#### Abstract

We focus on the behavior of solutions of the difference equation , , where () is a fixed sequence of complex numbers, and () is a fixed sequence in a complex Banach space. We give the general solution of this difference equation. To examine the asymptotic behavior of solutions, we compute the spectra of operators which correspond to such type of difference equations. These operators are represented by upper triangular or lower triangular infinite banded Toeplitz matrices.

#### 1. Introduction

The theory of difference equations is one of the most important representations of real world problems. The situation of an event at a fixed time usually depends on the situations of the event in the history. One of the ways to mathematically model such an event is to find a difference equation that directly or asymptotically describes the dependence of the situation at a time to the situations of the event in the history.

Let be a complex Banach space. In this work, we are interested in a difference equation of the form , where and are fixed sequences in and , respectively, and . The difference equation (1.1) is a generalization of the difference equations investigated by Copson [1], Popa [2], and Stević [3]. We will give the general solution of the system of equations (1.1) and examine the different types of stability conditions by determining the spectra of related matrix operators. We will also examine the system of equations which correspond to the transpose of these matrices.

Let and be Banach spaces and be a bounded linear operator. By , we denote the range of , that is, ; . By , we 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 and be a linear operator with domain . With , we associate the operator
where is a complex number, and is the identity operator on . If has an inverse, which is linear, we denote it by , that is,
and call it the *resolvent operator* of . Many properties of and depend on , and spectral theory is concerned with those properties. For instance, we will be interested in the set of all 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 . For our investigation of , , and , we need some basic concepts in spectral theory which are given as follows (see [4, page 370-371]).

Let be a complex normed space, and let be a linear operator with domain . A *regular value * of is a complex number such that (R1) exists, (R2) is bounded, (R3) is defined on a set which is dense in .

The *resolvent set * of is the set of all regular values of . Its complement in the complex plane is called the *spectrum* of . Furthermore, the spectrum is partitioned into three disjoint sets as follows. The *point spectrum * is the set such that does not exist. A is called an *eigenvalue* of . The *continuous spectrum * is the set such that exists and satisfies (R3) but not (R2). The *residual spectrum * is the set such that exists but does not satisfy (R3).

We will write , and for the spaces of all bounded, convergent, and null sequences, respectively. By , we denote the space of all -absolutely summable sequences, where . Let and be two sequence spaces and be an infinite matrix of real or complex numbers , where , . Then, we say that defines a matrix mapping 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 (1.4) converges for each and every , and we have for all .

Several authors have studied the spectrum and fine spectrum of linear operators defined by matrices over some sequence spaces. Rhoades [5] examined the fine spectra of the weighted mean operators. Reade [6] worked on the spectrum of the Cesàro operator over the sequence space . Gonzáles [7] studied the fine spectrum of the Cesàro operator over the sequence space . Yıldırım [8] examined the fine spectra of the Rhally operators over the sequence spaces and . Akhmedov and Başar [9] have determined the fine spectrum of the difference operator over . Later, Bilgiç et al. [10] worked on the spectrum of the operator , defined by a triple-band lower triangular matrix, over the sequence spaces and . Recently, Altun and Karakaya [11] determined the fine spectra of Lacunary operators.

Let which is the left shift operator, and let the transpose of be which is the right shift operator. Let be the unit disc .

Let . A lower triangular Toeplitz matrix corresponding to is in the form And an upper triangular Toeplitz matrix corresponding to is in the form .

Lemma 1.1. *Let . Then if and only if . Moreover .*

* Proof. * Let us do the proof for . The proof for or is similar. Let denote the norm of . Firstly, we have
Now, fix , and let be a sequence such that
then we have
Hence, .

Now, let , and let denote the norm of . We have
On the other hand,
So, we have the same norm .

*Remark 1.2. *We have for the sequence spaces in Lemma 1.1 since these spaces are BK spaces.

We also have an version of the last lemma, for which we leave the proof to the reader.

Lemma 1.3. *Let . if and only if . Moreover .*

For any sequence , let us associate the function .

#### 2. Spectra of the Operators

Theorem 2.1. *Let . Then and .*

* Proof. * Let us do the proof for . The proof for can be done similarly. Let . Then
So by Lemma 1.1,
Hence,
and so .

Theorem 2.2. *Let be one of the sequence spaces , or with . Then is a bounded linear operator over with and .*

* Proof. * Let us do the proof first for for . Let ,
and , hence . In a similar way, we can show that also for . This means the spectral radius is less or equal to 1 and so
for .

Now, let us examine the eigenvalues for . If , then
If , then for all . So, let , then
Hence, for any with , the sequence is an eigenvector for . Hence, . Combining this with (2.5), we have
for , since the spectrum is a closed set.

Theorem 2.3. *Let be one of the sequence spaces or with . Then is a bounded linear operator over with and .*

* Proof. * The boundedness of the operator can be proved as in the proof of Theorem 2.2. Now, 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 and with . Hence,
It is known by Cartlidge [12] that if a matrix operator is bounded on , then . Then we also have

Theorem 2.4. *Let be a sequence such that is holomorphic in a region containing . Then .*

* Proof. * By the spectral mapping theorem for holomorphic functions (see, e.g., [13, page 569]) we have

Theorem 2.5. *Let be a sequence space. If is not a multiple of the identity mapping, then
*

* Proof. * Suppose is an eigenvalue and is an eigenvector which corresponds to . Then and so we have the following linear system of equations.
Let be the first nonzero entry of . Then the system of equations reduces to
From the first equation we get , and using the other equations in the given order we get for . This means there is no solution if there exists any with .

#### 3. Applications to the System of Equations

Let us consider the evolutionary difference equation . Here is a sequence of complex numbers. The condition is needed to make the system of equations (3.1) solvable. By solvability of a system of equations we mean that for any given sequence of complex numbers there exists a unique sequence of complex numbers satisfying the equations. Equation (3.1) may be written in the form for , and , by the change of variables for and where is a function of variables with .

Let for the following theorem.

Theorem 3.1. *The solution of the difference equation (3.2) is
**, with
**, where the summation is over nonnegative integers satisfying .*

* Proof. * We have . For , let us show that
. We can see that the left side of (3.5) can be written in the form
where the summation is over nonnegative integers satisfying . To show (3.5), let us fix and the numbers such that . Then for the coefficient , we have two cases.*Case 1 (). *, and then
*Case 2 (). *By induction we have

So, by Cases 1 and 2, we have

Now, we will prove the theorem by induction over . and so (3.3) is true for . Suppose (3.3) is true for for a positive integer . We have
Hence, (3.3) is true for .

A special case of (3.1) is the one where consists of finitely many nonzero terms. So there exists a fixed such that the equations turn into the form .

We give the following theorem, which is a direct consequence of Lemma 1.1, to compare the results of it with the results of the next theorem.

Theorem 3.2. *Let be a sequence of complex numbers such that the system of difference equations (3.1) hold for the complex number sequences and . Then the following are equivalent:*(i)*boundedness of always implies boundedness of ,*(ii)*convergence of always implies convergence of ,*(iii)* always implies ,*(iv)* always implies ,*(v)*.*

Theorem 3.3. *Suppose is a nonconstant holomorphic function on a region containing . Let be a sequence of complex numbers such that the system of difference equations (3.1) hold for the complex number sequences and . Then the following are equivalent:*(i)*boundedness of always implies boundedness of ,*(ii)*convergence of always implies convergence of ,*(iii)* always implies ,*(iv)* always implies ,*(v)* has no zero in the unit disc .*

* Proof. * Let us prove only . We will omit the proofs of , , since they are similarly proved. Since is a holomorphic function on a region containing , we have which means is bounded by Lemma 1.1. Suppose boundedness of implies boundedness of . Then the operator is onto. We have and since is not constant is one to one by Theorem 2.5. Hence, is bijective and by the open mapping theorem is continuous. This means that is not in the spectrum , so .

For the inverse implication, suppose has no zero on the unit disc . So, is in the resolvent set . Hence by Lemma 7.2-3 of [4] is defined on the whole space , which means that the boundedness of implies the boundedness of .

Corollary 3.4. *Let a polynomial be given such that the system of difference equations (3.11) hold for the complex number sequences and . Then the following are equivalent:*(i)*boundedness of always implies boundedness of ,*(ii)*convergence of always implies convergence of ,*(iii)* always implies ,*(iv)* always implies ,*(v)*all zeros of are outside the unit disc .*

Now consider the system of equations . Here is a sequence of complex numbers.

A special case of (3.12) is the one where consists of finitely many nonzero terms. So there exists a fixed such that the equations turn into the form .

Now, we again give a theorem, which is a direct consequence of Lemma 1.3, to compare the results of it with the results of the next theorem.

Theorem 3.5. *Let be a sequence of complex numbers such that the system of difference equations (3.12) hold for the complex number sequences and . Then the following are equivalent:*(i)*boundedness of always implies boundedness of ,*(ii)*convergence of always implies convergence of ,*(iii)* always implies ,*(iv)* always implies ,*(v)*.*

Theorem 3.6. *Suppose that is a holomorphic function on a region containing . Let be a sequence of complex numbers such that the system of difference equations (3.12) hold for the complex number sequences and . Then the following are equivalent:*(i)*boundedness of always implies a unique bounded solution ,*(ii)*convergence of always implies a unique convergent solution ,*(iii)* always implies a unique solution with ,*(iv)* always implies a unique solution with ,*(v)* has no zero on the unit disc .*

* Proof. * Let us prove only . We will omit the proofs of , , since they are similarly proved. Suppose boundedness of implies a unique bounded solution . Then the operator is bijective. Since is a holomorphic function on a region containing , we have which means is bounded by Lemma 1.3. By the open mapping theorem is continuous. This means that is not in the spectrum , so .

For the inverse implication, suppose that has no zero on the unit disc . So, is in the resolvent set . Hence by Lemma 7.2-3 of [4] is defined on the whole space , which means that the boundedness of implies a bounded unique solution .

Corollary 3.7. *Let a polynomial be given such that the system of difference equations (3.13) hold for the complex number sequences and . Then the following are equivalent:*(i)*boundedness of always implies a bounded unique solution ,*(ii)*convergence of always implies a convergent unique solution ,*(iii)* always implies a unique solution with ,*(iv)* always implies a unique solution with ,*(v)*all zeros of are outside the unit disc .*

*Remark 3.8. *We see that the unit disc also has an important role in examining the different types of stability conditions of difference equations or system of equations related with holomorphic functions.