskii and others. The problems of existence of two bases of eigen and associated vectors of the operator  when , complex numbers were considered in the [4],[9].. In these works under some additional conditions was proved the existence of two kind of the bases of the eigen and associated vectors of operator.

Later, in the work [7] Allakhverdiev proved the existence of two kind of multiple completeness system of eigen and associated vectors of the operator pencil, rationally depending on parameter.

Theorems about multiple expansions of the root subspaces of the operator pencil were proved in the works of RM Dzhabarzadeh [2], V.N. Vizitey, A.S. Marcus [3] with the proviso that operators are bounded if  and operators are completely continuous if

Through are designated the sequence of the characteristic values of the operator, in order of increasing various modules taking into account their multiplicities.

In the work [6] it is proved the summation of the series on eigen and associated vectors of the operator pencil (1) by the method of Abel.

The work [5] is devoted also to the questions of multiple summation of series on eigen and associated vectors of operator pencil (1) under the conditions that the resolvent of operator pencil on closed expanding indefinitely contours is uniformly bounded .

Below it is presented a theorem asserting the existing of the multiple basis of eigen and associated vectors of the operator pencil

(2)

where ,  We show the summation of the root subspaces of the operator pencil, We remind

Definition1.The sequence of subspacesis called a basis of subspaces if the any vector of the spaceis expanded in the seriesby unique manner.

2. Preliminary Definitions

Let ,,in (2) be completely continuous linear operators, acting in a separable Hilbert space. Each of operator pencils  and  is known in the spectral theory of operators as a pencil of Keldysh.

Definition2.Eigenvalue  of is called a complex number for which there exists a non-zero element such that

This element is called an eigenvector corresponding to the eigenvalue.

Definition3. There is an associated vector to the eigenvector if the following series of equations

(3)

are satisfied.. Keldysh built the derivative systems according to the rule:

(4)

Definition4. [1] Under canonical system of e.a. vectors for

eigenvalueof the operator pencil (1) are understood the

system

(5)

possessing the following properties: elements  form basis of a eigen subspace ; there is  eigenvector which multiplicity reaches a possible maxima;  is  eigenvector which is not expressing linearly though which sum of multiplicities reaches a possible maxima . .

Let's designate though  a subspace spanned by eigen and associated vectors of the system (2), corresponding to an eigenvalue .

Linearly-independent elements form a chain of a set (5) of eigen and associated (e.a) vectors of (1). The multiplicity of eigenvaluedesignates the greatest number of associated vectors to  a plus 1.

The sum  is a multiplicity of an eigen value .

Definition5. Under -multiple completeness of eigen and associated vectors of the operator pencil  in space it is understood the possibility of approaching anyelementof the space by linear combinations of elements, respectively, with accuracy given in advance, with the same coefficients for all elements .

Remark1.[1].If at least for one point of the operator invertible, then the set of eigenvalues ​​of the pencil consists of isolated points of finite algebraic multiplicity ([1] and [4)].

3. The Spectral Theorem on Multiple Basis of Eigen and Associated Vectors of the Operator Pencil

The study of the spectral properties of the equation in a Hilbert space is reduced to the study of the spectral properties of the equations and in the direct sum of copies of the Hilbert space .

The equation

(6)

can be represented as a system of equations:

and the equation ,where

,

.

can be represent as a system of equations

And equation

where

Operators act in direct sum of copies of the Hilbert space

Consider the operator

Operator is the completely continuous in the space(direct sum of copies of the space) so the operators are completely continuous in the space

So the operator is a selfadjoint, completely continuous, then the characteristic values of lie on the n rays emanated from origin, norms of characteristic values of operatorsand coincide. is a normal completely continuous operator.

Theorem. Suppose that the following conditions are satisfied:

a) operators,andare complete continuous;

b) there is the sequence of closed concentric circumferences with radii tending to infinity, in which for all we have

(6)

Then the eigen values of the operator can be grouped so that the subspaces spanned by their eigen and associated vectors corresponding to the eigen values form -multiple basis with brackets in the space. All eigen values are divided on two sequences; one of them has the limit in infinity, other has the limit point in zero.

Proof of Theorem.

We evaluate the resolvent-operator for the values. Letthen.

(7)

The characteristic numbers of the operator  lie on the rays with the arguments.The equalities .  if  So for all  then on the contours  norms of the operators satisfy the conditions (6) for all. Further,  when, (see[4]).

So , then starting with some .

(8)

(9)

By analogy we evaluate the resolvent of the operator when  Denote  then we obtain the equation  where  Then starting with somefor

and

(10)

Denote though the domain of the complex plain bounded by contour and and though the contour, bounded the domain .

We introduce the operator by the formula

(11)

is the border of the of the domain ..It is known that the meaning of this integral is equal to the sum of residues respectively of all poles of integrand, lying within the domain.

Within the domain, bounded by the contours  and , the integrand has the poles in characteristic values of the operator  . Letbe a pole .of the integrand in (11). The general part of the resolution of the operator at the neighborhood of its isolated characteristic valuehas the form

In the representation

(12)

is the canonical system of eigen and associated vectors of the operator pencil,is a canonical system of eigen and associated system of adjoint to operator, product is the operator which on the element is -defined by the rule . In the neighborhood of the isolated eigen value  of the operator the general part of resolution of integrand be and residue of integrand in its polesis

Evaluate the integral

(13)

On the contours we have

In fact it is possible to choose the contourssuch manner that

(14)

(15)

The contours are the border of .It is not difficult to see all poles of integrand in (15), starting with some lie in the domainso they are the characteristic values of the operator Then the poles of integrand of (13) and (15) in the , starting with some coincide. Moreover, the same residues of the corresponding same poles also coincide. We can choose the sequence of contours  for which, thus of the subspaces spanned by eigen and associates vectors, corresponding to the eigen of values from  form the basis with brackets in the space spanned by the eigen and associated vectors starting with sufficiently great index.

Introduce the contours andwhereis the set of numbers whereand is the set of numbers Denote though  the domain bounded by the contoursand and though thethe contour bounded the domain .

The sum of residues of all poles of integrand in (15) coincides with the subspace spanned by the eigen and associated vectors corresponding to all eigen values lying in the domain.

So we have two sequences of eigen values of the operator one sequence of eigenvalues of tends to infinity, and another tends to zero. Denote though  the area of the complex plain, bounded by the contours.and. It is known that in there are the finite number of eigen values of each operatorsand and also.

The point zero does not enter the domain . The sum of all residues of the all poles integrand in the domain contains the finite number of terms .

Thus we have  and, Operator is the project of the on the subspace spanned by the eigen and associated vectors of the operator, lying in the . We add to the considering set of subspaces the residue of the  and the subspace spanned by the eigen and associated vectors, corresponding to all characteristic values from. Last means the existence of n multiple basis in the space  For the definition the residue of integrand in zero in the small neighborhood with the border of the point  consider the integral

(16)

when

We have  and . Substituted the last expression into (16) we obtain

=

The last expansion in the representation of the integrand means that the residue in the point is equal to for Thus we have

-the assertion about possibility of expansion of elements from range of operatoron eigen and associated vectors of the operator . For the completing of the proof of this theorem we need in the statement of connections between the eigen and associated vectors of operators and

It is not difficult to state that the first components of eigen and associated vectors of the operator coincide with the eigen and associated vectors of the equation (1), correspondingly.

If is the system of eigen and associated vectors of (2) thenis the system of eigen and associated vectors of the operator Components  of eigen and associated vectors of the system of the operator are defined with the help of the formulas

where  is the chain of eigen and associated vectors of the operator. So the closure of the range of the operatorcoincides with the whole space, then the completeness of the eigen and associated vectors of the operatorin the space takes place. The last means the multiple completeness of eigen and associated vectors of the pencil  in the Hilbert space . Moreover, element from the range of the operator  is expanded on the system of the eigen and associated vectors of the operator  in the space .So we proved the possibility of multiple expansions on the system of eigen and associated vectors of the operator with brackets for any elements of the space

The Theorem is proved.

4. Conclusion

It is proved the convergence of n series on eigen and associated vectors of operator pencil rationally depending on parameters in the Hilbert space.

References

1. Keldysh M.V. About completeness of eigen functions of some class linear non-selfadjoint operators. Journal:Success of Mathematical Science , 1971, т.27, issue.4, pp.15-47
2. Dzhabarzadeh R.M. On expansions series on eigen and associated vectors of operator pencils, Journal: Scientific notes of Azerb.State University,1964, №3,pp.75-81.
3. Vizitei V.N., Markus A.S.. On convergence of multiple expansions on the system of eigen and associated vectors of polynomial pencils Mathematical collection,1965,т.66, №2,pp..287-320
4. Gokhberg I. Ts., Kreyn M.Q. Introduction to the theory of linear non-selfadjoint operators in the Hilbert space.Moscow, 1964, pp 1-433
5. Allakhverdiev J.E., Dzhabarzadeh R. M. // Spectral theory of operator pencil in the Hilbert space.ДAN of Azerbaijan - 2011, т.LXVII,№ 4.- pр.3-10
6. Allakhverdiev J.E. Dzhabarzadeh R.M. Оn summation of multiple series on eigen and associated vectors operator pensilw by Abel?s method. ДАN Аz. SSR, 1979,т.35, № 7, p 19-23..
7. Allakhverdiev J E. The evolution of resolvents and the theorems on completeness of oerators, depending on spectral parameters. Transactionof AN of Azerb. SSR , seria of physics -tekhnics and mathematical sciences,1074,6,pp.3-36
8. Lidskii. About summation of the series on the general vectors of the non-selfadjoint operators. Proceeding of Moscow Scientific Society, t.11,1962
9. Askerov N.Q.,Kreyn S.Q., Laptev Q.I. On some class of non-selfadjoint boundary problems. DAN, 155, 3 (1964),499-502

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