Upload
others
View
15
Download
0
Embed Size (px)
Citation preview
International Mathematical Forum, Vol. 9, 2014, no. 17, 773 - 784
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/imf.2014.4355
The Fine Spectrum of Upper
Triangular Double-Band Matrices U(r,s)
over the Class of Convergent Series
Osman Yılmaz
Department of Mathematics
54187, Sakarya University, Sakarya, Turkey
Merve Abay
Department of Mathematics,
54187, Sakarya University, Sakarya, Turkey
Selma Altundağ
Department of Mathematics,
54187, Sakarya University, Sakarya, Turkey
Copyright © 2014 Osman Yılmaz, Merve Abay and Selma Altundağ. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Abstract
The main purpose of this paper is to determine the fine spectrum of the upper triangular
double-band matrices ,U r s over the class of convergent series and we also determined the
the approximate point spectrum, defect spectrum and compression spectrum.
Keywords: Spectrum of an operator, Infinite matrices, Sequence spaces.
774 Osman Yılmaz, Merve Abay and Selma Altundağ
1. Introduction
In the existing literature of spectrum, there are many studies concerning with some particular
limitation matrices over some sequence spaces. We give a short summary about the existing
literature. For example;
Wenger [7] have studied the fine spectra of Hölder summability operators. The spectrum of
the Cesáro operator over some sequence spaces such as pl , 0c , bv were calculated by
Gonzàlez [8], Reade [9], Okutoyi [10], and Akhmedov and Başar [11], respectively. Altay
and Başar [12] studied the fine spectrum of the difference operator over 0c and c and later
Akhmedov and Başar also determined the fine spectrum of the difference operator over
pbv where 1 p in [14]. The fine spectrum of the generalized difference operator over
the sequence spaces 0c and c were computed by Altay and Başar [13]. Besides, Karakaya
and Altun [15] investigated the fine spectra of upper triangular double-band matrices on 0c
and c . Later, Karaisa and Başar [17] also determined the fine spectra of upper triangular
triple-band matrix on , 0pl p . Finally, Dutta and Tripathy [16] have determined fine
spectrum of the generalized difference operator over the class of convergent series.
2.Preliminaries, Background and Notations
Let X and Y be the Banach spaces and :T X Y also be a bounded linear operator. By
R T , we denote the range of T , i.e.,
: ,R T y Y y Tx x X .
By B X , we also denote the set of all bounded linear operators on X into itself. If X is
any Banach space and T B X , then the adjoint T of T is a bounded linear operator on
the dual X of X defined by T x Tx for all X and x X with T T .
Let X be a complex normed space and :T T R be a linear operator with domain
T X . By T , associate the operator
T T I ,
where is a complex number and I is the identity operator on T . If T has an inverse,
which is linear, we denote it by 1T
, that is
Fine spectrum of upper triangular double-band matrices 775
11T T I ,
and it is called to be the resolvent operator of T .
The name resolvent is appropriate, since 1T
helps to solve the equation T x y . Thus,
1x T y
provided 1T
exist. More important, the investigation of properties of 1T
will be
basic for an understanding of the operator T itself. Naturally, many properties of T and 1T
depend on , and spectral teory is concerned with those properties. For instance, we shall be
interested in the set of all in the complex plane such that 1T
exist. Boundedness of 1T
is
another property that will be essential. We shall also ask for what ’s the domain of 1T
is
dense in X , to name just a few aspects. For our investigaton of T , T and 1T
, we need
some basic concepts in spectral theory which are given as follows (see[1 pp. 370-371]):
Definition 1.1. Let X be a complex normed space and :T T X also be a linear
operator with domain T X . A reguler value of of T is a complex number such that
(R1) 1T
exist
(R2) 1T
is bounded
(R3) 1T
is defined on a set which is dense in X .
The resolvent set T of T is the set of all reguler values of T . Its complement
T T in the complex plane is called the spectrum of T . The spectrum T
is partitioned into three disjoint sets as follows:
The point spectrum p T is the set such that 1T
does not exist. A p T is called an
eigenvalue of T .
The continuous spectrum c T is the set such that 1T
exist and satisfies (R3) but not (R2).
The residual spectrum c T is the set such that 1T
exist (and may be bounded or not) but
not satisfy (R3).
From Goldberg (see [5, pp. 58–71]), if X is a Banach space and T B X , then there are
three possibilities for R T and 1T :
776 Osman Yılmaz, Merve Abay and Selma Altundağ
(I) R T X ,
(II) R T R T X ,
(III) R T X
and
(1) 1T exists and is continuous,
(2) 1T exists but is discontinuous,
(3) 1T does not exist.
If these possibilities are combined in all possible ways, nine different states are created. These
are labelled by: 1I , 2I , 3I , 1II , 2II , 3II , 1III , 2III and 3III . If is a complex number such
that 1T I or 1T II then is in the resolvent set ,T X of T is the set of all regular
values of T on X . The other classifications give rise to the fine spectrum of T . If an
operator is in state 2II for example, then R T R T X and 1T exists but is
discontinuous and we write 2,T X II .
Furthermore, following Appell in [6], we define the three more subdivisions of the spectrum
called as the approximate point spectrum, defect spectrum, and compression spectrum.
Given a bounded linear operator T in a Banach space X , we call a sequence kx in X as a
Weyl sequence for T if 1kx and 0kTx , as k .
In what follows, we call the set
, :ap T X thereexist aWeyl sequence for I T (2.1)
the approximate point spectrum of T . Moreover, the subspectrum
, :T X I T isnot surrjective (2.2)
is called defect spectrum of T .
The two subspectra given by (2.1) and (2.2) form a (not necessarily disjoint) subdivision
Fine spectrum of upper triangular double-band matrices 777
, , ,apT X T X T X (2.3)
of the spectrum. There is another subspectrum
, :co T X R I T X , (2.4)
which is often called compression spectrum in the literature. The compression spectrum gives
rise to another (not necessarily disjoint) decomposition
, , ,ap COT X T X T X (2.5)
of the spectrum. Clearly, , ,p apT X T X and , ,co T X T X . Moreover,
comparing these subspectra with
, , , ,p c rT X T X T X T X (2.6)
we note that
, , \ ,
, , \ , , .
r co c
c p co
T X T X T X
T X T X T X T X
(2.7)
Propositions. (see [6, Proposition 1.3, p. 28]). Spectra and subspectra of an operator
T B X and its adjoint T B X are related by the following relations:
(a) , ,T X T X ,
(b) , ,c apT X T X .
(c) , ,ap T X T X .
(d) , ,apT X T X .
(e) , ,p coT X T X .
(f) , ,co pT X T X .
(g) , , , , ,p ap p apT X T X T X T X T X .
The relations (c)-(f) show that the approximate point spectrum is in a certain sense dual to
defect spectrum, and the point spectrum dual to the compression spectrum.
778 Osman Yılmaz, Merve Abay and Selma Altundağ
The equality (g) implies, in particular, that , ,apT X T X if X is a Hilbert space and
T is normal. Roughly speaking, this shows that normal (in particular, self-adjoint) operators
on Hilbert spaces are most similar to matrices in finite-dimensional spaces (see [6]).
By w , we shall denote the space of all real valued sequences. Any vector subspace of w is
called as a sequence space. We shall write l , c , 0c and bv for the spaces of all bounded,
convergent, null and bounded variation sequences, respectively.
Let X and Y be two sequence spaces and nkA a be an infinite matrix of real or complex
numbers nka , where , 1,2,3,n k . Then, we say that A defines a matrix mapping
from X into Y , and we denote it by writing :A X Y , if for every sequence kx x X
the sequence n nAx Ax
, the A -transform of x , is in Y , where
nk knk
Ax a x n . (2.8)
By :X Y , we denote the class of all matrices A such that :A X Y . Thus, :A X Y if
and only if the series on the right side of (2.8) converges for each n and every x X ,
and we have n nAx Ax Y
for all x X .
By well-established convention, we define the space of all convergent series is defined by
0
( ) :n
k k
k
x x w x c
. (2.9)
An upper triangular double-band infinite matrix is of the form
0 0 0
0 0 0( , )
0 0 0
r s
r sU r s
r s
.
If :T is a bounded linear operator with the matrix 𝐴, then its adjoint operator
:T is defined by transpose of the matrix 𝐴 and is isomorphic to 1l with the norm
1 1:0
kl lk
x x
.
Fine spectrum of upper triangular double-band matrices 779
Lemma 2.1. ([4] p.253) A matrix nkA a gives rise to a bounded linear operator 1T B l
from 1l to itself if and only if the supremum of 1l norms of the columns of A is bounded.
Lemma 2.2. ([5] p.59) T has a dense range if and only if T is one to one.
Lemma 2.3. ([5] p.60) T has a bounded inverse if and only if T is onto.
3. Fine spectra of the upper triangular double-band matrices U r,s over
sequence space
In this section, we give the main result about the spectrum and the subdivisions of spectrum.
The spectral results are clear when 0s , so in what follows we will have 0s .
Theorem 3.1. , :U r s is a bounded linear operator with :
( , ) .r s U r s r s
Proof. It is not hard to prove that ,U r s is linear from , :U r s to itself and so we
omit it.
Since 1, k kU r s x rx sx for any x and for all k then we have
1 1:
0 0 0
, sup supn n n
k k k kn nk k k
U r s x rx sx r x s x
1
0 0
sup supn n
k kn nk k
r x s x r s x
. (3.1)
Let us take 2e . Then, since
2( , ) , ,0,0,0,U r s e s r , we find that
2
:
2:
:
,
,
U r s e
U r s s re
. (3.2)
Thus, we have the result from (3.1) and (3.2).
Theorem 3.2. ( , ), :p U r s r s .
780 Osman Yılmaz, Merve Abay and Selma Altundağ
Proof. Consider ( , )U r s x x for ( , )U r s x x . Then, by solving the system of equations,
0 1 0
1 2 1
2 3 2
1n n n
rx sx x
rx sx x
rx sx x
rx sx x
. (3.3)
we find that 0
n
n
rx x
s
for n . (3.4)
If 0x is the firs no-zero term of the sequence nx x , then r and 0ix for all i
which contradicts to the fact that x .
Now, let r . Due to the fact that we take x , 0
n
k
k
x
is in the convergent sequence
space c . That is, 0 0
limn
k kn
k k
x x
.
On the other hand, since 0 0
k
k k
rx
s
if and only if r s , we can say that
0
n
k
k
x c
for r s . Thus, , ,p U r s if and only if r s . This
completes the proof.
Theorem 3.3. , ,p U r s .
Proof. Suppose that ,U r s x x for 1x l . Then, consider the system of the
linear equations
0 0
0 1 1
1 2 2
1k k k
rx x
sx rx x
sx rx x
sx rx x
. (3.5)
Fine spectrum of upper triangular double-band matrices 781
If 0x is the no-zero term of the sequence nx x , then r and we obtain from (3.5)
1 0ix for any i which is a contradiction.
If jx is the non-zero term of the sequence nx x for 1 j n and n , then we obtain
from (3.5) 1j j
sx x
r
which implies that 0jx . This contradicts the fact that 0jx .
Thus, the proof is completed.
Corollary 3.4. ( , ),r U r s .
Theorem 3.5. ( , ), :U r s r s .
Proof. Let 1ky y l and by consider the equation ( , )U r s I x y , we have
0 0
0 1 1
1 2 2
r x y
sx r x y
sx r x y
(3.6)
and we obtain from (3.6) 0
1k ik
k i
i
sx y
r r
for all k . One can see that
0 0
1k k
k k
i i
x yr s
for s r
and all k . If k goes to infinity, then we get
1
l ll lx y
r s
.
Thus, ( , )U r s I is onto for s r and by Lemma 2.3, ( , )U r s I has a bounded
inverse, that is, ( , ), :c U r s r s for s r . Combining this with
Theorem 3.2 and Corollary 3.4, we get
: ( , ), :r s U r s r s . (3.7)
Since the spectrum of any bounded linear operator is closed, we have the result from (3.7).
782 Osman Yılmaz, Merve Abay and Selma Altundağ
Theorem 3.6. ( , ), :c U r s r s .
Proof. Since , , , ,p c rU U U U , then we have
( , ), :c U r s r s .
Goldberg Classifications for the operator ( , )U r s are given at following two theorems:
Theorem 3.7. If r s then 3,U I .
Proof. To prove this theorem, we must show that 11 ( , )U U r s I does not exist and
11 ( , )U U r s I is surjective for r s ,that is, ( )R U .
Let S . Then, since we know that ( , )p U by theorem 2, 1U
does not exist.
Now, let us prove that 1U
is surjective for r s .
Let 0 1 2, , ,y y y y and by considering the equation U x y , we have
0 1 0
1 2 1
2 3 2
r x sx y
r x sx y
r x sx y
. (3.8)
If we choose 0 1 2( , , )x x x x such that
0 0x , then we obtain from (3.8)
11
0
1k ik
k i
i
rx y
s s
for 1k .
One can see that
1 1
0 0
1k k
i i
i i
x ys r
(3.9)
because of
11
0
k ik
i
sr
s s r
for all 1k and r s .
Taking limits of both sides of the inequality in (3.9) for k , we get
Fine spectrum of upper triangular double-band matrices 783
1 1
0 0
1lim lim
k k
i ik k
i i
x ys r
and so x . This comletes the proof.
Theorem 3.8. If r s then 2,U II .
Proof. Let r s . It is easy to show that 1U
is not surjective.
On the other hand, if r s then ,c U . Hence, we can write ( )R U for
r s . This completes the proof.
Finally, we give the approximate spectrum, the defect spectrum and the compression
spectrum as a corollary:
Corollary 3.9. The following statements hold:
i) ( , ), :ap U r s r s ,
ii) ( , ), :U r s r s ,
iii) ( , ),co U r s .
Proof. (i) By [17], since 1( , ), ( , ), \ ( , ),ap U r s U r s U r s III
from Table 1 and we have 1 2( , ), ( , ),U r s III U r s III by Corollary 3.4 then we
reach the result.
At the same way, since
3( , ), ( , ), \ ( , ),U r s U r s U r s I (3.10)
and
1 2 3( , ), ( , ), ( , ), ( , ),co U r s U r s III U r s III U r s III , (3.11)
it is seen the results in (b) and (c) by helping Corollary 3.4, Theorem 3.7 and Theorem 3.3,
respectively.
784 Osman Yılmaz, Merve Abay and Selma Altundağ
References
[1] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons
Inc, New York, 1978.
[2] A. Wilansky, Summability Through Functional Analysis, in Mathematics Studies, vol.
85, North-Holland, Amsterdam, New York, Oxford, 1984.
[3] I. J. Maddox, Element of Functional Analysis, Cambridge University Press, London,
UK, 1970.
[4] B. Choudhary, S. Nanda, Functional Analysis with Applications, John Wiley & Sons
Inc, New York, Chishester, Brisbane, Toronto, Singapore, 1989.
[5] Goldberg S., Unbounded Linear Operators, Dover Publications, Inc. New York, 1985.
[6] J. Appell, E. Pascale, and A. Vignoli, Nonlinear Spectral Theory, vol. 10 of de Gruyter
Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany,
2004.
[7] M. Gonzàlez, The fine spectrum of the Cesàro operator in 1pl p , Arch. Math. 44
(1985) 355–358.
[8] R.B. Wenger, The fine spectra of Hölder summability operators, Indian J. Pure Appl.
Math. 6 (1975) 695–712.
[9] J.B. Reade, On the spectrum of the Cesàro operator, Bull. Lond. Math. Soc. 17 (1985)
263–267.
[10] J.T. Okutoyi, On the spectrum of 1C as an operator on bv , Commun. Fac. Sci. Univ.
Ank. Sér. A1 Math. Stat. 41 (1992) 197–207.
[11] A.M. Akhmedov, F. Başar, On spectrum of the Cesàro operator, Proc. Inst. Math. Mech.
Natl. Acad. Sci. Azerb. 19 (2004) 3–8.
[12] B. Altay and F. Başar, On the fine spectrum of the difference operator on 0c and c ,
Information Sciences, vol. 168, no. 1–4, (2004) pp. 217–224.
[13] B. Altay, F. Başar, On the fine spectrum of the generalized difference operator ,B r s
over the sequence spaces 0c and c , Int. J. Math. Math. Sci. 18 (2005) 3005–3013.
[14] A.M. Akhmedov, F. Başar, The fine spectra of the difference operator over the
sequence , 1pbv p , Acta. Math. Sin. (Engl. Ser.) 23 (2007) 1757-1768.
[15] V. Karakaya and M. Altun, Fine spectra of upper triangular double-band matrices,
Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1387-1394,
2010.
[16] A. J. Dutta, B. C. Tripathy, Fine spectrum of the generalized difference operator over the
class of convergent series, International Journal of Analysis, (2013), 1–4.
Received: March 25, 2014