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. Ì.:Mir.-1971
2. Otelbaev Ì.Î., Mynbaev Ê.Ò. Weighted
functional spaces and the spectrum of differential operator. Ì.:Nauka.-1988