R E S E A R C H
Open Access
Spectra and fine spectra of certain lower
triangular double-band matrices as operators
on
c
Saad R El-Shabrawy
**Correspondence:
[email protected] Mathematics Department, Faculty of Science, Damietta University, New Damietta, Egypt
Abstract
In this paper we determine the fine spectrum of the generalized difference operator
a,bdefined by a lower triangular double-band matrix over the sequence spacec0.
The class of the operator
a,bcontains as special cases many operators that have
been studied recently in the literature. Illustrative examples showing the advantage of the present results are also given.
MSC: Primary 47A10; secondary 47B37
Keywords: spectrum of an operator; infinite lower triangular matrices; sequence spaces
1 Introduction
Several authors have studied the spectrum and fine spectrum of linear operators defined by lower and upper triangular matrices over some sequence spaces [–].
LetXbe a Banach space. ByR(T),T∗,X∗,B(X),σ(T,X),σp(T,X),σr(T,X) andσc(T,X), we denote the range ofT, the adjoint operator ofT, the space of all continuous linear functionals onX, the space of all bounded linear operators onXinto itself, the spectrum of
TonX, the point spectrum ofTonX, the residual spectrum ofTonXand the continuous spectrum ofT onX, respectively. We shall writecandcfor the spaces of all convergent and null sequences, respectively. Also bylwe denote the space of all absolutely summable sequences.
We assume here some familiarity with basic concepts of spectral theory and we refer to Kreyszig [, pp.-] for basic definitions such as spectrum, point spectrum, residual spectrum, and continuous spectrum of linear operators in normed spaces. Also, we refer to Goldberg [, pp.-] for Goldberg’s classification of spectra.
Now, let (ak) and (bk) be two convergent sequences of nonzero real numbers with
lim
k→∞ak=a and klim→∞bk=b= . ()
We consider the operatora,b:c→c, which is defined as follows:
a,bx=a,b(xk) = (akxk+bk–xk–)∞k= withx–=b–= . ()
It is easy to verify that the operatora,bcan be represented by a lower triangular double-band matrix of the form
a,b=
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
a · · ·
b a · · · b a · · ·
..
. ... ... . ..
⎞ ⎟ ⎟ ⎟ ⎟
⎠. ()
We begin by determining when a matrixAinduces a bounded linear operator fromc to itself.
Lemma .(cf.[, p.]) The matrix A= (ank)gives rise to a bounded linear operator
T∈B(c)from cto itself if and only if
() the rows ofAare inland theirlnorms are bounded,
() the columns ofAare inc.
The operator norm of T is the supremum of the lnorms of the rows.
As a consequence of the above lemma, we have the following corollary for the bounded linearity of the operatora,bon the spacec.
Corollary . The operatora,b:c−→cis a bounded linear operator with the norm
a,bc=supk(|ak|+|bk–|).
The rest of the paper is organized as follows. In Section , we analyze the spectrum of the operatora,bon the sequence spacec. In Section we give some illustrative examples. Finally, Section concludes with remarks and some special cases.
2 Fine spectrum of the operator
a,bonc0
In this section we examine the spectrum, the point spectrum, the residual spectrum and the continuous spectrum of the operatora,bon the sequence spacec.
Theorem . Let D={λ∈C:|λ–a| ≤ |b|}and E={ak:k∈N,|ak–a|>|b|}.Then
σ(a,b,c) =D∪E.
Proof First, we prove that (a,b–λI)–exists and is inB(c) forλ∈/D∪E and then the operatora,b–λIis not invertible forλ∈D∪E.
Letλ∈/D∪E. Then|λ–a|>|b|andλ=ak, for allk∈N. So, (a,b–λI) is triangle and hence (a,b–λI)–exists. We can calculate that
(a,b–λI)–= (skj) =
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
(a–λ) · · · –b
(a–λ)(a–λ)
(a–λ) · · · bb
(a–λ)(a–λ)(a–λ)
–b (a–λ)(a–λ)
(a–λ) · · · ..
. ... ... . ..
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
.
Now, for eachk∈N, the seriesSk=
Sincelimk→∞|abk–kλ|=|
b
a–λ|< , then there existk∈Nandq< such that| bk
ak–λ|<q,
for allk≥k. Then, for eachk>k,
Sk=
|ak–λ|
+ |bk–|
|ak––λ|
+ |bk–||bk–|
|ak––λ||ak––λ|
+· · ·+ |bk–||bk–| · · · |bk|
|ak––λ||ak––λ| · · · |ak–λ|
+· · ·+ |bk–||bk–| · · · |bk+||bk| · · · |b|
|ak––λ||ak––λ| · · · |ak+–λ||ak–λ| · · · |a–λ|
≤ |a
k–λ|
+q+q+· · ·+q k–k +q
k–k
|bk–|
|ak––λ| +qk–k
|bk–||bk–|
|ak––λ||ak––λ|
+· · ·+qk–k
|bk–||bk–| · · · |b|
|ak––λ||ak––λ| · · · |a–λ|
.
Therefore
Sk≤
|ak–λ|
+q+q+· · ·+q k–k mk
,
where
mk= +
|bk–|
|ak––λ|
+ |bk–||bk–|
|ak––λ||ak––λ|
+· · ·+ |bk–||bk–| · · · |b|
|ak––λ||ak––λ| · · · |a–λ| .
Thenmk> and so
Sk≤
mk
|ak–λ|
+q+q+· · ·+q k–k
.
But there existk∈Nand a real numberq<|b|such that |ak–λ|<q, for allk≥k. Then
Sk≤
qmk –q ,
for allk>max{k,k}. ThussupkSk<∞.
Also, it is easy to see thatlimk→∞|skj|= , for allj∈N, since
lim k→∞ sk+,j sk,j = lim k→∞ bk
ak+–λ
= b
a–λ
< .
So, the sequence (sj,sj,sj, . . .) converges to zero, for each j∈N. This shows that the columns of (a,b–λI)–are inc. Then, from Lemma ., (a,b–λI)–∈B(c) and so,
λ∈/σ(a,b,c). Thusσ(a,b,c)⊆D∪E.
Conversely, suppose thatλ∈/σ(a,b,c). Then (a,b–λI)–∈B(c). Since the (a,b–
λI)–transform of the unite sequencee= (, , , . . .) is inc, we havelimk→∞|ak+bk–λ|=
| b
a–λ| ≤ andλ=ak, for allk∈N. Then{λ∈C:|λ–a|<|b|} ⊆σ(a,b,c) and{ak:k∈
N} ⊆σ(a,b,c). Butσ(a,b,c) is a compact set, and so it is closed. ThenD={λ∈C:
|λ–a| ≤ |b|} ⊆σ(a,b,c) andE={ak:k∈N,|ak–a|>|b|} ⊆σ(a,b,c). This completes
Theorem . σp(a,b,c) =E∪K,where
K=
aj:j∈N,|aj–a|=|b|,
∞
i=m
bi–
aj–ai
diverges to zero for some m∈N
.
Proof Supposea,bx=λxfor anyx∈c. Then we obtain
(a–λ)x= and bkxk+ (ak+–λ)xk+= , for allk∈N.
If the sequence (ak) is constant, then we can easily see thatx=θ and so,σp(a,b,c) =∅ and the result follows immediately. Now, if the sequence (ak) is not constant, then for all
λ∈ {/ ak:k∈N}, we havexk= for allk∈N. So,λ∈/σp(a,b,c). Also, we can easily prove thata∈/σp(a,b,c). Thusσp(a,b,c)⊆ {ak:k∈N}\{a}. Now, we will prove that
λ∈σp(a,b,c) if and only if λ∈E∪K.
Ifλ∈σp(a,b,c), thenλ=aj= afor somej∈Nand there existsx∈c,x=θ such that
a,bx=ajx. Then
lim k→∞
xk+
xk
= b
a–aj
≤.
Thenλ=aj∈Eor|aj–a|=|b|. In the case when|aj–a|=|b|, we have
xk=
bk–bk–· · ·bm– (aj–ak)(aj–ak–)· · ·(aj–am)
xm–=xm– k
i=m
bi–
aj–ai
, k≥m.
Then∞i=m bi–
aj–aidiverges to , sincex∈c. Thereforeλ∈E∪K. Thusσp(a,b,c)⊆E∪K.
Conversely, letλ∈E∪K. Ifλ∈E, then there existsi∈Nsuch thatλ=ai=aand so we can takex=θ such thata,bx=aixand
lim k→∞
xk+
xk
= b
a–ai
< ,
that is,x∈c. Also, ifλ∈K, then there existsj∈Nsuch thatλ=aj=aand|aj–a|=|b|,
∞
i=m bi–
aj–ai diverges to , for somem∈N. Then we can takex∈c,x=θsuch thata,bx= ajx. ThusE∪K⊆σp(a,b,c). This completes the proof.
Theorem . σp(∗a,b,c∗) ={λ∈C:|λ–a|<|b|} ∪E∪H,where
H=
λ∈C:|λ–a|=|b|, ∞
k=
k
i=
λ–ai
bi
<∞
Proof Suppose that∗a,bf =λf forf= (f,f,f, . . .)=θ inc∗∼=l. Then, by solving the sys-tem of linear equations
af+bf=λf,
af+bf=λf,
af+bf=λf,
.. .
we obtain
fk+=
λ–ak
bk
fk, k∈N.
Then we must takef= since otherwise we would havef =θ. It is clear that for allk∈N, the vectorf = (f,f, . . . ,fk, , , . . .) is an eigenvector of the operator∗a,bcorresponding to the eigenvalueλ=ak, wheref= and fn=λ–abn–n–fn–, for alln= , , , . . . ,k. Then{ak:k∈
N} ⊆σp(∗a,b,c∗). Also, ifλ= akfor allk∈N, thenfk= , for allk≥ and
∞
k=|fk|<∞if limk→∞|fkf+k |=|λ–ab |< . Also, if|λ–a|=|b|, we can easily see that
|fk|=
(λ–a)(λ–a)· · ·(λ–ak–)
bb· · ·bk–
|f|=|f| k–
i=
λ–ai
bi
, for allk≥,
and so∞k=|fk|<∞if
∞
k=|
k
i= λ–ai
bi |<∞. This implies thatH⊆σp(
∗
a,b,c∗). Thus
λ∈C:|λ–a|<|b|∪E∪H⊆σp
∗a,b,c∗.
The second inclusion can be proved analogously.
The following lemma is required in the proof of the next theorem.
Lemma .[, p.] T has a dense range if and only if T∗is one to one.
Theorem . σr(a,b,c) =σp(a,b∗ ,c∗)\σp(a,b,c).
Proof Forλ∈σp(∗a,b,c∗)\σp(a,b,c), the operatora,b–λIis one to one and hence has an inverse. But∗a,b–λIis not one to one. Now, Lemma . yields the fact that the range of the operatora,b–λIis not dense inc. This implies thatλ∈σr(a,b,c).
Theorem . σr(a,b,c) ={λ∈C:|λ–a|<|b|} ∪(H\K).
Proof The proof follows immediately from Theorems ., ., and ..
Theorem . σc(a,b,c) ={λ∈C:|λ–a|=|b|}\H.
Proof Since σ(a,b,c) is the disjoint union of the parts σp(a,b,c), σr(a,b,c) and
σc(a,b,c) we must haveσc(a,b,c) ={λ∈C:|λ–a|=|b|}\H.
Theorem . σc(a,b,c) =σ(a,b,c)\σp(∗a,b,c∗).
Proof The proof is obvious and so is omitted.
With respect to Goldberg’s classification of the spectrum of an operator (see [, pp.-]), the spectrum is partitioned into nine states, which areI,I,I,II,II,II,III,III, andIII. For the operatora,b:c→c, we have
Iσ(a,b,c) =IIσ(a,b,c) =∅,
sinceσp(a,b,c)⊆σp(∗a,b,c∗). Also,Iσ(a,b,c) =∅, by the closed graph theorem. Thus we have to discuss the statesII,III,III, andIII.
Theorem . λ∈σp(a,b,c)if and only ifλ∈IIIσ(a,b,c).
Proof The proof is obvious and so is omitted.
Theorem . λ∈σc(a,b,c)if and only ifλ∈IIσ(a,b,c).
Proof Letλ∈σc(a,b,c). By Theorem .,∗a,b–λIis one to one. By Lemma .,a,b–
λIhas dense range. Additionally,λ∈/σp(a,b,c) implies that the operatora,b–λIhas inverse. Therefore,λ∈IIσ(a,b,c) orλ∈Iσ(a,b,c). ButIσ(a,b,c) =∅. Thusλ∈
IIσ(a,b,c).
Theorem . λ∈σr(a,b,c)if and only ifλ∈IIIσ(a,b,c)∪IIIσ(a,b,c).
Proof Letλ∈σr(a,b,c). By Theorem .,∗a,b–λIis not one to one. By Lemma .,
a,b–λI has not a dense range. Additionally,λ∈/σp(a,b,c) implies that the operator
a,b–λIhas inverse. Therefore,λ∈IIIσ(a,b,c)∪IIIσ(a,b,c).
3 Illustrative examples
In this section we provide some illustrative examples in support of our new results.
Example . Consider the sequences (ak) and (bk) defined by the following recurrence relations:
a=
√
, ak+=
+ak,
b=
√
, bk+=
bk,
for allk∈N. Then (ak) and (bk) are monotonically increasing sequences andlimk→∞ak=
a= andlimk→∞bk=b= . Also,ak≥bkfor allk∈N. Thus, for allλ∈Cwith|λ– |= , one can prove that|λ–ak
bk | ≥ for allk∈N. This implies thatH=∅. Also, we can prove that E=K=∅. Using Theorems ., ., ., and ., we have
σ(a,b,c) =
λ∈C:|λ– | ≤,
σr(a,b,c) =
λ∈C:|λ– |< ,
σc(a,b,c) =
λ∈C:|λ– |= .
Example . Let ak = k+k+ and bk = k+k+ for all k ∈N. Then limk→∞ak =a= and limk→∞bk=b= . Similarly, as in Example ., we can prove thatE=K=H=∅and so
σ(a,b,c) =
λ∈C:|λ– | ≤,
σp(a,b,c) =∅,
σr(a,b,c) =
λ∈C:|λ– |< ,
σc(a,b,c) =
λ∈C:|λ– |= .
Example . Consider the sequences (ak) and (bk) defined by the following recurrence
relations:
a= , a= , ak= fork≥,
b= , b= , bk=
k+
k
fork≥.
Therefore,limk→∞ak=a= andlimk→∞bk=b= . ThenE={},K=∅andH={λ∈C:
|λ– |= }, and so
σ(a,b,c) =
λ∈C:|λ– | ≤∪ {},
σp(a,b,c) ={},
σr(a,b,c) =
λ∈C:|λ– | ≤,
σc(a,b,c) =∅.
Remark . From the above examples, we note that the spectrum of the operatora,b
on the spacecmay include also a finite number of points outside a region enclosed by a circle. Also, we may haveσp(a,b,c)=∅.
Example . Let the sequences (ak) and (bk) be taken such thatak= –bk= (k+)
(k+)+(k+),
k∈N. Then we can prove thatE=K=H=∅and so we have
σ(a,b,c) =
λ∈C:λ–
≤
,
σp(a,b,c) =∅,
σr(a,b,c) =
λ∈C:λ–
<
,
σc(a,b,c) =
λ∈C:λ–
=
4 Remarks and some special cases
In this section we are going to give some special cases of the operatora,bwhich has been studied recently. More precisely, we show that special conditions on the sequences (ak) and (bk) characterize certain special cases of the operatora,b.
The difference operator: Ifak= andbk= –for allk∈N, then the operatora,b
reduces to the backward difference operator(cf.[]).
The generalized difference operatorB(r,s): Ifak=randbk=s= for allk∈N, then
the operatora,breduces to the operatorB(r,s)(cf.[]).
The generalized difference operatorv: Ifak= –bk=vkfor allk∈N, then the operator
a,breduces to the operatorv(cf.[]).
The generalized difference operatoruv: If(ak)is a sequence of positive real numbers
such thatak= for allk∈Nwithlimk→∞ak=U= and(bk)is either constant or
strictly decreasing sequence of positive real numbers withlimk→∞bk=V= and supkak<U+V, then the operatora,breduces to the operatoruv(cf.[]).
Remark . If (ak) and (bk) are convergent sequences of nonzero real numbers such that
lim
k→∞ak=a> , ()
lim
k→∞bk=b; |b|=a, ()
and
sup k
ak≤a, bk≤ak for allk∈N, ()
then we can prove thatH=∅and so we have:
σ(a,b,c) =
λ∈C:|λ–a| ≤a∪ak:k∈N,|ak–a|>a
,
σp(a,b,c) =
ak:k∈N,|ak–a|>a
,
σp
∗a,b,c∗=λ∈C:|λ–a|<a∪ak:k∈N,|ak–a|>a
,
σr(a,b,c) =
λ∈C:|λ–a|<a,
σc(a,b,c) =
λ∈C:|λ–a|=a.
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 spacec. 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 [, ].
Competing interests
The author declares that he has no competing interests.
Acknowledgements
I wish to express my thanks to Prof. Ali M Akhmedov, Baku State University, Faculty of Mechanics & Mathematics, Baku, Azerbaijan, for his kind help, careful reading, and making useful comments on the earlier version of this paper. Also, I thank the editor and the anonymous referees for their careful reading and making some useful comments which improved the presentation of the paper.
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doi:10.1186/1029-242X-2014-241
