#### Abstract

The main purpose of this paper is to determine the fine spectrum with respect to Goldberg's
classification of the operator defined by the lambda matrix over the sequence spaces and *c*. As a new
development, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix
operator on the sequence spaces and *c*. Finally, we present a Mercerian theorem. Since the matrix is
reduced to a regular matrix depending on the choice of the sequence having certain properties and its
spectrum is firstly investigated, our work is new and the results are comprehensive.

#### 1. Introduction

Let and be Banach spaces, and let also be a bounded linear operator. By , we denote the range of ; that is,
By , we also denote the *set of all bounded linear operators* on into itself. If is any Banach space and then the *adjoint * of is a bounded linear operator on the dual of defined by for all and .

Let be a nontrivial complex normed space and a linear operator defined on a subspace . We do not assume that is dense in or that has closed graph . By the statement “* is invertible*,” it is meant that there exists a bounded linear operator for which on and , such that is necessarily uniquely determined and linear; the boundedness of means that must be *bounded below*, in the sense that there is for which for all . Associated with each complex number, is the perturbed operator
defined on the same domain as . The *spectrum * consists of those , the complex field, for which is not invertible, and the *resolvent* is the mapping from the complement of the spectrum into the algebra of bounded linear operators on defined by .

#### 2. The Subdivisions of Spectrum

In this section, we define the parts of spectrum called point spectrum, continuous spectrum, residual spectrum, approximate point spectrum, defect spectrum, and compression spectrum. There are many different ways to subdivide the spectrum of a bounded linear operator. Some of them are motivated by applications to physics, in particular, quantum mechanics.

##### 2.1. The Point Spectrum, Continuous Spectrum, and Residual Spectrum

The name resolvent is appropriate since helps to solve the equation . Thus, provided that exists. More importantly, the investigation of properties of will be basic for an understanding of the operator itself. Naturally, many properties of and depend on , and the spectral theory is concerned with those properties. For instance, we are interested in the set of all 's in the complex plane such that exists. Boundedness of is another property that will be essential. We will also ask for what 's the domain of is dense in , to name just a few aspects. A *regular value * of is a complex number such that exists and is bounded and whose domain is dense in . For our investigation of , , and , we need some basic concepts in the spectral theory which are given, as follows (see [1, pages 370-371]).

The *resolvent set * of is the set of all regular values of . Furthermore, the spectrum is partitioned into the following three disjoint sets.

The *point (discrete) spectrum * is the set such that does not exist. An is called an *eigenvalue* of .

The *continuous spectrum * is the set such that exists and is unbounded, and the domain of is dense in .

The *residual spectrum * is the set such that exists (and may be bounded or not) but the domain of is not dense in .

Therefore, these three subspectra form a disjoint subdivision To avoid trivial misunderstandings, let us say that some of the sets defined above may be empty. This is an existence problem which we will have to discuss. Indeed, it is well known that and the spectrum consists of only the set in the finite-dimensional case.

##### 2.2. The Approximate Point Spectrum, Defect Spectrum, and Compression Spectrum

In this subsection, following Appell et al. [2], we define three more subdivisions of the spectrum called *approximate point spectrum*, *defect spectrum*, and *compression spectrum*.

Given a bounded linear operator in a Banach space , we call a sequence in a *Weyl sequence* for if and , as . Then, the *approximate point spectrum ** of * is defined by
Moreover, the subspectrum
is called the *defect spectrum* of .

The two subspectra given by (4) and (5) form a (not necessarily disjoint) subdivision
of the spectrum. There is another subspectrum,
which is often called *compression spectrum* in the literature. The compression spectrum gives rise to another (not necessarily disjoint) decomposition
of the spectrum. Clearly, and . Moreover, comparing these subspectra with those in (3), we note that

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 are also useful.

Proposition 1 (see [2, Proposition 1.3, page 28]). *Spectrum and subspectrum of an operator and its adjoint are related by the following relations: *(a), (b), (c), (d), (e), (f), (g) = ) ) = ,,).

The relations (c)–(f) show that the approximate point spectrum is in a certain sense dual to the defect spectrum, and the point spectrum is dual to the compression spectrum. The equality (g) implies, in particular, that if is a Hilbert space and is normal. Roughly speaking, this shows that normal (in particular, self-adjoint) operators on the Hilbert spaces are most similar to matrices in finite dimensional spaces (see [2]).

##### 2.3. Goldberg's Classification of Spectrum

If is a Banach space and , then there are three possibilities for : (A), (B), (C), and (1) exists and is continuous, (2) exists but is discontinuous, (3) does not exist.

If these possibilities are combined in all possible ways, nine different states are created. These are labelled by: , , , , , , , , and . If an operator is in state , for example, then and exists but is discontinuous (see [3]). Figure 1 due to Wenger [4] may be useful for the readers.

If is a complex number such that or , then . All scalar values of not in comprise the spectrum of . The further classification of gives rise to the fine spectrum of . That is, can be divided into the subsets , , , , , , and . For example, if is in a given state, (say), then we write .

By the definitions given above, we can illustrate subdivision (3) in Table 1.

Observe that the case in the first row and the second column cannot occur in a Banach space , by the closed graph theorem. If we are not in the third column, that is, if is not an eigenvalue of , we may always consider the resolvent operator (on a possibly “thin” domain of definition) as “algebraic inverse” of .

By a *sequence space*, we understand a linear subspace of the space of all complex sequences which contain , the set of all finitely nonzero sequences, where . We write , , , and for the spaces of all bounded, convergent, null, and bounded variation sequences which are the Banach spaces with the sup-norm and , respectively, while is not a Banach space with respect to any norm. Also by , we denote the space of all -absolutely summable sequences which is a Banach space with the norm , where .

Let and be two sequence spaces, and let be an infinite matrix of complex numbers , where . Then, we say that defines a matrix transformation from into , and we denote it by writing if for every sequence , the sequence , the -transform of , is in , where By , we denote the class of all matrices such that . Thus, if and only if the series on the right side of (10) converges for each and each , and we have for all .

Throughout this paper, let be a strictly increasing sequence of positive reals tending to infinity; that is,
Following Mursaleen and Noman [20], we define the matrix of weighted mean relative to the sequence by
for all . It is easy to show that the matrix is regular and is reduced, in the special case for all , to the matrix of Cesàro mean of order one. Introducing the concept of *-strong convergence*, several results on -strong convergence of numerical sequences and Fourier series were given by Móricz [21]. Since we have
in the special case for all , the matrix is also reduced to the Riesz means with respect to the sequence . We say that a sequence is -convergent if . In particular, we say that is a -null sequence if and we say that is -bounded if .

Lemma 2 (see [22, Theorem , page 6]). *The matrix = gives rise to a bounded linear operator from to itself if and only if *(1)*the rows of are in and their norms are bounded; *(2)*the columns of are in ; *(3)*the sequence of row sums of is in . *

The operator norm of is the supremum of the norms of the rows.

Corollary 3. * is a bounded linear operator with the norm .*

Lemma 4 (see [22, Example , page 129]). *The matrix gives rise to a bounded linear operator from to itself if and only if *(1)*the rows of are in and their norms are bounded, *(2)*the columns of are in . *

The operator norm of is the supremum of the norms of the rows.

Corollary 5. * is a bounded linear operator with the norm . *

We give a short survey concerned with the spectrum of the linear operators defined by some triangle matrices over certain sequence spaces. Wenger [4] examined the fine spectrum of the integer power of the Cesàro operator in and Rhoades [5] generalized this result to the weighted mean methods. The fine spectrum of the operator on the sequence space was studied by González [23], where . The spectrum of the Cesàro operator on the sequence spaces and were also investigated by Reade [6], Akhmedov and Başar [7], and Okutoyi [8], respectively. The fine spectrum of the Rhaly operators on the sequence spaces and were examined by Yıldırım [9]. Furthermore, Coşkun [10] has studied the spectrum and fine spectrum for -Cesàro operator acting on the space . Besides, de Malafosse [11] and Altay and Başar [12], respectively, studied the spectrum and the fine spectrum of the difference operator on the sequence spaces and , , where denotes the Banach space of all sequences normed by , . Altay and Karakuş [24] determined the fine spectrum of the Zweier matrix which is a band matrix as an operator over the sequence spaces and . In 2010, Srivastava and Kumar [16] determined the spectra and the fine spectra of the double sequential band matrix on , where is defined by and for all , under certain conditions on the sequence and they have just generalized these results by the double sequential band matrix defined by for all (see [18]). Altun [25] studied the fine spectra of the Toeplitz operators, which are represented by upper and lower triangular -band infinite matrices, over the sequence spaces and . Later, Karakaya and Altun determined the fine spectra of upper triangular double-band matrices over the sequence spaces and , in [26]. Quite recently, Akhmedov and El-Shabrawy [15] obtained the fine spectrum of the double sequential band matrix , defined as a double-band matrix with the convergent sequences and having certain properties, over the sequence space . The fine spectrum with respect to Goldberg’s classification of the operator defined by a triple band matrix over the sequence spaces and with has recently been studied by Furkan et al. [14]. Quite recently, Karaisa and Başar [19] have determined the fine spectrum of the upper triangular triple band matrix over the sequence space , where . At this stage, Table 2 may be useful.

In this work, our purpose is to determine the fine spectrum of the operator over the sequence spaces and with respect to Goldberg’s classification. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator over the spaces and . Finally, we state and prove a Mercerian theorem.

#### 3. The Fine Spectrum of the Operator on the Sequence Space

In this section, we examine the spectrum, the point spectrum, the continuous spectrum, the residual spectrum, the fine spectrum, the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator on the sequence space . For simplicity in the notation, we write throughout that for all and we use this abbreviation with other letters.

Theorem 6. *. *

*Proof. *Let . Since is triangle, exists, and solving the matrix equation for in terms of gives the matrix , where
for all . Thus, we observe that
The inequality is equivalent to , where . For all ,
holds for all . So, .

Firstly we take . Since , we have . Therefore and for .

Secondly we get . Since , . So,
Therefore, we have
that is, . But for ,
that is, is not in . This completes the proof.

Theorem 7. *Define and by and . Then,
*

*Proof. *Let and for any . Then,
So we have,
Note that if and only if
where . So, one can see that
which is equivalent to the inequality
For inequality (26) to be true for all sufficiently large , it is sufficient to have
We can write and . Therefore,
since the function defined by is monotone increasing in for .

For (27) to be true for all sufficiently large , it is sufficient to have satisfying
which is equivalent to
Therefore, for all , for some fixed ,
which diverges in the light of (29).

If for any , then clearly lies in the spectrum of . This completes the proof.

Theorem 8. *. *

*Proof. *Let be fixed and satisfy the inequality
and for any . We will show that . From Theorem 6, we need consider only those values of satisfying ; that is, . Under the assumption on , we wish to verify that
for all sufficiently large . It will be sufficient to show that
that is,
which is equivalent to (33).

Define the function by . has a minumum at . The above inequality is equivalent to and is also equivalent to
Therefore, for those values of satisfying (37), is monotone increasing. Let and small. Then = , where = . Note that for small , since is monotone increasing for , we will now show that . From (37),
which is equivalent to
But = , so we have = . Now choose and so small that = = . Then, by the definition of , there exists an such that implies , so that = . Using (23),
for all . Therefore is monotone decreasing in for each , so that has bounded columns. It remains to show that has finite norm.

For being used, from (29), we can enlarge , if necessary, to ensure that for . From (23),
where is a constant independent of . Further
Hence, has a finite norm.

Corollary 9. *Let exist. Then,
*

If with the matrix , then it is known that the adjoint operator is dened by the transpose of the matrix . It should be noted that the dual space of is isometrically isomorphic to the Banach space of absolutely summable sequences normed by .

Theorem 10. *Let be defined as in Corollary 9. Then, .*

*Proof. *Suppose that for in . Then, by solving the system of linear equations
we can write = ())[()()]. Let or and [)()]. One can see that for all sufficiently large if and only if
Then, we have, from the discussion in Theorem 7 and the hypothesis on ,
for all sufficiently large , so is convergent. Since is bounded, it follows that is convergent, so that has nonzero solutions. Therefore, the proof is completed.

Theorem 11. *Let be defined as in Corollary 9. Then
*

*Proof. *Let be any diagonal entry satisfying . Let be the smallest integer such that . By setting for , , the system reduces to a homogeneous linear system of equations in unknowns, so that nontrivial solutions exist. Therefore .

is not one to one for and so . This step concludes the proof.

Lemma 12 (see [3, page 59]). * has a dense range if and only if is one to one.*

Theorem 13. *. *

*Proof. *For , the operator is triangle, so has an inverse. But is not one to one by Theorem 10. Therefore by Lemma 12, , and this step concludes the proof.

Theorem 14. *Let be defined as in Corollary 9 and for all sufficiently large . Then,
*

*Proof. *Fix , , and satisfying . Since the operator is triangle, it has an inverse. Consider the adjoint operator . As in Theorem 11, is arbitrary and
for all positive . From the hypothesis, there exists a positive integer such that implies . This fact, together with the condition on , implies that for . Thus, for , where is a positive constant independent of . We can write
Therefore ; that is, is one to one. From Lemma 12, the range of is dense in . This completes the proof.

Lemma 15 (see [3, page 60]). * has a bounded inverse if and only if is onto.*

Theorem 16. *Let be defined as in Corollary 9 and less than 1. If satisfies and , then . *

*Proof . *First of all is a triangle; hence . Therefore . To verify that it is sufficient to show that is onto by Lemma 15.

Suppose , where . Then, and
Choose and solve (51) for in terms of to get
For example, substituting (52) into (53), with , yields
so that . For ,

Continuing this process, the entries of the matrix such that are calculated as
and otherwise.

To show that , it is sufficient to establish that is finite independent of . . We may write . Also, . Therefore,
and, for ,

Since , the series in inequality (24) is absolutely convergent from Theorem 7. Therefore, is finite.

Because of is bounded, it is continuous, and . This completes the proof.

Theorem 17. *Let be defined as in Corollary 9 and . If or for all and , then . *

*Proof. *First assume that has distinct diagonal entries and fix . Then the system implies that for , and for
The system (59) yields the following recursion relation:
which can be solved for to yield

Since , the argument of Theorem 7 applies and (24) is true. Therefore implies and is , so that .

Clearly . It remains to show that is onto.

Suppose that , where . By choosing we can solve for in terms of . As in Theorem 16, the remaining equations can be written in the form , where the nonzero entries of are as follows
From (62), we have
For ,
Using , one can convert (63) and (64) the similar expressions to (57) and (58), and therefore is finite.

Suppose that does not have distinct diagonal entries. The restriction on guarantees that no zero diagonal entries are being considered. Let be any diagonal entry which occurs more than once, and let denote, respectively, the smallest and largest integers for which . From (61) it follows that for . Also, for . Therefore the system becomes
*Case 1*. Let . Then (65) reduces to the single equation
which implies that , since and . Therefore .*Case 2*. Let . From (65) one can obtain the recursion formula with . Since it then follows that for . Using (65) with yields and so again .

To show that is onto, suppose , where . By choosing one can solve for in terms of . As in Theorem 16, the remaining equations can be written in the form , where the nonzero entries of are as in (62) with the other entries of clearly zero.

Since , there are two cases to consider.*Case 1*. If , then the proof proceeds exactly as in the argument following (62). *Case 2*. If , then from (62), at least for . If there are other values of with for which , then additional entries of will be zero. These zero entries do not affect the validity of the argument showing that (63) converges.

If , then does not lie inside the disc, and so it is not considered in this theorem.

Let . If for each , all sufficiently large, then the argument of Theorem 16 applies and . If for some , then the proof of Theorem 17 applies with replaced by and again, .

Therefore, in all cases, .

Theorem 18. *If .*

*Proof . *For and is not one to one. Therefore by Lemma 12. This concludes the proof.

Theorem 19. *The statement holds. *

*Proof. *Let be defined as in Corollary 9 and for all sufficiently large , then and follow from Corollary 9, Theorems 14, and 16–18.

Define the set by where is as in Theorem 7.

We will consider , that is, for which the main diagonal entries converge, where as in Corollary 9.

Theorem 20. *The following results hold:*(a)* = ,*(b)* = ,*(c)* = .*

*Proof . *(a) Since the relation
holds by Theorems 16 and 17 and from Table 1, . Therefore, we have .

(b) Since from Table 1 and by Theorem 19, we have .

(c) Since the equality