Fine spectra of tridiagonal Toeplitz matrices
UDC 514.74 The fine spectra of $n$-banded triangular Toeplitz matrices and $(2n+1)$-banded symmetric Toeplitz matrices were computed in ( M. Altun, Appl. Math. and Comput. – 2011. – 217. – P. 8044 – 8051) and ( M. Altun, Abstr. and Appl. Anal. – 2012. – Article ID 932785). As a continuation of the...
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| author | Altun, M. Bilgiç, H. Алтун, М. Билгич, Х. |
| author_facet | Altun, M. Bilgiç, H. Алтун, М. Билгич, Х. |
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| description | UDC 514.74
The fine spectra of $n$-banded triangular Toeplitz matrices and $(2n+1)$-banded symmetric Toeplitz matrices were computed
in ( M. Altun, Appl. Math. and Comput. – 2011. – 217. – P. 8044 – 8051) and ( M. Altun, Abstr. and Appl. Anal. – 2012. –
Article ID 932785). As a continuation of these results, we compute the fine spectra of tridiagonal Toeplitz matrices. These
matrices are, in general, not triangular and not symmetric. |
| first_indexed | 2026-03-24T02:06:25Z |
| format | Article |
| fulltext |
UDC 514.74
H. Bilgiç (Sütçü İmam Univ., Kahramanmaraş, Turkey), M. Altun (Kayseri, Turkey)
FINE SPECTRA OF TRIDIAGONAL TOEPLITZ MATRICES
ТОНКI СПЕКТРИ ТРИДIАГОНАЛЬНИХ МАТРИЦЬ ТЬОПЛIЦА
The fine spectra of n-banded triangular Toeplitz matrices and (2n+1)-banded symmetric Toeplitz matrices were computed
in ( M. Altun, Appl. Math. and Comput. – 2011. – 217. – P. 8044 – 8051) and ( M. Altun, Abstr. and Appl. Anal. – 2012. –
Article ID 932785). As a continuation of these results, we compute the fine spectra of tridiagonal Toeplitz matrices. These
matrices are, in general, not triangular and not symmetric.
Тонкi спектри n-смугових трикутних матриць Тьоплiца та (2n+ 1)-смугових симетричних матриць Тьоплiца було
отримано в ( M. Altun, Appl. Math. and Comput. – 2011. – 217. – P. 8044 – 8051) та ( M. Altun, Abstr. and Appl. Anal. –
2012. – Article ID 932785). Як продовження цих результатiв розраховано тонкi спектри тридiагональних матриць
Тьоплiца. В загальному випадку цi матрицi не є анi трикутними, анi симетричними.
1. Introduction and preliminaries. The spectrum of an operator over a Banach space is partitioned
into three parts, which are the point spectrum, the continuous spectrum and the residual spectrum.
Some other parts also arise by examining the surjectivity of the operator and continuity of the inverse
operator. Such subparts of the spectrum are called the fine spectra of the operator.
The spectra and fine spectra of linear operators defined by some particular limitation matrices
over some sequence spaces were studied by several authors. We introduce the knowledge in the
existing literature concerning the spectrum and the fine spectrum. Wenger [21] examined the fine
spectrum of the integer power of the Cesàro operator over c and Rhoades [17] generalized this result
to the weighted mean methods. Reade [16] worked on the spectrum of the Cesàro operator over the
sequence space c0. Gonzàlez [12] studied the fine spectrum of the Cesàro operator over the sequence
space \ell p . Okutoyi [15] computed the spectrum of the Cesàro operator over the sequence space bv.
Recently, Rhoades and Yıldırım [18] examined the fine spectrum of factorable matrices over c0 and
c. Akhmedov and Başar [1, 2] have determined the fine spectrum of the Cesàro operator over the
sequence spaces c0, \ell \infty , and \ell p. Altun and Karakaya [8] computed the fine spectra of lacunary
matrices over c0 and c. Furkan, Bilgiç and Altay [10] determined the fine spectrum of B(r, s, t) over
the sequence spaces c0 and c, where B(r, s, t) is a lower triangular triple-band matrix. Later, Altun
[6, 7] computed the fine spectra of triangular and symmetric Toeplitz matrices over c0 and c.
Recently, Akhmedov and El-Shabrawy [3] have obtained the fine spectrum of the generalized
difference operator \Delta a,b , defined as a double band matrix with the convergent sequences \~a = (ak)
and \~b = (bk) having certain properties, over c. In 2010, Srivastava and Kumar [19] have determined
the spectra and the fine spectra of the generalized difference operator \Delta \nu on \ell 1 , where \Delta \nu is defined
by (\Delta \nu )nn = \nu n and (\Delta \nu )n+1,n = - \nu n for all n \in \BbbN , under certain conditions on the sequence
\nu = (\nu n) and they have just generalized these results by the generalized difference operator \Delta uv
defined by \Delta uvx = (unxn + vn - 1xn - 1)n\in \BbbN (see [20]).
In this work, our purpose is to determine the spectra of the operator, for which the corresponding
matrix is a tridiagonal Toeplitz matrix, over the sequence spaces \ell 1, c0, c, and \ell \infty . We will also give
the fine spectral results for the spaces \ell 1, c0, and c.
c\bigcirc H. BİLGİÇ, M. ALTUN, 2019
748 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
FINE SPECTRA OF TRIDIAGONAL TOEPLITZ MATRICES 749
Let X and Y be Banach spaces and U : X \rightarrow Y be a bounded linear operator. By \scrR (U) we
denote the range of U , i.e.,
\scrR (U) = \{ y \in Y : y = Ux;x \in X\} .
By B(X) we denote the set of all bounded linear operators on X into itself. If X is any Banach
space and U \in B(X), then the adjoint U\ast of U is a bounded linear operator on the dual X\ast of X
defined by (U\ast \phi )(x) = \phi (Ux) for all \phi \in X\ast and x \in X . Let X \not = \{ \theta \} be a complex normed
space and U : \scrD (U) \rightarrow X be a linear operator with domain \scrD (U) \subseteq X . With U we associate the
operator
U\lambda = U - \lambda I,
where \lambda is a complex number and I is the identity operator on \scrD (U). If U\lambda has an inverse, which
is linear, we denote it by U - 1
\lambda , that is
U - 1
\lambda = (U - \lambda I) - 1
and call it the resolvent operator of U\lambda . If \lambda = 0 we will simply write U - 1 . Many properties of U\lambda
and U - 1
\lambda depend on \lambda , and spectral theory is concerned with those properties. For instance, we shall
be interested in the set of all \lambda in the complex plane such that U - 1
\lambda exists. Boundedness of U - 1
\lambda is
another property that will be essential. We shall also ask for what \lambda ’s the domain of U - 1
\lambda is dense in
X . For our investigation of U, U\lambda , and U - 1
\lambda , we need some basic concepts in spectral theory which
are given as follows (see [14, p. 370, 371]):
Let X \not = \{ \theta \} be a complex normed space and U : \scrD (U) \rightarrow X be a linear operator with domain
\scrD (U) \subseteq X . A regular value \lambda of U is a complex number such that
(\bfR 1) U - 1
\lambda exists,
(\bfR 2) U - 1
\lambda is bounded,
(\bfR 3) U - 1
\lambda is defined on a set which is dense in X .
The resolvent set \rho (U) of U is the set of all regular values \lambda of U . Its complement \sigma (U) =
= \BbbC \setminus \rho (U) in the complex plane \BbbC is called the spectrum of U . Furthermore, the spectrum \sigma (U)
is partitioned into three disjoint sets as follows: The point spectrum \sigma p(U) is the set such that U - 1
\lambda
does not exist. A \lambda \in \sigma p(U) is called an eigenvalue of U . The continuous spectrum \sigma c(U) is the set
such that U - 1
\lambda exists and satisfies (\mathrm{R}3) but not (\mathrm{R}2). The residual spectrum \sigma r(U) is the set such
that U - 1
\lambda exists but does not satisfy (\mathrm{R}3).
A triangle is a lower triangular matrix with all of the principal diagonal elements nonzero. We
shall write \ell \infty , c, and c0 for the spaces of all bounded, convergent and null sequences, respectively,
that is
\ell \infty =
\biggl\{
x = (xk) : \mathrm{s}\mathrm{u}\mathrm{p}
k
| xk| < \infty
\biggr\}
,
c =
\biggl\{
x = (xk) : \mathrm{l}\mathrm{i}\mathrm{m}
k
xk exists
\biggr\}
,
c0 =
\biggl\{
x = (xk) : \mathrm{l}\mathrm{i}\mathrm{m}
k
xk = 0
\biggr\}
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
750 H. BİLGİÇ, M. ALTUN
By \ell p we denote the space of all p-absolutely summable sequences, where 1 \leq p < \infty . In
particular \ell 1 denotes the space of all absolutely summable sequences, that is
\ell 1 =
\Biggl\{
x = (xk) :
\sum
k
| xk| < \infty
\Biggr\}
,
\ell p =
\Biggl\{
x = (xk) :
\sum
k
| xk| p < \infty
\Biggr\}
.
Let \mu and \gamma be two sequence spaces and A = (ank) be an infinite matrix of real or complex numbers
ank , where n, k \in \BbbN . Then we say that A defines a matrix mapping from \mu into \gamma , and we denote
it by writing A : \mu \rightarrow \gamma , if for every sequence x = (xk) \in \mu the sequence Ax = \{ (Ax)n\} , the
A-transform of x, is in \gamma , where
(Ax)n =
\sum
k
ankxk, n \in \BbbN . (1)
By (\mu , \gamma ) we denote the class of all matrices A such that A : \mu \rightarrow \gamma . Thus, A \in (\mu , \gamma ) if and only
if the series on the right-hand side of (1) converges for each n \in \BbbN and every x \in \mu , and we have
Ax = \{ (Ax)n\} n\in \BbbN \in \gamma for all x \in \mu .
A tridiagonal nonsymmetric infinite matrix is of the form
T = T (q, r, s) =
\left[
q r 0 0 0 0 \cdot \cdot \cdot
s q r 0 0 0 \cdot \cdot \cdot
0 s q r 0 0 \cdot \cdot \cdot
0 0 s q r 0 \cdot \cdot \cdot
0 0 0 s q r \cdot \cdot \cdot
0 0 0 0 s q \cdot \cdot \cdot
...
...
...
...
...
...
. . .
\right]
.
The spectral results when T is triangular can be found in [6], so for the sequel we will have s \not = 0
and r \not = 0.
Let R be the right shift operator
R =
\left[
0 0 0 0 0 \cdot \cdot \cdot
1 0 0 0 0 \cdot \cdot \cdot
0 1 0 0 0 \cdot \cdot \cdot
...
...
...
...
...
. . .
\right]
and L be the left shift operator
L = Rt = R - 1.
Let us call Q(z) = sz+q+rz - 1 as the associated function of the operator T . Let P be the function
P (z) = rz + q + sz - 1 . Clearly, the roots of Q(z) are nonzero. Let \alpha 1 and \alpha 2 be roots of Q(z). It
is easy to verify that \alpha - 1
1 and \alpha - 1
2 are roots of P (z). We also have
T = s(I - \alpha 1L)(R - \alpha 2I). (2)
Let D be the unit disc \{ z \in \BbbC : | z| \leq 1\} ; \partial D be the unit circle \{ z \in \BbbC : | z| = 1\} and D\circ be the
open unit disc \{ z \in \BbbC : | z| < 1\} .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
FINE SPECTRA OF TRIDIAGONAL TOEPLITZ MATRICES 751
Theorem 1.1 (cf. [22]). Let U be an operator with the associated matrix A = (ank).
(i) U \in B(c) if and only if
| | A| | := \mathrm{s}\mathrm{u}\mathrm{p}
n
\infty \sum
k=1
| ank| < \infty , (3)
ak := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
ank exists for each k, (4)
a := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\infty \sum
k=1
ank exists. (5)
(ii) U \in B(c0) if and only if (3) and (4) with ak = 0 for each k .
(iii) U \in B(\ell \infty ) if and only if (3).
In these cases, the operator norm of U is
| | U | | (\ell \infty ,\ell \infty ) = | | U | | (c,c) = | | U | | (c0,c0) = | | A| | .
(iv) U \in B(\ell 1) if and only if
| | At| | = \mathrm{s}\mathrm{u}\mathrm{p}
k
\infty \sum
n=1
| ank| < \infty . (6)
In this case the operator norm of U is | | U | | (\ell 1,\ell 1) = | | At| | .
Corollary 1.1. T \in B(\mu ) for \mu \in \{ c0, c, \ell 1, \ell \infty \} and
\| T\| (\mu ,\mu ) = | q| + | r| + | s| .
Lemma 1.1. For the linear system of equations
qx0 + rx1 = 0,
sx0 + qx1 + rx2 = 0, (7)
sx1 + qx2 + rx3 = 0,
. . . . . . . . . . . . . . . . .
the general solution is
xn =
\left\{
C
\biggl(
\alpha 2
\alpha n
1
- \alpha 1
\alpha n
2
\biggr)
, if \alpha 1 \not = \alpha 2,
C
1 + n
\alpha n
1
, if \alpha 1 = \alpha 2,
(8)
where C \in \BbbC is a general constant.
Proof. To solve (7) we have x1 = - (q/r)x0 and
rxn + qxn - 1 + sxn - 2 = 0 for n \geq 2.
This is a linear recurrence relation with the characteristic equation
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
752 H. BİLGİÇ, M. ALTUN
0 = rz2 + qz + s = zP (z) = r(z - \alpha - 1
1 )(z - \alpha - 1
2 )
which has roots \alpha - 1
1 and \alpha - 1
2 . By the theory of recurrence relations, the general solution of (7) is
xn =
\left\{
C1
\alpha n
1
+
C2
\alpha n
2
, if \alpha 1 \not = \alpha 2,
C3 + C4n
\alpha n
1
, if \alpha 1 = \alpha 2,
with the restriction from the first line, that is x1 = - (q/r)x0 . Notice that, since \alpha 1 and \alpha 2 are roots
of Q, we have \alpha 1 + \alpha 2 = - q/s and \alpha 1\alpha 2 = r/s.
If \alpha 1 \not = \alpha 2 , then we obtain
C1
\alpha 1
+
C2
\alpha 2
= x1 = - q
r
x0 = - q
r
(C1 + C2) =
\alpha 1 + \alpha 2
\alpha 1\alpha 2
(C1 + C2).
Therefore, C1\alpha 1 + C2\alpha 2 = 0 which implies C1 = C\alpha 2 and C2 = - C\alpha 1 for a general constant C .
If \alpha 1 = \alpha 2 , then we get
C3 + C4
\alpha 1
= x1 = - q
r
x0 = - q
r
C3 =
\alpha 1 + \alpha 2
\alpha 1\alpha 2
C3 =
2
\alpha 1
C3.
Therefore, C3 = C4 = C for a general constant C .
Theorem 1.2.
(i) T \in (c0, c0) is one-to-one if and only if Q has a root in the unit disc.
(ii) T \in (\ell 1, \ell 1) is one-to-one if and only if Q has a root in the unit disc.
(iii) T \in (c, c) is one-to-one if and only if Q has a root in D \setminus \{ 1\} or 1 is a double root of Q.
(iv) T \in (\ell \infty , \ell \infty ) is one-to-one if and only if Q has a root in D\circ or a double root on \partial D.
Proof. (i) T \in (c0, c0) is not one-to-one if and only if there exists x = (x0, x1, x2, . . .) \not = \theta
in c0 such that Tx = \theta . Tx = \theta for nonzero x = (xn) \in c0 if and only if x satisfies system of
equations (7). Hence, by Lemma 1.1, Tx = \theta for nonzero x = (xn) \in c0 if and only if (8) holds for
\theta \not = x \in c0 .
For the case \alpha 1 \not = \alpha 2 , (8) holds for \theta \not = x \in c0 if and only if | \alpha 1| > 1 and | \alpha 2| > 1. Similarly,
for the case \alpha 1 = \alpha 2 , (8) holds for \theta \not = x \in c0 if and only if | \alpha 1| > 1.
Hence, Tx = \theta with x \not = \theta if and only if roots of Q are outside the unit disc. Equivalently,
T \in (c0, c0) is one-to-one if and only if Q has a root in the unit disc.
(ii) T \in (\ell 1, \ell 1) is not one-to-one if and only if there exists x = (x0, x1, x2, . . .) \not = \theta in \ell 1 such
that Tx = \theta . Tx = \theta for nonzero x = (xn) \in \ell 1 if and only if x satisfies system of equations (7).
Hence, by Lemma 1.1, Tx = \theta for nonzero x = (xn) \in \ell 1 if and only if (8) holds for \theta \not = x \in \ell 1 .
For the case \alpha 1 \not = \alpha 2 , (8) holds for \theta \not = x \in \ell 1 if and only if | \alpha 1| > 1 and | \alpha 2| > 1. Similarly,
for the case \alpha 1 = \alpha 2 , (8) holds for \theta \not = x \in \ell 1 if and only if | \alpha 1| > 1.
So, Tx = \theta with \theta \not = x \in \ell 1 if and only if roots of Q are outside the unit disc. Equivalently,
T \in (\ell 1, \ell 1) is one-to-one if and only if Q has a root in the unit disc.
(iii) Before we begin the proof, we remind that; if zn = 1/zn is a complex sequence we have
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
FINE SPECTRA OF TRIDIAGONAL TOEPLITZ MATRICES 753
zn =
1
zn
\left\{
converges to 0, if | z| > 1,
converges to 1, if z = 1,
diverges, otherwise.
In particular, if | z| = 1 and z \not = 1, the sequence 1/zn “spins” around the unit circle; i.e., diverges.
T \in (c, c) is not one-to-one if and only if there exists x = (x0, x1, x2, . . .) \not = \theta in c such that
Tx = \theta . Tx = \theta for nonzero x = (xn) \in c if and only if x satisfies system of equations (7). Hence,
by Lemma 1.1, Tx = \theta for nonzero x = (xn) \in c if and only if (8) holds for \theta \not = x \in c.
For the case \alpha 1 \not = \alpha 2 , (8) holds for \theta \not = x \in c if and only if the three cases: | \alpha 1| > 1 and
| \alpha 2| > 1 or | \alpha 1| > 1 and \alpha 2 = 1 or \alpha 1 = 1 and | \alpha 2| > 1. For the case \alpha 1 = \alpha 2 , (8) holds for
\theta \not = x \in c if and only if | \alpha 1| > 1.
So, Tx = \theta with \theta \not = x \in c if and only if roots of Q are outside the unit disc or one of the roots
is outside the unit disc and the other one is 1. Equivalently, T \in (c, c) is one-to-one if and only if Q
has a root in D \setminus \{ 1\} or 1 is a double root of Q.
(iv) T \in (\ell \infty , \ell \infty ) is not one-to-one if and only if there exists x = (x0, x1, x2, . . .) \not = \theta in
\ell \infty such that Tx = \theta . Tx = \theta for nonzero x = (xn) \in \ell \infty if and only if x satisfies system of
equations (7). Hence, by Lemma 1.1, Tx = \theta for nonzero x = (xn) \in \ell \infty if and only if (8) holds
for \theta \not = x \in \ell \infty .
For the case \alpha 1 \not = \alpha 2 , (8) holds for \theta \not = x \in \ell \infty if and only if | \alpha 1| \geq 1 and | \alpha 2| \geq 1. For the
case \alpha 1 = \alpha 2 , (8) holds for \theta \not = x \in \ell \infty if and only if | \alpha 1| > 1.
So, Tx = \theta with \theta \not = x \in \ell \infty if and only if | \alpha 1| \geq 1 and | \alpha 2| \geq 1 with \alpha 1 \not = \alpha 2 or \alpha 1 = \alpha 2
with | \alpha 1| > 1. Equivalently, T \in (\ell \infty , \ell \infty ) is one-to-one if and only if Q has a root in D\circ or Q has
a double root on \partial D.
We have the following two lemmas as a consequence of the corresponding results in [13] and
[4], respectively.
Lemma 1.2. (I - \alpha L) \in (c0, c0) is onto if and only if \alpha is not on the unit circle.
Lemma 1.3. (R - \alpha I) \in (c0, c0) is onto if and only if \alpha is outside the unit disc.
If U : \mu \rightarrow \mu (\mu is \ell 1 or c0) is a bounded linear operator represented by the matrix A, then it
is known that the adjoint operator U\ast : \mu \ast \rightarrow \mu \ast is defined by the transpose At of the matrix A. It
should be noted that the dual space c\ast 0 of c0 is isometrically isomorphic to the Banach space \ell 1 and
the dual space \ell \ast 1 of \ell 1 is isometrically isomorphic to the Banach space \ell \infty .
Lemma 1.4 [11, p. 59]. U has a dense range if and only if U\ast is one-to-one.
Corollary 1.2. If U \in (\mu , \mu ), then \sigma r(U, \mu ) = \sigma p(U
\ast , \mu \ast ) \setminus \sigma p(U, \mu ).
Theorem 1.3. T \in (c0, c0) is onto if and only if roots of Q are not on the unit circle and at
least one root of Q is outside the unit disc.
Proof. We will use the representation (2) of T :
T = s(I - \alpha 1L)(R - \alpha 2I) = s(I - \alpha 2L)(R - \alpha 1I).
Suppose T \in (c0, c0) is onto. An operator of the form I - \alpha L or R - \alpha I is in (c0, c0) for any
\alpha \in \BbbC . So the operators I - \alpha 1L and I - \alpha 2L are both onto. Therefore, by Lemma 1.2, \alpha 1 and \alpha 2
are not on the unit circle. Let us assume here that both \alpha 1 and \alpha 2 are in D\circ . Then the associated
function of the adjoint operator T \ast \in (\ell 1, \ell 1), which is represented by the transpose T t , is P . Both
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
754 H. BİLGİÇ, M. ALTUN
roots of P are outside the unit disc. This means, by Theorem 1.2, T \ast is not one-to-one and, by
Lemma 1.4, T does not have a dense range and so T is not onto. Then, our assumption is not true,
so at least one root of Q is outside the unit disc.
For the inverse implication, suppose the roots \alpha 1 and \alpha 2 of Q are not on the unit circle and
at least one root, say \alpha 2 , is outside the unit disc. Then, by Lemma 1.2, I - \alpha 1L is onto and, by
Lemma 1.3, R - \alpha 2I is onto. So, T = s(I - \alpha 1L)(R - \alpha 2I) is onto.
Corollary 1.3. T \in (c, c) is onto if and only if roots of Q are not on the unit circle and at least
one root of Q is outside the unit disc.
Proof. We prove by showing that ontoness of T in (c0, c0) and (c, c) are equivalent. Suppose
T is onto over (c0, c0). Then, by Theorem 1.3, \gamma := Q(1) = s+ q + r \not = 0. Then\left[
q r 0 0 \cdot \cdot \cdot
s q r 0 \cdot \cdot \cdot
0 s q r \cdot \cdot \cdot
0 0 s q \cdot \cdot \cdot
0 0 0 s \cdot \cdot \cdot
0 0 0 0 \cdot \cdot \cdot
...
...
...
...
. . .
\right]
\left[
1
1
1
1
1
1
...
\right]
=
\left[
q + r
s+ q + r
s+ q + r
s+ q + r
s+ q + r
s+ q + r
...
\right]
= (q + r + s)
\left[
1
1
1
1
1
1
...
\right]
- s
\left[
1
0
0
0
0
0
...
\right]
.
Now, by letting e = (1, 1, . . .) and e1 = (1, 0, 0, . . .), we have the equation
Te = \gamma e - se1.
Since T is onto over (c0, c0) there exists x \in c0 such that Tx = se1 . Then, by linearity of T, we get
T (e+ x) = Te+ Tx = \gamma e - se1 + se1 = \gamma e.
Then, for e\prime := (e + x)/\gamma , we obtain Te\prime = e. Let y = (yk) be an arbitrary element of c with
\delta = \mathrm{l}\mathrm{i}\mathrm{m} yk . Then clearly y - \delta e \in c0 and so there exists x\prime \prime \in c0 such that Tx\prime \prime = y - \delta e. Hence,
T (x\prime \prime + \delta e\prime ) = Tx\prime \prime + \delta Te\prime = y - \delta e+ \delta e = y.
Now x\prime \prime + \delta e\prime \in c, since
x\prime \prime + \delta e\prime = x\prime \prime +
\delta
\gamma
(e+ x) \rightarrow 0 +
\delta
\gamma
(1 + 0) =
\delta
\gamma
\in \BbbC .
So, T is onto over (c, c).
For the inverse implication, suppose T is onto over (c, c). Then Te = \gamma e - se1 /\in c0 , because
if Te \in c0, then we have the contradiction T \in (c, c0). So, \gamma \not = 0. Let y = (yk) be an arbitrary
element of c0 . Since c0 \subset c we have y \in c and since T is onto over (c, c), there exists x = (xk) \in c
such that Tx = y . Let \delta = \mathrm{l}\mathrm{i}\mathrm{m}xk . Then x - \delta e \in c0 . By Theorem 1.1 (ii) T \in (c0, c0), and so we
must have T (x - \delta e) \in c0 . By linearity of T we get
T (x - \delta e) = Tx - \delta Te = y - \delta (\gamma e - se1) \in c0.
Now since y \in c0 and (\gamma e - se1) /\in c0 we must have \delta = 0 and so x \in c0 . Therefore, T is onto
over (c0, c0).
The following theorem gives a general result about the resolvent set of a bounded operator over
a Banach space. (For a proof see, e.g., [7].)
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FINE SPECTRA OF TRIDIAGONAL TOEPLITZ MATRICES 755
Theorem 1.4. Let X be a Banach space and U \in B(X). Then \lambda \in \rho (U,X) if and only if U\lambda
is bijective.
2. The spectra and fine spectra.
Theorem 2.1.
\sigma (T, c0) =
\left\{
Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
, if | r| \leq | s| ,
Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
, if | r| > | s| .
Proof. Suppose | r| \leq | s| and \lambda \in \sigma (T, c0). By Theorem 1.4, T - \lambda I is not onto or is not
one-to-one. The associated function of T - \lambda I is (Q - \lambda )(z) = sz + q - \lambda + rz - 1 . The product of
the roots of Q - \lambda is r/s. Since | r/s| \leq 1, at least one root is in D, which means, by Theorem 1.2,
T - \lambda I is one-to-one. So T - \lambda I is not onto. Now, by Theorem 1.3, Q - \lambda has a root on the
unit circle or both roots are in D\circ . Suppose \beta is a root of Q - \lambda . If both roots are in D\circ , then
| r/s| < | \beta | < 1 and \lambda = Q(\beta ), which means \lambda \in Q
\Bigl(
D\circ \setminus r
s
D
\Bigr)
. If Q - \lambda has a root on the
circle, then | \beta | = 1 or | \beta | = | r/s| with \lambda = Q(\beta ), which means \lambda \in Q
\Bigl(
\partial D \cup r
s
\partial D
\Bigr)
. So, we have
\lambda \in Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
. Therefore, \sigma (T, c0) \subseteq Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
.
For the reverse inclusion, suppose | r| \leq | s| and \lambda \in Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
. Then \lambda = Q(\beta ) with
| r/s| \leq | \beta | \leq 1. Then both roots of Q - \lambda are in D\circ or Q - \lambda has a root on the unit circle.
Hence, by Theorem 1.3, T - \lambda I is not onto and, by Theorem 1.4, \lambda \in \sigma (T, c0). So, we have
Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
\subseteq \sigma (T, c0). Hence, for | r| \leq | s| we have \sigma (T, c0) = Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
.
Now, suppose | r| > | s| and \lambda \in \sigma (T, c0). By Theorem 1.4, T - \lambda I is not onto or is not
one-to-one. If T - \lambda I is not onto, by Theorem 1.3, Q - \lambda has a root on the unit circle or both
roots are in D\circ . But, both roots cannot be in D\circ , since the product of the roots, r/s, is absolutely
greater than 1. So, Q - \lambda has a root on the unit circle, which means \lambda = Q(\beta ) for some \beta with
| \beta | = 1 or | \beta | = | r/s| . Then \lambda \in Q
\Bigl(
\partial D \cup r
s
\partial D
\Bigr)
. If T - \lambda I is not one-to-one, by Theorem 1.2,
both roots of Q - \lambda are outside D. Let \beta be a root of Q - \lambda . If both roots are outside D, then
1 < | \beta | < | r/s| and \lambda = Q(\beta ), which means \lambda \in Q
\Bigl( r
s
D\circ \setminus D
\Bigr)
. So, we have \lambda \in Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
.
Therefore, \sigma (T, c0) \subseteq Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
.
For the reverse inclusion, suppose | r| > | s| and \lambda \in Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
. Then \lambda = Q(\beta ) with
1 \leq | \beta | \leq | r/s| . Then, both roots of Q - \lambda are outside D or Q - \lambda has a root on the unit circle. Hence,
by Theorems 1.2 and 1.3, T - \lambda I is not one-to-one or not onto, and, by Theorem 1.4, \lambda \in \sigma (T, c0).
So, we have Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
\subseteq \sigma (T, c0). Hence, for | r| > | s| we get \sigma (T, c0) = Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
.
Theorem 2.2. For \mu \in \{ \ell 1, c, \ell \infty \} ,
\sigma (T, \mu ) =
\left\{
Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
, if | r| \leq | s| ,
Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
, if | r| > | s| .
Proof. We will use the fact that the spectrum of a bounded operator over a Banach space is
equal to the spectrum of the adjoint operator. The adjoint operator is the transpose of the matrix for
c0 . So \sigma (T, \ell 1) = \sigma (T \ast , c\ast 0) = \sigma (T t, c0). The associated function of T t is P (z) = rz + q + sz - 1 .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
756 H. BİLGİÇ, M. ALTUN
So, by Theorem 2.1, we get
\sigma (T, \ell 1) =
\left\{
P
\Bigl(
D \setminus s
r
D\circ
\Bigr)
, if | s| \leq | r| ,
P
\Bigl( s
r
D \setminus D\circ
\Bigr)
, if | s| > | r| .
We can see that, for | s| \leq | r| , we have P
\Bigl(
D \setminus s
r
D\circ
\Bigr)
= Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
, and, for | s| > | r| , we get
P
\Bigl( s
r
D \setminus D\circ ) = Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
. Hence, \sigma (T, \ell 1) = \sigma (T, c0).
We know by Cartlidge [9] that if a matrix operator U is bounded on c, then \sigma (U, c) = \sigma (U, \ell \infty ).
Hence, we have \sigma (T, c) = \sigma (T, \ell \infty ) = \sigma (T \ast \ast , c\ast \ast 0 ) = \sigma (T, c0).
Theorem 2.3. For \mu \in \{ \ell 1, c0\} ,
\sigma p(T, \mu ) =
\left\{ Q
\Bigl( r
s
D\circ \setminus D
\Bigr)
, if | r| > | s| ,
\varnothing , if | r| \leq | s| .
Proof. Suppose \lambda \in \sigma p(T, \mu ) for \mu \in \{ \ell 1, c0\} . Then T - \lambda I is not one-to-one. By Theorem 1.2,
T - \lambda I is not one-to-one if and only if roots of Q - \lambda are outside D. The product of the roots
of Q - \lambda is r/s. So, if \beta is a root of Q - \lambda , then 1 < | \beta | < | r/s| and Q(\beta ) = \lambda . Hence,
\lambda \in Q
\Bigl( r
s
D\circ \setminus D
\Bigr)
. So, we have \sigma p(T, \mu ) \subseteq Q
\Bigl( r
s
D\circ \setminus D
\Bigr)
.
For the reverse inclusion, suppose \lambda \in Q
\Bigl( r
s
D\circ \setminus D
\Bigr)
. Then there exists \beta \in \BbbC with 1 <
< | \beta | < | r/s| such that Q(\beta ) = \lambda . Now, \beta is a root of Q - \lambda and is outside D. The other root
is r/(s\beta ) which is also outside D, since | r/(s\beta )| > 1. So both roots of Q - \lambda are outside D,
which means, by Theorem 1.2, that T - \lambda I is not one-to-one. Hence, \lambda \in \sigma p(T, \mu ). So, we have
Q
\Bigl( r
s
D\circ \setminus D
\Bigr)
\subseteq \sigma p(T, \mu ).
The following two theorems can be proved by using similar arguments, so we give them without
proof.
Theorem 2.4. \sigma p(T, c) =
\left\{ Q
\Bigl( r
s
D\circ \setminus D
\Bigr)
\cup Q(\{ 1\} ), if | r| > | s| ,
\varnothing , if | r| \leq | s| .
Theorem 2.5. \sigma p(T, \ell \infty ) =
\left\{
Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
, if | r| > | s| ,
Q
\biggl(
\partial D \setminus
\biggl\{
\pm
\sqrt{}
r
s
\biggr\} \biggr)
, if | r| = | s| ,
\varnothing , if | r| < | s| .
Theorem 2.6. \sigma r(T, c0) =
\left\{ \varnothing , if | r| \geq | s| ,
Q
\Bigl(
D\circ \setminus r
s
D
\Bigr)
, if | r| < | s| .
Proof. Let us do the proof only for the case | r| < | s| . When | r| < | s| , \sigma p(T
t, \ell 1) =
= P
\Bigl( s
r
D\circ \setminus D
\Bigr)
= Q
\Bigl(
D\circ \setminus r
s
D
\Bigr)
by Theorem 2.3. Now, using Corollary 1.2, we have \sigma r(T, c0) =
= \sigma p(T
\ast , c\ast 0) \setminus \sigma p(T, c0) = \sigma p(T
t, \ell 1) \setminus \sigma p(T, c0) = Q
\Bigl(
D\circ \setminus r
s
D
\Bigr)
.
If U : c \rightarrow c is a bounded matrix operator represented by the matrix A, then U\ast : c\ast \rightarrow c\ast acting
on \BbbC \oplus \ell 1 has a matrix representation of the form
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
FINE SPECTRA OF TRIDIAGONAL TOEPLITZ MATRICES 757\biggl[
\chi 0
b At
\biggr]
,
where \chi is the limit of the sequence of row sums of A minus the sum of the limits of the columns of
A, and b is the column vector whose kth entry is the limit of the kth column of A for each k \in \BbbN .
Then, for T : c \rightarrow c, the matrix T \ast is of the form\biggl[
s+ q + r 0
0 T t
\biggr]
=
\biggl[
Q(1) 0
0 T t
\biggr]
.
Theorem 2.7. \sigma r(T, c) =
\left\{
\varnothing , if | r| > | s| ,
Q(\{ 1\} ), if | r| = | s| ,
Q
\Bigl(
D\circ \setminus r
s
D
\Bigr)
\cup Q(\{ 1\} ), if | r| < | s| .
Proof. Let us do the proof only for the case | r| < | s| . The other cases can be proved similarly,
so we omit them. Let x = (x0, x1, . . .) \in \BbbC \oplus \ell 1 be an eigenvector of T \ast corresponding to the
eigenvalue \lambda . Then we have (s+q+r)x0 = \lambda x0 and Tx\prime = \lambda x\prime , where x\prime = (x1, x2, . . .). If x0 \not = 0,
then \lambda = s+ q + r, and s+ q + r is an eigenvalue, since T \ast (1, 0, 0, . . .) = (s+ q + r)(1, 0, 0, . . .).
If x0 = 0, then x\prime is an eigenvector of T t over \ell 1 and T tx\prime = \lambda x\prime . By Theorem 2.3, \lambda \in
\in \sigma p(T
t, \ell 1) = P
\Bigl( s
r
D\circ \setminus D
\Bigr)
= Q
\Bigl(
D\circ \setminus r
s
D
\Bigr)
. Hence, \sigma p(T \ast , c\ast ) = Q
\Bigl(
D\circ \setminus r
s
D
\Bigr)
\cup Q
\bigl(
\{ 1\}
\bigr)
. Then
\sigma r(T, c) = \sigma p(T
\ast , c\ast ) \setminus \sigma p(T, c) = Q
\Bigl(
D\circ \setminus r
s
D
\Bigr)
\cup Q(\{ 1\} ).
As a consequence of Theorems 2.3 and 2.5, we have the following result.
Theorem 2.8. \sigma r(T, \ell 1) =
\left\{ \varnothing , if | r| > | s| ,
Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
\setminus Q
\biggl( \biggl\{
\pm
\sqrt{}
r
s
\biggr\} \biggr)
, if | r| \leq | s| .
The spectrum \sigma is the disjoint union of \sigma p, \sigma r, and \sigma c, so we obtain the following theorem as a
consequence of Theorems 2.3, 2.6, 2.7 and 2.4.
Theorem 2.9. We have
\sigma c(T, c0) = Q (\partial D) \cup Q
\Bigl( r
s
\partial D
\Bigr)
,
\sigma c(T, c) = Q (\partial D) \setminus Q(\{ 1\} ),
\sigma c(T, \ell 1) =
\left\{
Q(\partial D), if | r| > | s| ,
Q
\biggl( \biggl\{
\pm
\sqrt{}
r
s
\biggr\} \biggr)
, if | r| \leq | s| .
3. The resolvent operator and some applications.
Theorem 3.1. Let \mu \in \{ c0, c, \ell 1, \ell \infty \} . The resolvent operator T - 1 over \mu exists, is continuous
and the domain of T - 1 is the whole space \mu if and only if one of the roots of the function Q is in
D\circ , and the other root is outside D. In this case, say | \alpha 1| < 1 < | \alpha 2| , the matrix representation of
T - 1 is S = (snk) defined by
snk =
1
s(\alpha 1 - \alpha 2)
\left\{
\Bigl(
\alpha k+1
2 - \alpha k+1
1
\Bigr)
\alpha - n - 1
2 , if n \geq k,\bigl(
\alpha - n - 1
1 - \alpha - n - 1
2
\bigr)
\alpha k+1
1 , if n < k.
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758 H. BİLGİÇ, M. ALTUN
Proof. By Theorem 1.4, the resolvent operator T - 1 over \mu exists, is continuous and the domain
of T - 1 is the whole space \mu if and only if 0 is in the resolvent set \rho (T, \mu ). Hence, for | r| \leq | s| ,
0 /\in Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
and for | r| > | s| , 0 /\in Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
. In both cases, this is equivalent to saying,
one of the roots of Q is absolutely less than 1, and the other root is absolutely greater than 1.
Now, let us show that S(Tx) = x for all x = (x0, x1, . . .) \in \ell \infty . By definition we have
(Tx)k = sxk - 1 + qxk + rxk+1, where x - 1 = 0. Then, for n \in \BbbN ,
(S(Tx))n =
\infty \sum
k=0
snk(Tx)k =
\infty \sum
k=0
snk(sxk - 1 + qxk + rxk+1).
This sum is absolutely convergent since rows of S are in \ell 1 and x \in \ell \infty . So, we can change the
order of summation and get
(S(Tx))n =
\infty \sum
k=0
(ssn(k+1) + qsnk + rsn(k - 1))xk,
where sn( - 1) = 0. Now, it is not difficult to check that ssn(k+1) + qsnk + rsn(k - 1) = \delta nk for all
n, k \in \BbbN , where \delta nk is the Kronecker delta. Hence, we have (S(Tx))n = xn for all n \in \BbbN . So, we
get S(Tx) = x for all x \in \ell \infty .
Corollary 3.1. Let \mu \in \{ c0, c, \ell 1, \ell \infty \} . For \lambda /\in \sigma (T, \mu ) the matrix representation of (T - \lambda ) - 1
is V = (vnk) defined by
vnk =
1
s(\beta 1 - \beta 2)
\left\{
\Bigl(
\beta k+1
2 - \beta k+1
1
\Bigr)
\beta - n - 1
2 , if n \geq k,\bigl(
\beta - n - 1
1 - \beta - n - 1
2
\bigr)
\beta k+1
1 , if n < k,
where \beta 1 and \beta 2 are the roots of Q - \lambda satisfying | \beta 1| < 1 < | \beta 2| .
Example 3.1. When T is a symmetric tridiagonal matrix we have s = r and
T =
\left[
q r 0 0 0 0 \cdot \cdot \cdot
r q r 0 0 0 \cdot \cdot \cdot
0 r q r 0 0 \cdot \cdot \cdot
0 0 r q r 0 \cdot \cdot \cdot
...
...
...
...
...
...
. . .
\right] ,
then \sigma (T, \mu ) = Q
\Bigl(
D \setminus r
r
D\circ
\Bigr)
= Q(D \setminus D\circ ) = Q(\partial D) for \mu \in \{ \ell 1, c0, c, \ell \infty \} , where Q(z) =
= q + r(z + z - 1). Therefore,
\sigma (T, \mu ) = \{ q + 2r \mathrm{c}\mathrm{o}\mathrm{s} \theta : \theta \in [0, \pi ]\} = [q - 2r, q + 2r]
which is one of the main results of [5]; [q - 2r, q + 2r] is the closed line segment in the complex
plane with endpoints q - 2r and q + 2r.
Example 3.2. When | s| = | r| , it can be proved that the spectrum is always a closed line segment.
For example, let
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
FINE SPECTRA OF TRIDIAGONAL TOEPLITZ MATRICES 759
T =
\left[
1 i 0 0 0 0 \cdot \cdot \cdot
1 1 i 0 0 0 \cdot \cdot \cdot
0 1 1 i 0 0 \cdot \cdot \cdot
0 0 1 1 i 0 \cdot \cdot \cdot
...
...
...
...
...
...
. . .
\right] ,
where i is the complex number
\surd
- 1. Then
\sigma (T, \mu ) = Q
\Bigl(
D \setminus i
1
D\circ
\Bigr)
= Q(D \setminus D\circ ) = Q(\partial D) for \mu \in \{ \ell 1, c0, c, \ell \infty \} ,
where Q(z) = z + 1 + iz - 1 . Therefore,
\sigma (T, \mu ) = \{ 1 + (\mathrm{c}\mathrm{o}\mathrm{s} \theta + i \mathrm{s}\mathrm{i}\mathrm{n} \theta ) + i(\mathrm{c}\mathrm{o}\mathrm{s} \theta - i \mathrm{s}\mathrm{i}\mathrm{n} \theta ) : \theta \in [0, 2\pi ]\} =
= \{ 1 + (1 + i)(\mathrm{c}\mathrm{o}\mathrm{s} \theta + \mathrm{s}\mathrm{i}\mathrm{n} \theta ) : \theta \in [0, 2\pi ]\} =
=
\Bigl[
1 -
\surd
2(1 + i), 1 +
\surd
2(1 + i)
\Bigr]
.
Example 3.3. Let
T =
\left[
0 1 0 0 0 0 \cdot \cdot \cdot
2 0 1 0 0 0 \cdot \cdot \cdot
0 2 0 1 0 0 \cdot \cdot \cdot
0 0 2 0 1 0 \cdot \cdot \cdot
...
...
...
...
...
...
. . .
\right] .
Then \sigma (T, \mu ) = Q
\Bigl(
D \setminus 1
2
D\circ
\Bigr)
for \mu \in \{ \ell 1, c0, c, \ell \infty \} , where Q(z) = 2z + z - 1 . For the boundaries
we have
Q(\partial D) = Q
\biggl(
1
2
\partial D
\biggr)
= \{ 2(\mathrm{c}\mathrm{o}\mathrm{s} \theta + i \mathrm{s}\mathrm{i}\mathrm{n} \theta ) + (\mathrm{c}\mathrm{o}\mathrm{s} \theta - i \mathrm{s}\mathrm{i}\mathrm{n} \theta ) : \theta \in [0, 2\pi ]\} =
= \{ (3 \mathrm{c}\mathrm{o}\mathrm{s} \theta + i \mathrm{s}\mathrm{i}\mathrm{n} \theta : \theta \in [0, 2\pi ]\} =
\biggl\{
(x, y) \in \BbbR 2 :
x2
32
+ y2 = 1
\biggr\}
.
Hence, \sigma (T, \mu ) is the elliptical region
\Bigl\{
(x, y) \in \BbbR 2 :
x2
32
+ y2 \leq 1
\Bigr\}
.
Now, let us give an application of Theorem 2.2, related to the system of equations
yk = sxk - 1 + qxk + rxk+1, k = 0, 1, 2, . . . , (9)
where x - 1 = 0.
Theorem 3.2. Let r, q and s be complex numbers with r, s \not = 0, and Q(z) = sz + q + rz - 1
with roots \alpha 1, \alpha 2 satisfying | \alpha 1| \leq | \alpha 2| . Let the complex sequences x = (xn) and y = (yn) be
solutions of the system of equations (9). Then the following are equivalent:
(i) boundedness of (yn) always implies a unique bounded solution (xn),
(ii) convergence of (yn) always implies a unique convergent solution (xn),
(iii) yn \rightarrow 0 always implies a unique solution (xn) with xn \rightarrow 0,
(iv)
\sum
| yn| < \infty always implies a unique solution (xn) with
\sum
| xn| < \infty ,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
760 H. BİLGİÇ, M. ALTUN
(v) | \alpha 1| < 1 < | \alpha 2| .
Proof. Let | r| \leq | s| . The system of equations (9) hold, so we have Tx = y . Then Q is the
function associated with T . Let us prove only (i)\leftrightarrow (v) and omit the proofs of (ii)\leftrightarrow (v), (iii)\leftrightarrow (v),
(iv)\leftrightarrow (v) since they are similarly proved. Suppose boundedness of (yn) always implies a unique
bounded solution (xn). Then the operator T - 0I = T \in (\ell \infty , \ell \infty ) is bijective. This means, by
Theorems 1.4 and 2.2, 0 /\in \sigma (T, \ell \infty ) = Q
\Bigl(
D \setminus r
s
D\circ
\Bigr)
. This is equivalent to | \alpha 1| < 1 < | \alpha 2| . If
| r| > | s| , similarly we get 0 /\in Q
\Bigl( r
s
D \setminus D\circ
\Bigr)
, which is also equivalent to | \alpha 1| < 1 < | \alpha 2| .
For the reverse implication, suppose | \alpha 1| < 1 < | \alpha 2| . So, \lambda = 0 /\in \sigma (T, \ell \infty ). Now, by
Theorem 1.4, T = T - 0I is bijective on \ell \infty , which means that the boundedness of (yn) implies a
bounded unique solution (xn).
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Received 19.11.14,
after revision — 14.04.19
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
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| id | umjimathkievua-article-1473 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:25Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ce/746ab85b799a0ca9f17208692eed30ce.pdf |
| spelling | umjimathkievua-article-14732019-12-05T08:56:42Z Fine spectra of tridiagonal Toeplitz matrices Тонкi спектри тридiагональних матриць Тьоплiца Altun, M. Bilgiç, H. Алтун, М. Билгич, Х. UDC 514.74 The fine spectra of $n$-banded triangular Toeplitz matrices and $(2n+1)$-banded symmetric Toeplitz matrices were computed in ( M. Altun, Appl. Math. and Comput. – 2011. – 217. – P. 8044 – 8051) and ( M. Altun, Abstr. and Appl. Anal. – 2012. – Article ID 932785). As a continuation of these results, we compute the fine spectra of tridiagonal Toeplitz matrices. These matrices are, in general, not triangular and not symmetric. УДК 514.74 Тонкi спектри $n$-смугових трикутних матриць Тьоплiца та $(2n + 1)$-смугових симетричних матриць Тьоплiца було отримано в ( M. Altun, Appl. Math. and Comput. – 2011. – 217. – P. 8044 – 8051) та ( M. Altun, Abstr. and Appl. Anal. – 2012. – Article ID 932785). Як продовження цих результатiв розраховано тонкi спектри тридiагональних матриць Тьоплiца. В загальному випадку цi матрицi не є анi трикутними, анi симетричними. Institute of Mathematics, NAS of Ukraine 2019-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1473 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 6 (2019); 748-760 Український математичний журнал; Том 71 № 6 (2019); 748-760 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1473/457 Copyright (c) 2019 Altun M.; Bilgiç H. |
| spellingShingle | Altun, M. Bilgiç, H. Алтун, М. Билгич, Х. Fine spectra of tridiagonal Toeplitz matrices |
| title | Fine spectra of tridiagonal Toeplitz matrices |
| title_alt | Тонкi спектри тридiагональних матриць Тьоплiца |
| title_full | Fine spectra of tridiagonal Toeplitz matrices |
| title_fullStr | Fine spectra of tridiagonal Toeplitz matrices |
| title_full_unstemmed | Fine spectra of tridiagonal Toeplitz matrices |
| title_short | Fine spectra of tridiagonal Toeplitz matrices |
| title_sort | fine spectra of tridiagonal toeplitz matrices |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1473 |
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