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DESCRIPTION OF THE POINT SPECTRUM OF A 3x3 TRIDIAGONAL OPERATOR MATRIX WITH FREDHOLM OPERATORS Merajov N.I.1, Rasulov T.H.2 (Republic of Uzbekistan) Email: [email protected]
'Merajov Nursaid Ikrom o 'g 'li — Student; 2Rasulov Tulkin Husenovich — PhD in Mathematics, Head of Department, DEPARTMENT OF MATHEMATICS, BUKHARA STATE UNIVERSITY, BUKHARA, REPUBLIC OF UZBEKISTAN
Abstract: we consider a bounded self-adjoint 3 X 3 tridiagonal operator matrix T acting in the direct sum of three identical Hilbert spaces of square-integrable functions on [—ft, ft] . We analyze
the case where the matrix elements of T are Fredholm operators of rank '. The point spectrum of T are described. It is established that the number zero is an eigenvalue of infinite multiplicity. The Fredholm determinant whose zeros are eigenvalues of T is constructed. The number of the eigenvalues of T with finite multiplicity is identified.
Keywords: the Fredholm operator, tridiagonal operator matrix, point spectrum, eigenvalue, multiplicity, Fredholm determinant.
ОПИСАНИЕ ТОЧЕЧНОГО СПЕКТРА ТРИДАГОНАЛЬНОГО 3Х3 ОПЕРАТОРНОЙ МАТРИЦЫ С ФРЕДГОЛЬМСКИМИ ОПЕРАТОРАМИ Меражов Н.И.1, Расулов Т.Х.2 (Республика Узбекистан)
'Меражов Нурсаид Икром угли — студент; 2Расулов Тулкин Хусенович — кандидат физико-математических наук, заведующий кафедрой,
кафедра математики, Бухарский государственный университет, г. Бухара, Республика Узбекистан
Аннотация: рассматривается ограниченная самосопряженная 3 X 3 тридиагональная операторная матрица T , действующая в прямой сумме трех одинаковых гильбертовых пространств квадратично интегрируемых функций в [—ft, ft] . Мы анализируем случай,
когда матричные элементы оператора T являются фредгольмскими операторами ранга '.
Описан точечный спектр оператора T . Установлено, что число ноль есть бесконечнократное собственное значение. Построен определитель Фредгольма, нули которого
являются собственными значениями оператора T . Определено число конечнократных
собственных значений оператора T .
Ключевые слова: оператор Фредгольма, тридиагональная операторная матрица, точечный спектр, собственное значение, кратность, определитель Фредгольма.
The essential and discrete spectra of operator matrices [1] and Schroedinger operators on a lattice (see for example, [2-4]) are the most actively studied objects in operator theory. In both cases crucial role is played the Faddeev type system of integral equations (or matrix equation whose entries are the Fredholm operators) for eigenvectors. The construction and some spectral properties of the Faddeev type equations for the eigenvectors of the operator matrices in the cut subspaces of the standard Fock space are studied in many works, see for example [5-23]. In [5-13] using the Faddeev operator the location of the essential spectrum of operator matrices are studied and in [14-23] using such type operators the finiteness or infiniteness of the number of eigenvalues of operator matrices are investigated. In the present paper we a bounded self-adjoint 3 X 3 tridiagonal operator matrix and analyze the case where its matrix elements are Fredholm operators of rank 1.
Let ¿2[—ft', ft] be the Hilbert space of square integrable (complex) functions defined on [—ft, ft] and
l23)[-^]:= {/ /2, /3): fae L2[—a = 1,2,3}.
For the elements f = (/1, /2, /3 ) of the space lL2^ [—ft, ft] the norm is given by
1
=1 s h/k (t)i2 dt 12
k=1 -h
We consider a tridiagonal 3 x 3 operator matrix T acting in the Hilbert space Z2 [—ft; ft] as
fTii Tl2 0 1
T := * T12 T22 T23
v 0 * T23 T33 y
with the entries T : L2 [—h; h] ^ L2 [—h; h], i, j = 1,2,3, | i — j | < 1:
(Tjfj )(x) = j (x) J tj (s)f (s)ds, * < j ,
— ft
where the functions tj (•) are real-valued continuous functions on [—ft; ft]. Here,
j
T* (i < j) denotes the adjoint operator to Tj and
A h
(Tjjf )(x) = tj(x) J tj,(s)f(s)ds, f eL2[—h;h].
—h
Under these assumptions the operator T is bounded and self-adjoint in the Hilbert space
I'
'2
L23)[ — h;h]
Generally, study of arbitrary linear operators in infinite-dimensional spaces is a very complicated problem. However, some important classes of such operators can be described completely. One of such classes is so called compact operators. These operators are closed to finite-dimensional ones with respect to their properties also play very important role in many applications such as the Theory of
Integral Operators. Since all matrix elements Tij are the Fredholm integral operators of rank 1, the
'.J
operator T is a compact operator.
The first main result of the paper is the following theorem.
Theorem 1. The number a = 0 is an eigenvalue of T with multiplicity infinity. If for all j = 1,2,3 the non-zero function fj (•) g ft; ft] is an orthogonal to the functions tj (•) ,
i = 1,2,3, then the vector-function (f, f2, f3 ) is an eigenvector corresponding to the eigenvalue A = 0 of the operator T .
Denote by || • || and (•,•) the norm and scalar product in Z^t—■ft; ft], respectively. To formulate next main result of the present paper we define the following matrix valued function:
A(-):= det (Aj (o)T=f
where matrix elements Aj (A) are defined by
ЛП(Я):=Я— ||t„||2, А12(Я):—(fn,t2x);
А22(Я) := Я, Л2Э(Я) := —1| t121|2, Л24(Я) := —(fo, ^2), Л25(Я):=—(t12=> t32);
Аэ1(Я) := —(tn,t21), Л32(Я) := —1| ||2, Л33Я) := Я;
Л43(Я) := —fe t22), Л44(Я) := Я— || ^22 || , Л45(Я) := —(^22,^32);
Л55Я) := Я, Л5б(Я) := —1| t^B ||2, Л5у(Я) := —fe, 'ээ);
Лб3(Я):=— (t32,t12), Лб4(Я):=—(¿32,^22), Лб5(Я):= —1|?32||2,
Лбб(Я):=Я;
Л7б(Я) := —(t33,t23), Луу(Я) := Я—1| ^32 ||2; Л у (Я) = 0, otherwise.
Usually the function Л() is called the Fredholm determinant corresponding to the operator matrix T .
The spectrum, the point spectrum, the continuous spectrum, the essential spectrum and the discrete spectrum of a bounded self-adjoint operator will be denoted by <t("), &pp ('), Jcont() ,
°'ess О , and Jdisc (0, respectively. Second main result is the following assertion.
Theorem 2. The operator T has a purely point spectrum and for the point spectrum (Jpp (T) of T the equalities
j(T) = Jpp (T) = {0} u {Я e R: Л(Я) = 0}
hold.
From Theorem 2 one can conclude that J cont (T) = ^. By the definition the function
Л(') is a polynomial function of degree 7 with respect to Я . Therefore, the function Л(') has at most 7 zeros (counting with multiplicities), and hence by Theorem 2, these zeros are the discrete eigenvalues of the self-adjoint operator matrix T .
Remark. For the essential spectrum Jess (T) of T and the discrete spectrum Jdisc (T)
of T
we have
Jess (T) = {0}, Jdisc (T) = {Я E R: Л(Я) = 0}.
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