Spectral Theory of Operator Pencils in the Hilbert Spaces

Rakhshanda Dzhabarzadeh

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

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(Dzhabarzadeh R. M.)

To cite this article:

Rakhshanda Dzhabarzadeh. Spectral Theory 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. 22-26. doi: 10.11648/j.pamj.s.2015040401.15

Abstract: The theorem on possibility of multiple summation of the series on eigen and associated vectors of the operator pencil in the Hilbert space is proved. Research of multiple completeness and multiple expansions of eigen and associated vectors of such operator pencils are closely connected with the research of differential operator equation with the boundary conditions.

Keywords: Operator Pencil, Residue, Completely Continuous, Multiple

1. Introduction

Keldysh M.V. had proved the fundamental results [1] about the multiple completeness of the system of eigen and associated vectors (e.a) and properties of the eigen values for a wide class of polynomial pencils 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.

It is known that the general theory of equations

 .

where - completely continuous operators, based on the theory of operator pencils

(1)

Keldysh in [1] under the conditions that operators completely continuous; is completely continuous operator of finite order (series of the norms of eigenvalues in some positive order converges),  proved the multiple completeness of the eigen and associated vectors of the operator pencil (1) in the Hilbert space. Later, this result was generalized by many authors in different directions. It should be noted of the works of J.E.Allahverdiev , M.G.Gasymov, A.G.Kostyuchenko, G.Radzievskii and others.

Theorems about multiple expansions of the root subspaces of the operator pencil  are 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 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 under the conditions that the resolvent of operator pencil on closed expanding indefinitely contours uniformly bounded .

Below it is presented a theorem asserting on multiple summation over the root subspaces of the operator pencil, in other words, the possibility of multiple expansions with brackets on eigen and associated vectors of the operator.

2. Preliminary Definitions

Let ,in (1) be completely continuous linear operators, acting in a separable Hilbert space. The operator pencil (1) is known in the spectral theory of operators as a pencil of Keldysh.

Definition1.Eigenvalue  of is called a complex number such that there exists a non-zero element such that This element is called an eigenvector corresponding to the eigenvalue.

Definition2. There is an associated vector to the eigenvector if the following series of equations

(2)

are satisfied.

Keldysh built the derivative systems according to the rule:

(3)

Definition3. [1] Under canonical system of e.a. vectors for eigenvalueof the operator pencil (1) is understood the system

(4)

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 through  linear span of eigen and associated vectors of the system (1), corresponding to an eigenvalue .

Linearly-independent elements (4) form a chain of eigen and associated vectors of (1). The multiplicity of eigenvector is equal to the greatest number of associated vectors to  a plus 1.

The sum  is a multiplicity of an eigenvalue

Definition4. Under -multiple completeness of eigen and associated vectors of the operator pencil in spaceis understood the possibility of approaching anyelementof space by linear combinations of elements, respectively, with the same coefficients independent of the indices of the elements .

If at least for one point of the operator invertible, then the set of eigen values of the pencil consists of isolated points of finite algebraic multiplicity [1].

3. The Spectral Theorem on Multiple Summation of Series on Eigen and Associated Vectors of 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 equationin the direct sum of copies of the space .

Indeed,  can be represented as a system of equations:

Consider the equation

(5)

where

, , .

Operator is the completely continuous in the space so the operators  are completely continuous in the space

Operator is a normal completely continuous , characteristic values of lie on rays emanated from origin, norms of characteristic values of operatorsand coincide.

Theorem. Suppose that the following conditions are satisfied:

a) andare complete continuous operators, exists and bounded

b) there is a sequence of closed contours(circumferences) with radii, such that for all we have

(6)

Then we have the multiple basis of eigen and associated vectors with brackets in the range of the operator. If then the multiple completeness of the system of eigen and associated vectors of the operatorin the spacetakes place

Proof of Theorem.

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

(7)

Suppose the estimate holds for all .

The characteristic numbers of the operator  lie on the rays with the arguments .The equalities

.

 if

then accord to the condition of the theorem

By condition b) of Theorem there exists a family of contours on which norms of the operators satisfy the conditions (6).

Further, and the norms of operator are closed to zero, starting with some .(see[4],p309-310). Indeed, choose the arbitrary little number and represent operator as sum of two operatorsand,where is finite dimensional operator andis the operator arbitrary small norm.

 Let.

Then for sufficient large meanings we have

So the operatoris normal, completely continuous with the zero kernel, then

, where - complete orthonormal sequence of eigen vectors of operator, and -corresponding system of characteristic values of the operator.

For sufficiently large  and number

(8)

Let is the arbitrary element in

Then

In virtue of

and (8) we have

Put Series

converges and

(9)

By condition b) of Theorem there exists a family of contours on which norms of the operators satisfy the conditions

(10)

We introduce the operator by the formula

(11)

Introduce the domains  on the complex plain, bounded by the contours and.The contour, bounded the domain , is denoted . It is known that the meaning of this integrand is equal to the sum of residues respectively of all poles of integrand inside of domain. It is clear the contours may be chosen such manner to be satisfying the condition -of equality of sum all residues of integrand to zero.

Inside the domain bounded by the contour  the integrand has the poles in characteristic numbers of the operator

The residue of integrand in the point in domainis equal to . In the neighborhood of isolated characteristic value of operator the general part of resolution of the operator of  has the form

 .

In  is the canonical system of eigen and associated vectors of the operator pencil ,is 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 isolated characteristic value  residue of integrand is .

Always we may choose the sequence of contours  for which, In this case the sum of all residues of the integrand respectively of all its poles is equal to zero.In particular, residue of integrand , respectively the point 0 is the element .

Indeed, so  then the operator  in the small neighborhood of the point  is expanded in the converging series . The following equalities

+

hold. Substituting the last expansion in the representation of the integrand we define the residue in zero. Thus we have

We obtain 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

The eigen and associated vectors of the operator coincides with the eigen and associated vectors of the operator ,correspondingly. It is not difficult to state that the first components of eigen and associated vector 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 (5) 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

(12)

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  Therefore we proved the possibility of summation of any of elements of the space on the systems (12), correspondingly, with the brackets and the same coefficients.

The theorem is proved.

4. Conclusion

It is proved the convergence of n series on eigen and associated vectors of operator pencil of Keldysh 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,p.15-47
2. Dzhabarzadeh R.M. On expansions series on eigen and associated vectors of operator pencilsJournal: 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 of operator pencils by Abel’s method. ДАN Аz SSR, 1979,т.35, № 7,p 19-23.
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