Over the Class of Convergent Series

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.


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 p l , 0 c , 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 0 c and c and later Akhmedov and Başar also determined the fine spectrum of the difference operator  over

2.Preliminaries, Background and Notations
Let X and Y be the Banach spaces and : T X Y  also be a bounded linear operator.By

 
RT , we denote the range of T , i.e.,

   
By   BX, we also denote the set of all bounded linear operators on X into itself.If X is any Banach space and 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 1 T   , that is

775
  and it is called to be the resolvent operator of T .
The name resolvent is appropriate, since

 
T B X  , then there are three possibilities for   RT and 1 T  :

and
(1) 1  T  exists and is continuous, If these possibilities are combined in all possible ways, nine different states are created.These are labelled by: 1 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 In what follows, we call the set the approximate point spectrum of T .Moreover, the subspectrum is called defect spectrum of T .The two subspectra given by (2.1) and (2.2) 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, we note that .
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.
The equality (g) implies, in particular, that     ,, ap T 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 , 0 c and bv for the spaces of all bounded, convergent, null and bounded variation sequences, respectively.

Let
X and Y be two sequence spaces and By well-established convention, we define the space  of all convergent series is defined by 0 ( ) : (2.9) An upper triangular double-band infinite matrix is of the form 000 0 0 0 ( , ) 0 0 0 T   is a bounded linear operator with the matrix , then its adjoint operator :  is defined by transpose of the matrix  and  is isomorphic to Let us take   2 e   .Then, since , ,0,0,0, U r s e s r  , we find that Thus, we have the result from (3.1) and (3.2).
. Then, by solving the system of equations, and we obtain from (3. Since the spectrum of any bounded linear operator is closed, we have the result from (3.7).

   
Goldberg Classifications for the operator ( , ) U r s are given at following two theorems: Proof.To prove this theorem, we must show that    Taking limits of both sides of the inequality in (3.9) for k , we get

p bv where 1 p
   in [14].The fine spectrum of the generalized difference operator over the sequence spaces 0 c 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 0 c and c .Later, Karaisa and Başar [17] also determined the fine spectra of upper triangular triple-band matrix on  .Finally, Dutta and Tripathy [16] have determined fine spectrum of the generalized difference operator over the class of convergent series.
linear operator with domain   TX  .By T , associate the operator

2 II
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

x
is the firs no-zero term of the sequence for any i  which is a contradiction.If j x is the non-zero term of the sequence

Fine spectrum of upper triangular
In this section, we give the main result about the spectrum and the subdivisions of spectrum.
Proof.It is not hard to prove that   , U r s is linear from   ,:U r s   to itself and so we omit it. and for all k  then we have