Matrix Mappings on the Domains of Invertible Matrices

We focus on sequence spaces which are matrix domains of Banach sequence spaces. We show that the characterization of a random matrix operator T = (t nk ) ∈ (E A , F B ), where E A and F B are matrix domains with invertible matrices A and B, can be reduced to the characterization of the operator S = B ∘ T ∘ A−1 ∈ (E, F). As an application, the necessary and sufficient conditions for the matrix operators between invertible matrix domains of the classical sequence spaces and norms of these operators are given.


Introduction and Preliminaries
Let  denote the set of all complex sequences.Any subspace of  is called as a sequence space.We will write ℓ ∞ , , and  0 for the spaces of all bounded, convergent, and null sequences, respectively.By ℓ  , we denote the space of all  absolutely summable sequences, where 1 ≤  < ∞.
Let  and  be two sequence spaces and  = (  ) an infinite matrix of real or complex numbers   , where ,  ∈ N = {1, 2, 3, . ..}.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 Thus,  ∈ (, ) if and only if the series on the right side of (1) converges for each  ∈ N and every  ∈ , and we have  = {()  } ∈N ∈  for all  ∈ .
Let  = (  ) and  = (  ).Suppose the sums exist for all ,  ∈ N.Then, the product of  and  is defined by  = (  ).
If  is a subset of , then   = { ∈  :  ∈ } is the matrix domain of  in .We will say that a matrix  is invertible over a sequence space  if the operator  :   →  is bijective; that is, there exists an operator  −1 such that  −1 () =  for all  ∈   and ( −1 ) =  for all  ∈ .We will say  is invertible if  is invertible over .
A BK space is a Banach sequence space with continuous coordinates.The sequence spaces  0 , , ℓ  , and ℓ ∞ are the wellknown examples of BK spaces.
Several authors studied matrix mappings on sequence spaces that are matrix domains of the difference operator or of the matrices of some classical methods of summability in spaces such as ℓ  ,  0 , , or ℓ ∞ .For instance, some matrix domains of the difference operator were studied in [1,2], of the Cesàro matrices in [3,4], of the Euler matrices in [5][6][7], and of the Nörlund matrices in [8].A general approach was done in [9] reducing the characterizations of the classes (  , ) for arbitrary FK spaces  with AK and  ⊂  to those of the classes (, ) and (, ), where  is a triangle.Compact operators on matrix domains of triangles were examined in [10].The gliding hump properties of matrix domains were examined in [11].
In this work, our aim is to give some general results for matrix mappings between sequence spaces, which are matrix domains of invertible matrices of sequence spaces.Also, we give some applications of the results.An infinite matrix  = (  ) is said to be a triangle if   = 0 for  <  and   ̸ = 0 for  ∈ N. The following is a wellknown result about triangles.
Let  1 ,  2 be two matrices and  a sequence space.If  1 ( 2 ) = ( 1  2 ) holds for all  ∈ , then we will say  1 and  2 are associative over the space .If  1 and  2 are associative over the space , we will shortly say  1 and  2 are associative.For row finite matrices, we do not generally have an inverse.But we have associativity; that is, for any two row finite matrices  1 and  2 we have Moreover, we have the following result.
so the triangle inequality holds.Now, suppose that ‖‖   = 0.Then, ‖‖  = 0 and since ‖ ⋅ ‖  is a norm we have  = .Since  is invertible, we have  = .Theorem 7. Let  be a Banach sequence space and  a matrix.Then, the operator  :   →  is linear and continuous.
Proof.Let   be the operator that corresponds to the th row of the matrix operator ; that is,    = ∑ ∞ =1     for all  = (  ) ∈   .Let  = (  ) ∈   and  = (  ) ∈   .Clearly, we have    =    for any  ∈ C.
That means the operator   is linear for arbitrary , which implies the linearity of .
is continuous since

Main Results
Let  ∈  such that lim Theorem 9. Let  and  be invertible matrices over the BK spaces  and , respectively.Suppose that  ∈ (  ,   ).
By the bounded inverse theorem  −1 is continuous.Similarly the inverse  −1 of  :   →  is continuous.Now, we have that the matrix operator  =  ∘  ∘  −1 ∈ (, ) and is continuous by Theorem 1.Since  and  −1 are continuous, the operator  =  −1 ∘  ∘  is continuous.Theorem 10.Let  and  be invertible matrices over the sequence spaces  and , respectively.Then, for an operator ,  ∈ (  ,   ) if and only if  ∘  ∘  −1 ∈ (, ).
Corollary 11.Let  and  be sequence spaces and  and  triangles.Then, for an operator ,  ∈ (  ,   ) if and only if  ∘  ∘  −1 ∈ (, ).Theorem 12. Let  and  be two normed sequence spaces and  and  invertible matrices over  and , respectively.Then, for an operator ,  ∈ B(  ,   ) if and only if  ∘  ∘  −1 ∈ B(, ).In this case, one has Proof.It is enough to show the following equality:

Examples and Applications
Theorem 10, Corollary 11, and Theorem 12 have many applications, especially in the subject of characterization of matrices which act as operators between certain sequence spaces.We just give a taste by the following examples and theorems.
Example 13.Let  = (  ) be a sequence of nonzero scalars in C. For any  = (  ) ∈ , let  = (    ).Then, for a sequence space , the multiplier sequence space (), associated with the multiplier sequence , is defined as Then, the matrix Λ is invertible with and () =  Λ .So, all the Theorems in Section 2 are applicable for multiplier sequence spaces (see [13] for detailed applications and examples).
Example 14.Let  = (  ) be a row-finite matrix and  the operator with matrix representation is invertible and the inverse operator has the matrix representation and  −1 are triangles and so they are one to one. −1 is the operator (, ) of [14] with  = 1 and  = −1.Since both  and  −1 are row finite, we have  ∘  −1 =  −1 .Now, using Corollary 11 we have that  is in (  , ) if and only if  −1 is in (, ): so by the Kojima-Schur Theorem (see, e.g., Theorem 2.7 of [15]) we have that  ∈ (  , ) if and only if the following three conditions hold: (i) lim    −  (+1) exists for each ; (ii) lim  ∑    −  (+1) exists; Now, let us examine (,   ). is in (,   ) if and only if  is in (, ).We have where   is defined as the sum of the first  terms of the th column of the matrix ; that is, Now, by using the Kojima-Schur Theorem, we have,  ∈ (,   ) if and only if the following three conditions hold: (i) lim    exists for each ; (ii) lim  ∑    exists; Now, let us examine (  ,   ). is in (  ,   ) if and only if  −1 is in (, ).We have where  0 = 0 for all  ∈ N. Now, using Theorem 2.
Example 17.Let  = (  ) be a row finite matrix and  the Cesàro operator which has a matrix representation Then,   is the well-known space of Cesàro summable sequences.This is a sequence space which includes the space of convergent sequences . is invertible and the inverse operator has the matrix representation Since , , and  −1 are all row finite, the operator  ∘  ∘  −1 is represented by the matrix  −1 .Now, using Corollary 11, we have that  is in , where (ℓ ∞ )  denotes the space, introduced by Ng and Lee [3], of all sequences whose -transforms are in the space ℓ ∞ .Let   be defined as in the previous example.Then, after some calculations, we have where Proof.By Corollary 11,  ∈ (,   ) if and only if  ∘  ∈ (, ).
For the reverse implication, suppose that conditions (33)-(36) hold.Then, by the Kojima-Schur Theorem, conditions (34)-( 36 Finally, let one give an application on a matrix operator which is not a triangle. where   ̸ = 0 for all  ∈ N.Then, a matrix  ∈ ( 0Γ ,  0 ) if and only if the following three conditions hold: where  0 / 0 = 0.
Proof.Suppose that  ∈ ( 0Γ ,  0 ).The operator Γ :  Γ →  is one to one, because for Γ = 0 with  = (  ) ∈  we have the system of equations where  0 = 0.By a similar proof to the proof of Lemma 19, we can see that the sequence (   −1  ) ∞ =1 is bounded for each , when  ∈ ( 0Γ , ).Then, (49) Hence, we have all conditions (43).We leave the reverse implication part to the reader.