A decisive break in the theory of matrix transformations was in 1950, when Robinson considered the action of infinite matrices of linear operators from a Banach space on sequences of elements of that space . In the past years, many remarkable results    were yielded in this direction.
Let X and Y be topological vector spaces, and . For sequence families and , the matrix means that converges when , and for each .
In 2001, Li Ronglu depicted the nonlinear operator matrices transformation with some restrictive condition on topological vector spaces . In the next year, Li Ronglu gave some clear-cut characterizations of the matrix families and consisted of matrices of linear and some nonlinear operators between topological vector spaces . In this paper, we study the summability theory for a class of matrices of nonlinear mapping on Banach space, and discuss the characterization of the matrix classes:
, , .
All of the researches enrich the results on infinite matrices transformations, and have important meaning for the study of Banach space.
2. Preliminaries and Lemmas
In 1993, nonlinear Schur Theorem was given by Li Ronglu and C. Swartz, and broke the limitations of linear operator matrices.
Theorem A.  Let G be an Abelian topological group, , a matrix in such that for some and all . If i.e., exists for each , then the series converges uniformly with respect to both and , and exists for every . If, in addition, G is sequentially complete, then the converse implication is true.
As a special case, the following theorem is a nice result for the matrix family .
Theorem B.  Let be topological vector spaces and a mapping such that for every . If , then for every bounded , the series converges uniformly with respect to both and and exists for every . If, in addition, Y is sequentially complete, then the converse implication is true.
Note that theorem B exceeded the restriction of linear operators, and a characterization of was given. For Banach spaces , it is useful to discuss the characterization of a variety of matrix families, where the mapping need not be linear.
As preparation of the proves of the main results, we also need following lemma.
Lemma  if and only if for all .
3. Main Results
Unless otherwise noted below are Banach spaces, and the mapping we studied in this section need not be linear.
Theorem 1. Let for all , then if and only if
(i) exists for all and ;
(ii) For any , there exists such that for all natural number , , and with .
Proof. Necessity of condition (i) and (ii) is easy to prove by the theorem B in Introduction.
Now suppose that (i) and (ii) are hold, and , then for any , there exists such that for all by the condition (ii). And because of condition (i) there is , such that for all . Hence we have
for all . Therefore
So is a Cauchy sequence in Y. Therefore converges by the completeness of Y, and then . The sufficiency is proved. Q.E.D.
Since , we can get the next corollary by the theorem.
Corollary 1. Suppose that , then if and only if for all , and for any , there exists such that for all , and with .
Proof. Necessity is clear by above theorem and the definition of .
Conversely, let , then for any , there exists such that for all . Since for all and , there is , such that for all and .
Hence we have
So column converges to 0, and then . The sufficiency is proved. Q.E.D.
Theorem 2. Let with respect to , then if and only if
(i) for all ;
(ii) For any and , there exists such that for all and with .
Proof. ⇒) Suppose that , the condition (i) is clear.
Since for every , for every by lemma 1, that is . Hence, for every , we have . Therefore, by above corollary, for every and there is such that for all , , and , we have
condition (ii) is proved.
⇐) For every , and , we have by the condition (i). Because of the condition (ii), we have by the Corollary 1, and then for , we have is hold for every . Therefore by lemma 1, and then . Q.E.D.
Theorem 3. Let with respect to , then if and only if
(i) for all and ;
(ii) For any and , there exists such that
Proof. For , since , for any and , by condition (i). So condition (i) and (ii) is equivalent to by corollary 1.
Suppose that , then for all . By lemma 1, , for all and . Hence .
On the other hand, suppose that . For every , there exists and , such that , and so . Hence . Q.E.D.
Theorem 4. Let with respect to , then if and only if
(i) for all and ;
(ii) For any and , there exists , such that for all and .
Proof. By condition (i), for all and ,
By theorem 2, condition (i) and (ii) are equivalent to . Next, we prove that for all is equivalent to .
In fact, If , and let , then . Since for all by lemma, we have . So . On the other hand, suppose that . Since for any , there must be , , such that , we have
and then . Q.E.D.
In this paper, we first review the research history of infinite matrix transformation, and then we mainly study the summability of a class of nonlinear mapping matrices in Banach space.
And some new results about, matrix transformation theorems are obtained: we characterize the matrix classes such as , , , .
This research was supported by the Science and Technology Project of Jilin Provincial Department of Education (JJKH20180891KJ).
 Bani-Ahmad, F.A. and Bani-Domi, W. (2016) New Norm Equalities and Inequalities for Operator Matrices. Journal of Inequalities and Applications, 2016, Article number: 175.
 Li, R.L., Kang, S.M. and Swartz, C. (2002) Operator Matrices on Topological Vector Spaces. Journal of Mathematical Analysis and Applications, 274, 645-658.