Monashova A.Z.

L.N.Gumilyov Eurasian National University, Astana

THE SECOND ORDER NON SELF ADJOINT OPERATOR AHD ITS COMPACT PERTURBATION

We study the spectrum of non-self adjoint operator , which we defined in Hilbert space. In particular case it is useful to represent operator like the sum , where is a positive self adjoint operator in and is compact perturbation with respect to operator [1].

In this paper it is studied the following operator

, , (1)

where , are complex-valued local-integrable functions in .
We suppose that domain of operator : and the operator is generated by the boundary condition .

We represent the function in following way where and are the real imaginary paits, respectively.

Further we also assume that the following conditions hold:

I. The function is semi-bounded below function: and for almost all .

II. For every : .

III. .

Let be the Friedrechs extension of the following semi-bounded below minimal operator

if ,

where is the space of infinitely differentiable and finite functions in .

For every we define

.

Let be the complement of with respect to norm .

We consider Otelbaev function

, (2)

where .

The following limitary equalities hold:

, . (3)

We define . It is know that , ,

where [2].

Now we define the following negative norm

, (4)

Let be the complement of with respect to norm (4).

Theorem 1. The operator , if by continuity to a compact operator .

References

1. Lions J.-L., Magenrs E. Non homogeneous boundary value problems and their application. М.:Mir.-1971

2. Otelbaev М.О., Mynbaev К.Т. Weighted functional spaces and the spectrum of differential operator. М.:Nauka.-1988