Subdivision of spectra for some lower triangular double-band matrices as operators on $c_0$
The generalized difference operator $\Delta_{a,b}$ was defined by El-Shabrawy: $\Delta{a,b}x = \Delta_{a,b} (x_n) = (a_nx_n + b_{n-1}x_{n-1})^{\infty}_{n = 0}$ with $x_1 = b_1 = 0$, where $(a_k), (b_k)$ are convergent sequences of nonzero real numbers satisfying certain conditions. We completely det...
Збережено в:
| Дата: | 2018 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2018
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/1605 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507420789309440 |
|---|---|
| author | Durna, N. Дурна, Н. |
| author_facet | Durna, N. Дурна, Н. |
| author_sort | Durna, N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:20:38Z |
| description | The generalized difference operator $\Delta_{a,b}$ was defined by El-Shabrawy: $\Delta{a,b}x = \Delta_{a,b} (x_n) = (a_nx_n + b_{n-1}x_{n-1})^{\infty}_{n = 0}$
with $x_1 = b_1 = 0$, where $(a_k), (b_k)$ are convergent sequences of nonzero real numbers satisfying certain conditions.
We completely determine the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator
$\Delta_{a,b}$ in the sequence space $c_0$. |
| first_indexed | 2026-03-24T02:09:02Z |
| format | Article |
| fulltext |
UDC 517.9
N. Durna (Cumhuriyet Univ., Sivas, Turkey)
SUBDIVISION OF SPECTRA FOR SOME LOWER TRIANGULAR
DOUBLE-BAND MATRICES AS OPERATORS ON \bfitc \bfzero
ПIДРОЗДIЛ СПЕКТРIВ ДЛЯ ДЕЯКИХ НИЖНЬО-ТРИКУТНИХ
ДВОРЯДКОВИХ МАТРИЦЬ ЯК ОПЕРАТОРIВ НА \bfitc \bfzero
The generalized difference operator \Delta a,b was defined by El-Shabrawy: \Delta a,bx = \Delta a,b (xn) = (anxn + bn - 1xn - 1)
\infty
n=0
with x - 1 = b - 1 = 0, where (ak) and (bk) are convergent sequences of nonzero real numbers satisfying certain conditions.
We completely determine the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator
\Delta a,b in the sequence space c0.
Узагальнений рiзницевий оператор \Delta a,b було визначено Ель-Шабравi: \Delta a,bx = \Delta a,b (xn) = (anxn + bn - 1xn - 1)
\infty
n=0
при x - 1 = b - 1 = 0, де (ak) , (bk) — збiжнi послiдовностi ненульових дiйсних чисел, що задовольняють деякi умо-
ви. Повнiстю визначено наближений точковий спектр, дефектний спектр та стискувальний спектр оператора \Delta a,b
у просторi послiдовностей c0.
1. Introduction. Spectral theory is an important part of functional analysis. It has numerous appli-
cations in many parts of mathematics and physics including matrix theory, function theory, complex
analysis, differential and integral equations, control theory and quantum physics. For example, in
quantum mechanics, it may determine atomic energy levels and thus, the frequency of a laser or the
spectral signature of a star.
In numerical analysis, matrices from finite element or finite difference problems are often banded.
Such matrices can be viewed as descriptions of the coupling between the problem variables; the
bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances.
Such matrices can be further divided — for instance, banded matrices exist where every element in
the band is nonzero. These often arise when discretizing one-dimensional problems.
Problems in higher dimensions also lead to banded matrices, in which case the band itself also
tends to be sparse. For instance, a partial differential equation on a square domain (using central
differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension,
but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or
equivalently an LU decomposition) to such a matrix results in the band being filled in by many
nonzero elements. And so, the resolvent set of the band operators is important for solving such
problems.
Many problems in computational physics can be reduced to linear algebra problems. In this
laboratory you will use several fundamental techniques of computational linear algebra to solve
physics problems common in many different areas of science. In order to solve the problem with
some variations you will need to solve a system of linear equations by Gauss – Jordan Elimination,
“LU decomposition plus back substitution,” matrix inversion and matrix diagonalization.
In recent years, spectral theory has witnessed an explosive development. There are many types
of spectra, both for one or several commuting operators, with important applications, for example the
approximate point spectrum, Taylor spectrum, local spectrum, essential spectrum etc.
c\bigcirc N. DURNA, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7 913
914 N. DURNA
1.1. The spectrum. Let X and Y be the Banach spaces, and L : X \rightarrow Y also be a bounded
linear operator. By R(L), we denote the range of L, i.e.,
R(L) = \{ y \in Y : y = Lx, x \in X\} .
By B(X), we also denote the set of all bounded linear operators on X into itself. If X is any
Banach spaces and L \in B(X), then the adjoint L\ast of L is a bounded linear operator on the dual X\ast
of X defined by
\bigl(
L\ast f
\bigr)
(x) = f(Lx) for all f \in X\ast and x \in X.
Let L : D(L) \rightarrow X be a linear operator, defined on D(L) \subset X, where D(L) denotes the domain
of L and X is a complex normed linear space. For L \in B(X) we associate a complex number \lambda
with the operator (\lambda I - L) denoted by L\lambda defined on the same domain D(L), where I is the identity
operator. The inverse (\lambda I - L) - 1 , denoted by L - 1
\lambda is known as the resolvent operator of L\lambda .
A regular value of L is a complex number \lambda of L such that L - 1
\lambda exists, is bounded and, is
defined on a set which is dense in X.
The resolvent set of L is the set of all such regular values a of L, denoted by \rho (L,X). Its
complement is given by \BbbC \setminus \rho (L;X) in the complex plane \BbbC is called the spectrum of L, denoted
by \sigma (L,X). Thus the spectrum \sigma (L,X) consist of those values of \lambda \in \BbbC , for which L\lambda is not
invertible.
The spectrum \sigma (L,X) is union of three disjoint sets as follows: the point (discrete) spectrum
\sigma p(L,X) is the set such that L - 1
\lambda does not exist. Further \lambda \in \sigma p(L,X) is called the eigen value of
L. We say that \lambda \in \BbbC belongs to the continuous spectrum \sigma c(L,X) of L if the resolvent operator
L - 1
\lambda is defined on a dense subspace of X and is unbounded. Furthermore, we say that \lambda \in \BbbC
belongs to the residual spectrum \sigma r(L,X) of L if the resolvent operator L - 1
\lambda exists, but its domain
of definition (i.e., the range R(\lambda I - L) of (\lambda I - L) is not dense in X; in this case L - 1
\lambda may
be bounded or unbounded. Together with the point spectrum, these two subspectra form a disjoint
subdivision
\sigma (L,X) = \sigma p(L,X) \cup \sigma c(L,X) \cup \sigma r(L,X) (1)
of the spectrum of L.
Also the spectrum \sigma (L,X) is partitioned into three sets which are not necessarily disjoint as
follows:
we call a sequence (xk)k in X a Weyl sequence for L if \| xk\| = 1 and \| Lxk\| \rightarrow 0 as k \rightarrow \infty .
We call the set
\sigma ap(L,X) := \{ \lambda \in \BbbC : there exists a Weyl sequence for \lambda I - L\}
the approximate point spectrum of L. Moreover, the subspectrum
\sigma \delta (L,X) := \{ \lambda \in \BbbC : \lambda I - L is not surjective\}
is called defect spectrum of L. There exists another subspectrum,
\sigma co(L,X) = \{ \lambda \in \BbbC : R(\lambda I - L) \not = X\}
which is often called compression spectrum in the literature. Clearly, \sigma p(L,X) \subseteq \sigma ap(L,X) and
\sigma co(L,X) \subseteq \sigma \delta (L,X). Moreover, comparing these subspectra with those in (1) we note that
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
SUBDIVISION OF SPECTRA FOR SOME LOWER TRIANGULAR DOUBLE-BAND MATRICES . . . 915
\sigma r(L,X) = \sigma co(L,X)\setminus \sigma p(L,X)
and
\sigma c(L,X) = \sigma (L,X)\setminus [\sigma p(L,X) \cup \sigma co(L,X)].
Sometimes it is useful to relate the spectrum of a bounded linear operator to that of its adjoint.
Building on classical existence and uniqueness results for linear operator equations in Banach spaces
and their adjoints.
Proposition 1 ([4], Proposition 1.3). The spectra and subspectra of an operator L \in B(X) and
its adjoint L\ast \in B(X\ast ) are related by the following relations:
(a) \sigma (L\ast , X\ast ) = \sigma (L,X),
(b) \sigma c(L
\ast , X\ast ) \subseteq \sigma ap(L,X),
(c) \sigma ap(L
\ast , X\ast ) = \sigma \delta (L,X),
(d) \sigma \delta (L
\ast , X\ast ) = \sigma ap(L,X),
(e) \sigma p(L
\ast , X\ast ) = \sigma co(L,X),
(f) \sigma co(L
\ast , X\ast ) \supseteq \sigma p(L,X),
(g) \sigma (L,X) = \sigma ap(L,X) \cup \sigma p(L
\ast , X\ast ) = \sigma p(L,X) \cup \sigma ap(L
\ast , X\ast ).
1.2. Goldberg’s classification of spectrum. If T \in B(X), then there are three possibilities for
R(T ):
(I) R(T ) = X,
(II) R(T ) = X, but R(T ) \not = X,
(III) R(T ) \not = X
and three possibilities for T - 1 :
(1) T - 1 exists and continuous,
(2) T - 1 exists but discontinuous,
(3) T - 1 does not exist.
If these possibilities are combined in all possible ways, nine different states are created. These
are labelled by: (I1), (I2), (I3), (II1), (II2), (II3), (III1), (III2), (III3). If an operator is in state (III2) for
example, then R(T ) \not = X and T - 1 exists but is discontinuous (see [12]).
If \lambda is a complex number such that T = \lambda I - L \in (I1) or T = \lambda I - L \in (II1), then \lambda \in \rho (L,X).
All scalar values of \lambda not in \rho (L,X) comprise the spectrum of L. The further classification of
\sigma (L,X) gives rise to the fine spectrum of L. That is, \sigma (L,X) can be divided into the sub-
sets (I2)\sigma (L,X) = \varnothing , (I3)\sigma (L,X), (II2)\sigma (L,X), (II3)\sigma (L,X), (III1)\sigma (L,X), (III2)\sigma (L,X),
(III3)\sigma (L,X). For example, if T = \lambda I - L is in a given state, (III2) (say), then we write \lambda \in
\in (III2)\sigma (L,X).
By the definitions given above, we can write Table 1.
By w, we shall denote the space of all real or complex valued sequences. Any vector subspace
of w is called a sequence space. We shall write \ell \infty , c, c0 and bv for the space of all bounded,
convergent, null and bounded variation sequences, respectively. Also by \ell 1, \ell p, bvp we denote
the spaces of all absolutely summable sequences, p-absolutely summable sequences and p-bounded
variation sequences, respectively.
Several authors have studied the spectrum and fine spectrum of linear operators defined by some
particular limitation matrices over some sequence spaces. We summarize the knowledge in the
existing literature concerned with the spectrum and the fine spectrum. The fine spectrum of the
Cesaro operator on the sequence space \ell p for 1 < p < \infty has been studied by Gonzalez [13].
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
916 N. DURNA
Table 1
1 2 3
Possibility L - 1
\lambda exists L - 1
\lambda exists L - 1
\lambda
and is bounded and is unbounded does not exists
\lambda \in \sigma p(L,X)
I R(\lambda I - L) = X \lambda \in \rho (L,X) —
\lambda \in \sigma ap(L,X)
\lambda \in \sigma c(L,X) \lambda \in \sigma p(L,X)
II R(\lambda I - L) = X \lambda \in \rho (L,X) \lambda \in \sigma ap(L,X) \lambda \in \sigma ap(L,X)
\lambda \in \sigma \delta (L,X) \lambda \in \sigma \delta (L,X)
\lambda \in \sigma r(L,X) \lambda \in \sigma r(L,X) \lambda \in \sigma p(L,X)
III R(\lambda I - L) \not = X \lambda \in \sigma \delta (L,X)
\lambda \in \sigma ap(L,X) \lambda \in \sigma ap(L,X)
\lambda \in \sigma \delta (L,X) \lambda \in \sigma \delta (L,X)
\lambda \in \sigma co(L,X) \lambda \in \sigma co(L,X) \lambda \in \sigma co(L,X)
Also, Wenger [19] examined the fine spectrum of the integer power of the Cesaro operator over c,
and Rhoades [16] generalized this result to the weighted mean methods. Reade [15] worked the
spectrum of the Cesaro operator over the sequence space c0. The spectrum of the Rhaly operators
on the sequence spaces c0 and c is studied by Yildirim [17] and the fine spectrum of the Rhaly
operators on the sequence space c0 is studied by Yildirim [18]. In the last year, several authors
have investigated spectral divisions of generalized differance matrices. For example, Akhmedov and
El-Shabrawy [1, 2] have studied the spectrum and fine spectrum of the generalized lower triangle
double-band matrix \Delta v over the sequence spaces c0, c and \ell p, where 1 < p < \infty . The fine spectrum
of the difference operator \Delta over the sequence spaces \ell 1 and bv is investigated by Kayaduman and
Furkan [14] and c0 and c, is investigated by Altay and Başar [3] etc.
The above-mentioned articles, concerned with the decomposition of spectrum which defined by
Goldberg. However, in [8] Durna and Yildirim have investigated subdivision of the spectra for
factorable matrices on c0 and in [6] Basar, Durna and Yildirim have investigated subdivisions of the
spectra for genarilized difference operator over certain sequence spaces.
2. The fine spectrum of the operator \bfDelta \bfita ,\bfitb on \bfitc \bfzero . The generalized difference operator \Delta a,b
has been defined by El-Shabrawy [9]. Let (ak) and (bk) are two convergent sequences of nonzero
real numbers satisfying
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
ak = a > 0 and \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
bk = b \not = 0. (2)
We consider the operator \Delta a,b : c0 \rightarrow c0, which is defined as follows:
\Delta a,bx = \Delta a,b (xk) = (akxk + bk - 1xk - 1)
\infty
k=0 with x - 1 = b - 1 = 0.
It is easy to verify that the operator \Delta a,b can be represented by a lower triangular double-band matrix
of the form
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
SUBDIVISION OF SPECTRA FOR SOME LOWER TRIANGULAR DOUBLE-BAND MATRICES . . . 917
\Delta a,b =
\left(
a0 0 0 . . .
b0 a1 0 . . .
0 b1 a2 . . .
...
...
...
. . .
\right) .
Note that, if (ak) and (bk) are constant sequences, say ak = r \not = 0 and bk = s \not = 0 for all k \in \BbbN ,
then the operator \Delta a,b is reduced to the operator B (r, s) and the results for the subdivisions of the
spectra for generalized difference operator \Delta a,b over c0, c, \ell p and bvp have been studied in [6].
2.1. Subdivision of the spectrum of \bfDelta \bfita ,\bfitb on \bfitc \bfzero . If T : c0 \rightarrow c0 is a bounded linear operator
with matrix A, then it is known that the adjoint operator T \ast : \ell 1 \rightarrow \ell 1 is defined by the transpoze of
the matrix A. It is well known that the dual space c\ast 0 of c is isomorphic to \ell 1.
The spectra and the fine spectra of the operator \Delta a,b over the sequence space c0 has been studied
by El-Shabrawy [10]. In this subsection we summarize the main results.
Theorem 1 ([10], Theorem 2.1). Let D = \{ \lambda \in \BbbC : | \lambda - a| \leq | b| \} and E =
\bigl\{
ak : k \in \BbbN ,
| ak - a| > | b|
\bigr\}
. Then \sigma (\Delta a,b, c0) = D \cup E.
Theorem 2 ([10], Theorem 2.2). \sigma p (\Delta a,b, c0) = E \cup K, where
K =
\Biggl\{
aj : j \in \BbbN , | ak - a| = | b| ,
\infty \prod
i=m
bi - 1
aj - ai
diverges to zero for some m \in \BbbN
\Biggr\}
.
Theorem 3 ([10], Theorem 2.3). \sigma p
\bigl(
\Delta \ast
a,b, c
\ast
0
\bigr)
=
\bigl\{
\lambda \in \BbbC : | \lambda - a| < | b|
\bigr\}
\cup E \cup H, where
H =
\Biggl\{
\lambda \in \BbbC : | \lambda - a| = | b| ,
\infty \sum
k=0
\bigm| \bigm| \bigm| \bigm| \bigm|
k\prod
i=0
\lambda - ai
bi
\bigm| \bigm| \bigm| \bigm| \bigm| < \infty
\Biggr\}
.
Theorem 4 ([10], Theorem 2.5). \sigma r (\Delta a,b, c0) = \{ \lambda \in \BbbC : | \lambda - a| < | b| \} \cup (H\setminus K) .
Theorem 5 ([4], Theorem 2.6). \sigma c (\Delta a,b, c0) = \{ \lambda \in \BbbC : | \lambda - a| = | b| \} \setminus H.
Lemma 1 ([12], Theorem II 3.11). The adjoint operator T \ast is onto if and only if T has a
bounded inverse.
Lemma 2 ([12], Theorem II 3.7). A linear operator T has a dense range if and only if the
adjoint operator T \ast is one-to-one.
Lemma 3. If \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty ak = a \not = 1 for all k \in \BbbN where ak \not = 0 for all k \in \BbbN , then the product\prod
k
ak is divergent.
Lemma 4. For p, r \in \BbbN ,
\infty \sum
n=p
\Biggl(
n - r\sum
k=r
akbnk
\Biggr)
=
\infty \sum
k=r
ak
\Biggl( \infty \sum
n=p
bnk
\Biggr)
,
where (ak) and (bnk) are nonnegative real numbers and p \geq 2r.
Proof. We have
\infty \sum
n=p
\Biggl(
n - r\sum
k=r
akbnk
\Biggr)
=
p - r\sum
k=r
akbpk +
p+1 - r\sum
k=r
akbpk +
p+2 - r\sum
k=r
akbpk +
p+3 - r\sum
k=r
akbpk + . . . =
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
918 N. DURNA
= (arbpr + ar+1bp,r+1 + ar+2bp,r+2 + . . .+ ap - rbp,p - r)+
+ (arbp+1,r + ar+1bp+1,r+1 + ar+2bp+1,r+2 + . . .+ ap+1 - rbp+1,p+1 - r)+
+ (arbp+2,r + ar+1bp+2,r+1 + ar+2bp+2,r+2 + . . .+ ap+2 - rbp+2,p+2 - r) + . . . =
= ar (bpr + bp+1,r + bp+2,r + . . .) + ar+1 (bp,r+1 + bp+1,r+1 + bp+2,r+1 + . . .)+
+ar+2 (bp,r+2 + bp+1,r+2 + bp+2,r+2 + . . .) + . . . =
= ar
\infty \sum
n=p
bnr + ar+1
\infty \sum
n=p
bn,r+1 + ar+2
\infty \sum
n=p
bn,r+2 + . . . =
=
\infty \sum
k=r
ak
\Biggl( \infty \sum
n=p
bnk
\Biggr)
.
Theorem 6. (III1)\sigma (\Delta a,b, \ell p) = \{ ak : k \in \BbbN , | ak - a| < | b| \} \cup (H\setminus K) .
Proof. Let we investigate whether the operator (\lambda I - \Delta a,b)
\ast = \lambda I - \Delta \ast
a,b is surjective or not.
Does there exist x \in \ell 1 for y \in \ell 1 such that (\lambda I - \Delta \ast
a,b)x = y? If for y \in \ell 1, then (\lambda I - \Delta \ast
a,b)x = y
and we get
(\lambda - a0)x0 - b0x1 = y0,
(\lambda - a1)x1 - b1x2 = y1,
(\lambda - a2)x2 - b2x3 = y2,
. . . . . . . . . . . . . . . . . . . . . . . . . . .
(\lambda - an)xn - bnxn+1 = yn,
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Therefore, we obtain
x1 =
\lambda - a0
b0
x0 -
1
b0
y1,
x2 =
\lambda - a1
b1
x1 -
1
b1
y1 =
\lambda - a0
b0
\lambda - a1
b1
x0 -
1
b0
\lambda - a1
b1
y0 -
1
b1
y1,
x3 =
\lambda - a2
b2
x2 -
1
b2
y2 =
\lambda - a0
b0
\lambda - a1
b1
\lambda - a2
b2
x0 -
- 1
b0
\lambda - a1
b1
\lambda - a2
b2
y0 -
1
b1
\lambda - a2
b2
y1 -
1
b2
y2,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hence, we get
xn = x0
n - 1\prod
k=0
\lambda - ak
bk
+
n - 1\sum
k=1
yk - 1
bk - 1
n - 1\prod
i=k
\lambda - ai
bi
+
yn - 1
bn - 1
, n \geq 2.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
SUBDIVISION OF SPECTRA FOR SOME LOWER TRIANGULAR DOUBLE-BAND MATRICES . . . 919
Now, we must show that x \in \ell 1. That is, is the series
\sum \infty
n=0
| xn| covergent? We obtain
\infty \sum
n=0
| xn| = | x0| + | x1| +
\infty \sum
n=2
| xn| \leq
\leq | x0| + | x1| + | x0|
\infty \sum
n=2
\bigm| \bigm| \bigm| \bigm| \bigm|
n - 1\prod
k=0
\lambda - ak
bk
\bigm| \bigm| \bigm| \bigm| \bigm| +
\infty \sum
n=2
\bigm| \bigm| \bigm| \bigm| \bigm|
n - 1\sum
k=1
yk - 1
bk
n - 1\prod
i=k
\lambda - ai
bi
\bigm| \bigm| \bigm| \bigm| \bigm| +
\infty \sum
n=2
\bigm| \bigm| \bigm| \bigm| yn - 1
bn - 1
\bigm| \bigm| \bigm| \bigm| .
Let
\sum
1
=
\infty \sum
n=2
\bigm| \bigm| \bigm| \bigm| \bigm|
n - 1\prod
k=0
\lambda - ak
bk
\bigm| \bigm| \bigm| \bigm| \bigm| , \sum
2
=
\infty \sum
n=2
\bigm| \bigm| \bigm| \bigm| \bigm|
n - 1\sum
k=1
yk - 1
bk
n - 1\prod
i=k
\lambda - ai
bi
\bigm| \bigm| \bigm| \bigm| \bigm| , \sum
3
=
\infty \sum
n=2
\bigm| \bigm| \bigm| \bigm| yn - 1
bn - 1
\bigm| \bigm| \bigm| \bigm| .
Since \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty bk = b \not = 0 from (2), \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty
1
bk
=
1
b
and so
\biggl(
1
bk
\biggr)
is bounded. Hence, since
there exists M > 0 such that
\bigm| \bigm| \bigm| \bigm| 1bn
\bigm| \bigm| \bigm| \bigm| \leq M for all n \in \BbbN , (3)
the series
\sum
3
=
\infty \sum
n=2
\bigm| \bigm| \bigm| \bigm| yn - 1
bn - 1
\bigm| \bigm| \bigm| \bigm| \leq M
\infty \sum
n=2
| yn - 1| \leq M\| y\| \ell 1
is convergent.
If | \lambda - a| < | b| , then \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty
\bigm| \bigm| \bigm| \bigm| \lambda - ak
bk
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \lambda - a
b
\bigm| \bigm| \bigm| \bigm| < 1 \not = 1 and the product
\prod
k
\lambda - ak
bk
is
divergent from Lemma 3. Hence for \lambda \in \sigma r (\Delta a,b, c0) , the series
\sum
1
is covergent if and only if \lambda \in
\in (H\setminus K)\cup \{ ak : k \in \BbbN , | ak - a| < | b| \} . Now, let we investigate the series
\sum
2
to be convergent. If
\lambda \in \{ ak : k \in \BbbN , | ak - a| < | b| \} , then it is clear that the series
\sum
2
is convergent. Let \lambda \in (H\setminus K) .
Then, from (3) and triangle inequality, we get
\sum
2
=
\infty \sum
n=2
\bigm| \bigm| \bigm| \bigm| \bigm|
n - 1\sum
k=1
yk - 1
bk
n - 1\prod
i=k
\lambda - ai
bi
\bigm| \bigm| \bigm| \bigm| \bigm| \leq M
\infty \sum
n=2
\Biggl[
n - 1\sum
k=1
| yk - 1|
\bigm| \bigm| \bigm| \bigm| \bigm|
n - 1\prod
i=k
\lambda - ai
bi
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggr]
.
Therefore, if we take p = 2, r = 1, ak = | yk| and bnk =
\bigm| \bigm| \bigm| \bigm| \prod n - 1
i=k
\lambda - ai
bi
\bigm| \bigm| \bigm| \bigm| in Lemma 4, we have
\sum
2
\leq M
\infty \sum
k=2
| yk - 1|
\infty \sum
n=1
\bigm| \bigm| \bigm| \bigm| \bigm|
n - 1\prod
i=k
\lambda - ai
bi
\bigm| \bigm| \bigm| \bigm| \bigm| .
Since \lambda \in H,
\sum \infty
n=1
\bigm| \bigm| \bigm| \bigm| \prod n - 1
i=k
\lambda - ai
bi
\bigm| \bigm| \bigm| \bigm| is covergent. Setting L :=
\sum \infty
n=1
\bigm| \bigm| \bigm| \bigm| \prod n - 1
i=k
\lambda - ai
bi
\bigm| \bigm| \bigm| \bigm| , we obtain
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
920 N. DURNA
\sum
2
\leq ML
\infty \sum
k=2
| yk| \leq ML\| y\| \ell 1
and so
\sum
2
is covergent. That is, for \lambda \in \sigma r (\Delta a,b, c0) , the operator (\lambda I - \Delta a,b)
\ast is surjective
if and only if \lambda \in \{ ak : k \in \BbbN , | ak - a| < | b| \} \cup (H\setminus K) . Hence from Lemma 1, \lambda I - \Delta a,b has
bounded inverse.
Corollary 1. (III2)\sigma (\Delta a,b, c0) = \{ \lambda \in \BbbC : | \lambda - a| < | b| \} \setminus \{ ak : k \in \BbbN , | ak - a| < | b| \} .
Proof. It is clear from Theorems 4 and 6, since
(III2)\sigma (\Delta a,b, c0) = \sigma r (\Delta a,b, c0) \setminus (III1)\sigma (\Delta a,b, c0) .
Theorem 7 ([10], Theorem 2.8). (III3)\sigma (\Delta a,b, \ell p) = E \cup K.
Corollary 2. (I3)\sigma (\Delta a,b, c0) = (II3)\sigma (\Delta a,b, c0) = \varnothing dir.
Proof. Since from Table 1, \sigma p(\Delta a,b, c0) = (I3)\sigma (\Delta a,b, c0) \cup (II3)\sigma (\Delta a,b, c0) \cup (III3)\sigma (\Delta a,b,
c0) = E \cup K and (I3)\sigma (\Delta a,b, c0) \cap (II3)\sigma (\Delta a,b, c0) \cap (III3)\sigma (\Delta a,b, c0) = \varnothing the proof is finished
from Theorem 7.
Theorem 8. (a) \sigma ap (\Delta a,b, c0) = (D \cup E) \setminus [\{ ak : k \in \BbbN , | ak - a| < | b| \} \cup H] ,
(b) \sigma \delta (\Delta a,b, c0) = D \cup E,
(c) \sigma co (\Delta a,b, c0) = \{ \lambda \in \BbbC : | \lambda - a| < | b| \} \cup H \cup E.
Proof. (a) It is clear from Theorems 1 and 6.
(b) It is clear from Theorem 1 and Conclusion 2, since from Table 1, \sigma \delta (\Delta a,b, c0) =
= \sigma (\Delta a,b, c0) \setminus (I3)\sigma (\Delta a,b, c0) .
(c) Since from Table 1, \sigma co(\Delta a,b, c0) = (III1)\sigma (\Delta a,b, c0)\cup (III2)\sigma (\Delta a,b, c0)\cup (III3)\sigma (\Delta a,b, c0) =
= \sigma r(\Delta a,b, c0) \cup (III3)\sigma (\Delta a,b, c0), the proof is finished from Theorems 4 and 7.
Corollary 3. (a) \sigma ap
\Bigl(
\Delta \ast
a,b, \ell 1
\Bigr)
= D \cup E,
(b) \sigma \delta
\bigl(
\Delta \ast
a,b, \ell 1
\bigr)
= (D \cup E)\setminus
\bigl[
\{ ak : k \in \BbbN , | ak - a| < | b| \} \cup H
\bigr]
.
Proof. It is clear from Theorem 8 and Proposition 1 (c) and (d).
3. Remarks and some special cases. In recent years, some special cases of the operator \Delta a,b
has been studied. These special cases are related to sequences (ak) and (bk) . In here, we give some
cases:
If we take ak = 1 and bk = - 1 for all k \in \BbbN in the operator \Delta a,b, then it reduces to the
backward difference operator \Delta which has been studied in [7].
If we take ak = r and bk = s for all k \in \BbbN in the operator \Delta a,b, then it reduces to the generalized
difference operator B (r, s) which has been studied in [6].
If we take ak = - bk = vk for all k \in \BbbN in the operator \Delta a,b, then it reduces to the generalized
difference operator \Delta v which has been studied in [5].
If (ak) is a sequence of positive real numbers such that ak \not = 0 for all k \in \BbbN with \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty ak =
= U \not = 0 and (bk) is either constant or strictly decreasing sequence of positive real numbers with
\mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty bk = V \not = 0 and \mathrm{s}\mathrm{u}\mathrm{p}k ak < U + V, then the operator \Delta a,b reduces to the generalized
difference operator \Delta uv which has been studied in [11].
Remark 1. If (ak) and (bk) are convergent sequences of nonzero real numbers such that
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
ak = a > 0 and \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
bk = b, | b| = a,
and
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
SUBDIVISION OF SPECTRA FOR SOME LOWER TRIANGULAR DOUBLE-BAND MATRICES . . . 921
\mathrm{s}\mathrm{u}\mathrm{p}
k
ak \leq a and b2k \leq a2k for all k \in \BbbN ,
then we can prove that H = \varnothing and so we have
\sigma ap (\Delta a,b, c0) =
= (\{ \lambda \in \BbbC : | \lambda - a| \leq | b| \} \cup \{ ak : k \in \BbbN , | ak - a| > | b| \} ) \setminus \{ ak : k \in \BbbN , | ak - a| < | b| \} ,
\sigma \delta (\Delta a,b, c0) = \{ \lambda \in \BbbC : | \lambda - a| \leq | b| \} \cup \{ ak : k \in \BbbN , | ak - a| > | b| \} ,
\sigma co (\Delta a,b, c0) = \{ \lambda \in \BbbC : | \lambda - a| < | b| \} \cup \{ ak : k \in \BbbN , | ak - a| > | b| \} ,
\sigma ap
\bigl(
\Delta \ast
a,b, \ell 1
\bigr)
= \{ \lambda \in \BbbC : | \lambda - a| < | b| \} \cup \{ ak : k \in \BbbN , | ak - a| > | b| \} ,
\sigma \delta
\bigl(
\Delta \ast
a,b, \ell 1
\bigr)
=
= (\{ \lambda \in \BbbC : | \lambda - a| \leq | b| \} \cup \{ ak : k \in \BbbN , | ak - a| > | b| \} ) \setminus \{ ak : k \in \BbbN , | ak - a| < | b| \} .
4. Conclusion. Many researchers have determined the spectrum and the fine spectrum of a
matrix operator in some sequence spaces. Although the fine spectrum with respect to the Goldberg’s
classification of the generalized difference operator \Delta a,b over the sequence space c0 were studied
by El-Shabrawy [10], in the present paper, the concepts of the approximate point spectrum, defect
spectrum and compression spectrum are introduced, and given the subdivisions of the spectrum of
the generalized difference operator \Delta a,b over the sequence space c0, as the new subdivisions of
spectrum. It is immediate that our new results cover a wider class of linear operators which are
represented by infinite lower triangular double-band matrices on the sequence space c0. For this
reason, our study is more general and more comprehensive than the previous work. We note that our
new results in this paper improve and generalize the results which have been stated in [5 – 7].
References
1. Akhmedov A. M., El-Shabrawy S. R. The spectrum of the generalized lower triangle double-band matrix \Delta a over the
sequence space c // Al-Azhar Univ. Eng. J. Spec. Issue. – 2010. – 5, № 9. – P. 54 – 60.
2. Akhmedov A. M., El-Shabrawy S. R. On the fine spectrum of the operator \Delta v over the sequence space c and \ell p
(1 < p < \infty ) // Appl. Math. and Inform. Sci. – 2011. – 5, № 3. – P. 635 – 654.
3. Altay B., Başar F. On the fine spectrum of the difference operator on c0 and c // Inform. Sci. – 2004. – 168. –
P. 217 – 224.
4. Appell J., Pascale E. D., Vignoli A. Nonlinear spectral theory. – Berlin; New York: Walter de Gruyter, 2004.
5. Başar F., Durna N., Yildirim M. Subdivisions of the spectra for genarilized difference operator \Delta v on the sequence
space \ell 1 // Int. Conf. Math. Sci. – 2010. – P. 254 – 260.
6. Başar F., Durna N., Yildirim M. Subdivisions of the spectra for genarilized difference operator over certain sequence
spaces // Thai J. Math. – 2011. – 9, № 2. – P. 285 – 295.
7. Başar F., Durna N., Yildirim M. Subdivision of the spectra for difference operator over certain sequence spaces //
Malays. J. Math. Sci. – 2012. – 6. – P. 151 – 165.
8. Durna N., Yildirim M. Subdivision of the spectra for factorable matrices on c0 // GUJ Sci. – 2011. – 24, № 1. –
P. 45 – 49.
9. El-Shabrawy S. R. On the fine spectrum of the generalized difference operator \Delta a,b over the sequence space \ell p
(1 < p < \infty ) // Appl. Math. and Inform. Sci. – 2012. – 6, № 1. – P. 111 – 118.
10. El-Shabrawy S. R. Spectra and fine spectra of certain lower triangular double band matrices as operators on c0 // J.
Inequal. and Appl. – 2014. – 241, № 1. – P. 1 – 9.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
922 N. DURNA
11. Fathi J., Lashkaripour R. On the fine spectra of the generalized difference operator \Delta uv over the sequence space c0
// J. Mahani Math. Res. Cent. – 2012. – 1, № 1. – P. 1 – 12.
12. Goldberg S. Unbounded linear operators. – New York: McGraw Hill, 1966.
13. Gonzalez M. The fine spectrum of the Cesaro operator in \ell p (1 < p < \infty ) // Arch. Math. – 1985. – 44. – P. 355 – 358.
14. Kayaduman K., Furkan H. On the fine spectrum of the difference operator \Delta over the sequence spaces \ell 1 and bv //
Int. Math. Forum. – 2006. – 24, № 1. – P. 1153 – 1160.
15. Reade J. B. On the spectrum of the Cesaro operator // Bull. London Math. Soc. – 1985. – 17. – P. 263 – 267.
16. Rhoades B. E. The fine spectra for weighted mean operators // Pacif. J. Math. – 1983. – 104. – P. 263 – 267.
17. Yildirim M. On the spectrum of the Rhaly operators on c0 and c // Indian J. Pure and Appl. Math. – 1998. – 29. –
P. 1301 – 1309.
18. Yildirim M. The fine spectra of the Rhaly operators on c0 // Turkish J. Math. – 2002. – 26, № 3. – P. 273 – 282.
19. Wenger R. B. The fine spectra of Hölder summability operators // Indian J. Pure and Appl. Math. – 1975. – 6. –
P. 695 – 712.
Received 27.02.16
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
|
| id | umjimathkievua-article-1605 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:02Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d4/fd2a70e62df74b84e47849f2f94094d4.pdf |
| spelling | umjimathkievua-article-16052019-12-05T09:20:38Z Subdivision of spectra for some lower triangular double-band matrices as operators on $c_0$ Пiдроздiл спектрiв для деяких нижньо-трикутних дворядкових матриць як операторiв на $c_0$ Durna, N. Дурна, Н. The generalized difference operator $\Delta_{a,b}$ was defined by El-Shabrawy: $\Delta{a,b}x = \Delta_{a,b} (x_n) = (a_nx_n + b_{n-1}x_{n-1})^{\infty}_{n = 0}$ with $x_1 = b_1 = 0$, where $(a_k), (b_k)$ are convergent sequences of nonzero real numbers satisfying certain conditions. We completely determine the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator $\Delta_{a,b}$ in the sequence space $c_0$. Узагальнений рiзницевий оператор $\Delta_{a,b}$ було визначено Ель-Шабравi: $\Delta{a,b}x = \Delta_{a,b} (x_n) = (a_nx_n + b_{n-1}x_{n-1})^{\infty}_{n = 0}$ при $x_1 = b_1 = 0$, де $(a_k), (b_k)$ — збiжнi послiдовностi ненульових дiйсних чисел, що задовольняють деякi умови. Повнiстю визначено наближений точковий спектр, дефектний спектр та стискувальний спектр оператора $\Delta_{a,b}$ у просторi послiдовностей $c_0$. Institute of Mathematics, NAS of Ukraine 2018-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1605 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 7 (2018); 913-922 Український математичний журнал; Том 70 № 7 (2018); 913-922 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1605/587 Copyright (c) 2018 Durna N. |
| spellingShingle | Durna, N. Дурна, Н. Subdivision of spectra for some lower triangular double-band matrices as operators on $c_0$ |
| title | Subdivision of spectra for some lower triangular double-band matrices as operators
on $c_0$ |
| title_alt | Пiдроздiл спектрiв для деяких нижньо-трикутних
дворядкових матриць як операторiв на $c_0$ |
| title_full | Subdivision of spectra for some lower triangular double-band matrices as operators
on $c_0$ |
| title_fullStr | Subdivision of spectra for some lower triangular double-band matrices as operators
on $c_0$ |
| title_full_unstemmed | Subdivision of spectra for some lower triangular double-band matrices as operators
on $c_0$ |
| title_short | Subdivision of spectra for some lower triangular double-band matrices as operators
on $c_0$ |
| title_sort | subdivision of spectra for some lower triangular double-band matrices as operators
on $c_0$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1605 |
| work_keys_str_mv | AT durnan subdivisionofspectraforsomelowertriangulardoublebandmatricesasoperatorsonc0 AT durnan subdivisionofspectraforsomelowertriangulardoublebandmatricesasoperatorsonc0 AT durnan pidrozdilspektrivdlâdeâkihnižnʹotrikutnihdvorâdkovihmatricʹâkoperatorivnac0 AT durnan pidrozdilspektrivdlâdeâkihnižnʹotrikutnihdvorâdkovihmatricʹâkoperatorivnac0 |