Compact Operators for Almost Conservative and Strongly Conservative Matrices

and Applied Analysis 3 It is easy to see that for E ∈M l p χ (E) = lim n→∞ (sup


Introduction and Preliminaries
For some basic definitions and notations of this section we refer to [1,2].Let  denote the space of all complex sequences  = (  ), and let  be the set of all sequences that terminate in zeros.Let ℓ ∞ , , and  0 denote the spaces of all bounded, convergent, and null sequences, respectively.We will write  and ℓ 1 for the spaces of all convergent and absolutely convergent series, respectively.Further, we will use the conventions that  = (1, 1, 1, . ..) and  () = (0, 0, . . ., 1, 0, . ..)where 1 at the th place for each  ∈ N = {1, 2, 3, . ..}.
For arbitrary sequence spaces  and , we write (, ) for the class of all infinite matrices that map  into .Thus  ∈ (, ) if and only if   ∈   for all  ∈ N and  ∈  for all  ∈ .
The theory of  spaces is the most powerful tool in the characterization of matrix transformations between sequence spaces.
A sequence space  is called a  space if it is a Banach space with continuous coordinates   :  → C( ∈ N), where C denotes the complex field and   () =   for all  = (  ) ∈  and every  ∈ N.
The sequence spaces  0 , , and ℓ ∞ are  spaces with the usual sup norm given by ‖‖ ℓ ∞ = sup  |  |, where the supremum is taken over all  ∈ N. Also, the space ℓ 1 is a  space with the usual ℓ 1 -norm defined by If  ⊃  is a  space and  = (  ) ∈ , then we write Abstract and Applied Analysis provided the expression on the right exists and is finite which is the case whenever  ∈   , where   is the unit sphere in ; that is,   = { ∈  : ‖‖ = 1}.
A sequence (  ) ∞ =0 in a linear metric space (, ) is called a Schauder basis (or briefly basis) for  if for every  ∈  there exists a unique sequence (  ) ∞ =0 of scalars such that  = ∑ ∞ =1     ; that is, (,  [] ) → 0 ( → ∞), where  [] = ∑  =0     is known as the -section of .The series ∑      which has the sum  is called the expansion of , and (  ) is called the sequence of coefficients of  with respect to the basis (  ).
Let  and  be Banach spaces.Then, we write B(, ) for the set of all bounded linear operators  :  → , which is a Banach space with the operator norm given by ‖‖ = sup ∈  ‖()‖  for all  ∈ B(, ).A linear operator  :  →  is said to be compact if the domain of  is all of  and for every bounded sequence (  ) in , the sequence ((  )) has a subsequence which converges in .An operator  ∈ B(, ) is said to be of finite rank if dim () < ∞, where () denotes the range space of .An operator of finite rank is clearly compact.Further, we write C(, ) for the class of all compact operators from  to .Let us remark that every compact operator in C(, ) is bounded; that is, C(, ) ⊂ B(, ).More precisely, the class C(, ) is a closed subspace of the Banach space B(, ) with the operator norm.
Finally, the following known results are fundamental for our investigation.

The Hausdorff Measure of Noncompactness
Most of the definitions, notations, and basic results of this section are taken from [3].Throughout, we will write M  for the collection of all bounded subsets of a metric space (, ).
If  ∈ M  , then the Hausdorff measure of noncompactness of the set , denoted by (), is defined to be the infimum of the set of all reals  > 0 such that  can be covered by a finite number of balls of radii <  and centers in .This can equivalently be redefined as follows: The function  : If ,  1 , and  2 are bounded subsets of a metric space , then we have Further, if  is a normed space, then the function  has some additional properties connected with the linear structure; for example, Let  and  be Banach spaces and  1 and  2 be the Hausdorff measures of noncompactness on  and , respectively.An operator  :  →  is said to be ( Let  and  be Banach spaces and  ∈ B(, ).Then, the Hausdorff measure of noncompactness of , denoted by ‖‖  , can be determined by and we have that where  = lim sup  → ∞ ‖−  ‖ and the operator   :  → , defined for each  ∈ N by   () = ∑  =0   ()  ( ∈ ), is called the projector onto the linear span of { 0 ,  1 , . . .,   }.Besides, all operators   and  −   are equibounded, where  denotes the identity operator on .

Almost Conservative Matrices
A continuous linear functional  on ℓ ∞ is said to be a Banach limit if it has the following properties: (i) () = 0 if  = 0, (ii) () = 1, and (iii) () = (); where  is a shift operator defined by ()  =  +1 .A bounded sequence  = (  ) is said to be almost convergent (Lorentz [5]) to the value  if all of its Banach limits coincide; that is, () =  for all Banach limits .
Lorentz established the following characterization.
A sequence  = (  ) is almost convergent to the number  if and only if   () →  as  → ∞ uniformly in , where The number  is called the generalized limit of , and we write  = −lim .We denote the set of all almost convergent sequences by ; that is, Remark 6.Note that  ⊂  ⊂ ℓ ∞ and each inclusion is proper.Remark 7. Since  ⊂  ⊂ ℓ ∞ , we have ℓ 1 = ℓ  ∞ ⊂   ⊂   = ℓ 1 and hence   = ℓ 1 .Therefore, it is natural by ( 4) and Lemma 1 that ‖‖ *  = ‖‖ ℓ 1 for all  ∈ ℓ 1 .
Using the idea of almost convergence, King [7] defined and characterized the almost conservative and almost regular matrices.
An infinite matrix  = (  ) ∞ ,=1 is said to be almost conservative if  ∈  for all  ∈ , and we denote it by  ∈ (, ).If in addition  − lim  = lim , then  is called almost regular.
It is worth mentioning that the condition in ( 18) is only a sufficient condition for the operator   to be compact, where  is an almost conservative matrix.More precisely, the following example will show that it is possible for   to be compact while lim  → ∞ (∑ ∞ =1 |  |) ̸ = 0. Hence, in general, we have just "if " in (18) of Theorem 11.

Compact Operators for Strongly Conservative Matrices
An infinite matrix  = (  ) ∞ ,=1 is said to be strongly conservative if  ∈  for all  ∈ , and we denote it by  ∈ (, ).If in addition  − lim  = lim , then  is called strongly regular (cf.[5]).
In this final section, we establish some necessary and sufficient (or only sufficient) conditions for operators to be compact for matrix classes (, ), where  = ,  0 , ℓ ∞ .
We may begin with the following lemmas which will be needed in the sequel.