It is well known the Sherman-Morrison-Woodbury (for short SMW) formula gives an explicit form for the inverse of matrices of the form :
where A and G be and nonsingular matrices with . Also, let Y and Z be matrices such that is invertible. The SMW formula (1) is valid only if the matrices A and are invertible. Over the years, Generalizations (see   for example) have been considered in the case of singular or rectangular matrices using the concept of Moore-Penrose generalized inverses. Certain results on extending the SMW formula to operators on Hilbert spaces are also considered by many authors (see    ).
Let X and Y be Banach spaces, and be the Banach space consisting of all bounded linear operators from X to Y. For , let (resp. ) denote the kernel (resp. range) of A. It is well known that for , if and are topologically complemented in the spaces X and Y, respectively, then there exists a linear projector generalized inverse defined by and , where and are topologically complemented subspaces of and , respectively. In this case, is the projection from X onto along and is the projection from Y onto along . But, in general, we know that not every closed subspace in a Banach space is complemented, thus, the linear generalized inverse of A may not exist. In this case, we may seek other types of generalised inverses for T. Motivated by the ideas of linear generalized inverses and metric generalized inverses (cf.  ), by using the so-called homogeneous (resp. quasi-linear) quasi-linear projectors in Banach space, in  , the authors defined homogeneous (resp. quasi-linear) projector generalized inverse. Then, in   , the authors give a further study on this type of generalized inverse in Banach space. More important, from the results in  , we know that, in some reflexive Banach spaces X and Y, for an operator , there may exist a bounded homogeneous (quasi-linear) projector generalized inverse of T, which is generally neither linear nor metric generalized inverse of T. So, from this point of view, it is important and necessary to study homogeneous (resp. quasi-linear) generalized inverses in Banach spaces. From then on, many research papers about the Moore-Penrose metric generalized inverses have appeared in the literature.
The objectives of this paper are concerned with certain extensions of the so called Sherman-Morrison-Woodbury formula to operators between some Banach spaces. We consider the SMW formula in which the inverse is replaced by bounded homogeneous generalized inverse. More precisely, let be Banach spaces, and we denote the set of all bounded linear operators from X into Y by and by when . Let , , and , such that and exist, In the main part of this paper, we will develop some conditions under which the Sherman-Morrison-Woodbury formula can be represented as
where is a bounded homogeneous generalized inverse of T. As a consequence, some particular cases and applications will be also considered. Our results generalize the results of many authors for liner operator generalized inverses.
In this section, we recall some concepts and basic results will be used in this paper. We first present some facts about homogeneous operators. Let be Banach spaces. Denote by the set of all bounded homogeneous operators from X to Y. Equipped with the usual linear operations for , and for , the norm is defined by . then similar to the space of bounded linear operators, we can easily prove that is a Banach space. For a bounded homogeneous operator , we always assume that .
Definition 2.1 (  ). Let be a subset and let be a mapping. Then we call is quasi-additive on if satisfies
For a homogeneous operator , if is quasi-additive on , then we will simply say is a quasi-linear operator.
Definition 2.2 (  ). Let . If , we call is a homogeneous projector. In addition, if is also quasi-additive on , i.e., for any and any ,
then we call is a quasi-linear projector.
The following concept of bounded homogeneous generalized inverse is also a generalization of bounded linear generalized inverse.
Definition 2.3 (  ). Let . If there is such that
then we call is a bounded homogeneous generalized inverse of .
Definition 2.3 was first given in paper  for linear transformations and bounded linear operators. The existence condition of a homogeneous generalized inverse is also given in  .
3. Main Results
In this section, we mainly study the SMW formula for bounded homogeneous generalized inverses of a bounded linear operator in Banach spaces. In order to prove our main theorems, we first need to present some lemmas. The following result is well-known for bounded linear operators, we can generalize it to bounded homogeneous operators as follows.
Lemma 3.1 (  ). Let such that is quasi-additive on and is quasi-additive on , then is invertible if and only if is invertible. Specially, when and , if is quasi-additive on , then is invertible if and only if is invertible.
The following result is well-known for bounded linear operators, we generalize it to the bounded homogeneous operators and metric projections in the following form.
Lemma 3.2. Let . Let be a subspace and be the quasi-linear projection from onto .
1) if and only if ;
2) If is quasi-additive on , then if and only if .
Proof. Here, we only prove (1), and (2) can be proved in the same way. On the one hand, if , then . On the other hand, for any , since, , we can get that , thus, . This completes the proof. ,
Lemma 3.3. Let such that exists. Then and .
Proof. Since , we have
So, . Similarly, we also have
Therefore, . ,
Theorem 3.4. Let , , and , such that and exist, also, let and Math_123# such that and exist. Suppose that is quasi-additive on and , if
Proof. From (2) and Lemma 3.3,
Then, using Lemma 3.2, we obtain
Note that is quasi-additive on , thus, we have , and then
Similarly, by Lemma 3.2, and also note that , then, from , we get . Now, Since is also quasi-additive on and , thus
Now using Lemma 3.2 again, also note that and (4), we get
Now, using (5), by simple computation, we can obtain
This completes the proof. ,
In above Theorem 3.4, if and are all invertible, we have the following result.
Corollary 3.5. Let , , and such that and exist, also, let and such that and exist. Suppose that is quasi-additive on and , if
Furthermore, if is invertible and in Corollary 3.5, then we also have the following result.
Corollary 3.6 (  , Theorem 2.1). Let , , and such that is invertible. Then is invertible if and only if is invertible. Furthermore, when is invertible, then
Proof. Since exists, then, from Lemma 3.1, we see is invertible if and only if is invertible. Now, using the equality , we see is invertible if and only if is invertible. The formula (7) can be obtained by some simple computations. ,
Theorem 3.7. Let , , and , such that and exist, also, let and Math_187# such that and exist. Suppose that and are quasi-additive on and . If any of the following conditions holds:
Proof. For convenience, set , we will show that . Here, we only give the proof under the assumption (i). Another can be proved similarly. Note that, if , , then by Lemma 3.2, we have
Consequently, we obtain
Similarly, we can also check that . Thus, we have
From Definition 2.3, we have This completes the proof. ,
If we let and assume that is invertible in Theorem 3.7, then we can get the following result.
Corollary 3.8. Let , and such that exists. Suppose that is quasi-additive on . If is invertible and , then and
In this paper, we develop conditions under which the so-called Sherman-Morrison-Woodbury formula can be represented by the bounded homogeneous generalized inverse. More precisely, we will develop some conditions under which the Sherman-Morrison-Woodbury formula holds for the bounded homogeneous generalized inverse in Banach space. Note that this is the first related results about nonlinear generalized inverse. As a result, our results generalize the results of many authors for finite dimensional matrices and Hilbert space operators in the literature.
The author is supported by China Postdoctoral Science Foundation (No. 2015M582186), the Science and Technology Research Key Project of Education Department of Henan Province (No. 18A110018), Henan Institute of Science and Technology Postdoctoral Science Foundation (No. 5201029470209).
 Riedel, K.S. (1992) A Sherman-Morrison-Woodbury Identity for Rank Augmenting Matrices with Application to Centering. SIAM Journal on Matrix Analysis and Applications, 13, 659-662.