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 , where and are matrix domains with invertible matrices and , can be reduced to the characterization of the operator . 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.
1. Introduction and Preliminaries
Let denote the set of all complex sequences. Any subspace of is called as a sequence space. 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 Banach spaces. Then, is the set of all operators and is the set of all continuous linear operators . is a Banach space with the operator norm defined by .
Let and be two sequence spaces and 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 Thus, if and only if the series on the right side of (1) converges for each and every , and we have for all .
Let and . Suppose the sums exist for all , . 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 such that for all and 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 , , , and are the well-known 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 , , , or . 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 [5–7], and of the Nörlund matrices in . A general approach was done in  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 . The gliding hump properties of matrix domains were examined in .
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.
Theorem 1 (see [12, Theorem , page 57]). Matrix operators between BK spaces are continuous.
An infinite matrix is said to be a triangle if for and for . The following is a well-known result about triangles.
Theorem 2 (see [12, 1.4.8, page 9]). Every triangle has an inverse which also is a triangle, and for all .
Let be two matrices and a sequence space. If holds for all , then we will say and are associative over the space . If and are associative over the space , we will shortly say and 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 and we have Moreover, we have the following result.
Theorem 3 (see [12, Theorem , page 8]). Let be two matrices.(i)If rows of are in and , then and are associative over .(ii)If is row finite, then and are associative over .
Corollary 4. The set of matrices are associative over .
Remark 5. The associativity property does not hold in general (see, e.g., [12, Example ]).
Theorem 6. Let be an invertible matrix over a normed sequence space with norm . Then, is a normed sequence space with norm .
Proof. Let and . Then,
Secondly, for we have
so the triangle inequality holds.
Now, suppose that . Then, 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, for all . Let and . Clearly, we have for any . Let and . We have . That means the operator is linear for arbitrary , which implies the linearity of .
is continuous since
2. Main Results
Theorem 8. If is a Banach sequence space and is an invertible matrix over , then is a Banach space.
Proof. Let be a Banach sequence space. Then, is a normed space by the previous theorem. Now, let be a Cauchy sequence in . Let . Then is a sequence in and is Cauchy in since Let such that in . Let . Then,
Theorem 9. Let and be invertible matrices over the BK spaces and , respectively. Suppose that . If over , then the operator is continuous.
Proof. Let and let be the norm of . Since is invertible over , is bijective. Also, is continuous by Theorem 7.
By the bounded inverse theorem is continuous. Similarly the inverse of is continuous.
Now, we have that the matrix operator and is continuous by Theorem 1. Since and are continuous, the operator is continuous.
Theorem 10. Let and be invertible matrices over the sequence spaces and , respectively. Then, for an operator , if and only if .
Proof. Suppose that , and let . Then, clearly . Hence, we have and so .
For the inverse implication, suppose that , and let . Then, and so . Since is invertible, we have .
Corollary 11. Let and be sequence spaces and and triangles. Then, for an operator , if and only if .
Theorem 12. Let and be two normed sequence spaces and and invertible matrices over and , respectively. Then, for an operator , if and only if . In this case, one has
Proof. It is enough to show the following equality:
3. 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 . For any , let . Then, for a sequence space , the multiplier sequence space , associated with the multiplier sequence , is defined as Let Then, the matrix is invertible with and . So, all the Theorems in Section 2 are applicable for multiplier sequence spaces (see  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 are triangles and so they are one to one. is the operator of  with and . Since both and are row finite, we have . Now, using Corollary 11 we have that is in if and only if is in :
so by the Kojima-Schur Theorem (see, e.g., Theorem 2.7 of ) we have that if and only if the following three conditions hold:(i) exists for each ;(ii) exists;(iii).
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) exists for each ;(ii) exists;(iii).Now, let us examine . is in if and only if is in . We have Now, by using the Kojima-Schur Theorem we have that if and only if the following three conditions hold:(i) exists for each ;(ii) exists;(iii).
Theorem 15. An infinite matrix if and only if where denotes the collection of all finite subsets of .
Proof. First, we observe that , where is the difference operator of Example 14. The inverse of is . By Corollary 11, if and only if . Since , we have and so by the Schur Theorem (see, e.g., Theorem 2.11 of ) rows of are in . By Theorem 3 part (ii), and are associative over . So, over . We have where for all . Now, using Theorem 2.14 of , we have if and only if
Using Theorem 12, we have the following.
Corollary 16. If , then and
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 are all row finite, the operator is represented by the matrix . Now, using Corollary 11, we have that is in if and only if is in , where denotes the space, introduced by Ng and Lee , of all sequences whose -transforms are in the space .
Let be defined as in the previous example. Then, after some calculations, we have Then, is in if and only if by Theorem 2.6 of . So, we have if and only if (27) holds. Similarly, we have or if and only if (27) holds, where and denote the spaces and , studied by Şengönül and Başar , of all sequences whose -transforms are in the spaces and , respectively.
Theorem 18. Let C be the Cesàro matrix. Then, if and only if where .
Proof. By Corollary 11, if and only if . Since , we have that rows of are in . By Theorem 3 part (ii), and are associative over . So, over . We have Now, using Kojima-Schur Theorem, we have if and only if conditions (28) hold.
Lemma 19. If is convergent for all Cesàro summable sequences , then .
Proof. Suppose that . Then, there exist and a subsequence of such that . Without loss of generality, we can choose this subsequence such that . Let Then, the sequence is Cesàro summable because for we have On the other hand, and so we get to the contradiction for the Cesàro summable sequence . So, our assumption is not true.
Theorem 20. Let C be the Cesàro matrix. Then, if and only if where .
Proof. Suppose that . Then, by Lemma 19, we have (33). By Corollary 11, if and only if . First let us show that and are associative over . Let . We have that
is convergent in . Let be the th partial sum of this series. Then,
and is row finite. Then, by Theorem 3 part (ii), and are associative over . So, over . We have
Now, using Kojima-Schur Theorem, implies conditions (34)–(36).
For the reverse implication, suppose that conditions (33)–(36) hold. Then, by the Kojima-Schur Theorem, conditions (34)–(36) imply . By (33), over . By Part (ii) of Theorem 3, and are associative over , and so we have over . Hence, , and now by Corollary 11, .
Corollary 21. If , then and Finally, let one give an application on a matrix operator which is not a triangle.
Theorem 22. Let where for all . Then, a matrix if and only if the following three conditions hold: where .
Proof. Suppose that . The operator is one to one, because for with we have the system of equations
Now, let us show that is onto. Let . Then for with since Hence, is onto. So, is bijective. The inverse operator is then Now, by Theorem 10, if and only if .
For we have , and then for we have . Let be the th partial sum of this series. Then, where .
By a similar proof to the proof of Lemma 19, we can see that the sequence is bounded for each , when . Then, The conditions for to be in , by Example page 129 of , are Hence, we have all conditions (43). We leave the reverse implication part to the reader.
The author thanks the referees for their valuable comments and suggestions.
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