Hypo-EP Matrices of Adjointable Operators on Hilbert C∗-Modules

+e EP matrix, as an extension of the normal matrix, was proposed by Schwerdtfeger; a square matrix T over the complex field C is said to be an EP matrix if T and T∗ share the same range [1, 2]. +e notion of EP matrices was extended by Campbell and Meyer to operators with closed range on a Hilbert space in [3]. Let H be a complex Hilbert space and B(H) the collection of all bounded linear operators on H. Let T ∈B(H). Recall that T is called an EP operator if its range, R(T), is closed, and R(T) � R(T∗) [3]. It is well known that R(T) is closed if and only if the Moore–Penrose inverse T† of T exists and that T is an EP operator if and only if T†T � TT†. Sharifi [4] provided a generalization of the result for EP operators on Hilbert C∗-modules. +is has been studied by many other authors, see, e.g., [5–8] and references therein. More generally, T is said to be a hypo-EP operator if T†T≥TT† [9]. In fact, T is a hypo-EP operator if and only if R(T) is closed and R(T)⊆R(T∗). It is also shown that T is a hypo-EP operator if and only if T†T2T† � TT†. +e hypo-EP operator is our focus of attention in this paper, and it has been studied in [10, 11]. +e EP operator can be applied to the solution of operator equations, see Section 3 of this article. +e properties of hypo-EP and EP operators can find applications also in reverse order law [12] and core partial order [13] and will be useful in some other applied fields [14, 15]. In this note, we investigate the hypo-EP operators on Hilbert C∗-modules, and then we formulate some results of hypo-EP matrices of adjointable operators on Hilbert C∗-modules. As an application, the solvability conditions, and the general expression for the EP solution to the operator equations are given. Since the finite-dimensional spaces, Hilbert spaces, and C∗-algebras can all be regarded as Hilbert C∗-modules, one can study hypo-EPmodular operators in a unified way in the framework of Hilbert C∗-modules. Let us briefly recall some basic knowledge about Hilbert C∗-modules and adjointable operators. +roughout this paper, A is a C∗-algebra. A Hilbert A-module H is a right A-module equipped with an A-valued inner product 〈·, ·〉: H × H⟶ A such thatH is complete with respect to the induced norm


Introduction and Preliminaries
e EP matrix, as an extension of the normal matrix, was proposed by Schwerdtfeger; a square matrix T over the complex field C is said to be an EP matrix if T and T * share the same range [1,2]. e notion of EP matrices was extended by Campbell and Meyer to operators with closed range on a Hilbert space in [3]. Let H be a complex Hilbert space and B(H) the collection of all bounded linear operators on H. Let T ∈ B(H). Recall that T is called an EP operator if its range, R(T), is closed, and R(T) � R(T * ) [3]. It is well known that R(T) is closed if and only if the Moore-Penrose inverse T † of T exists and that T is an EP operator if and only if T † T � TT † . Sharifi [4] provided a generalization of the result for EP operators on Hilbert C * -modules. is has been studied by many other authors, see, e.g., [5][6][7][8] and references therein. More generally, T is said to be a hypo-EP operator if T † T ≥ TT † [9]. In fact, T is a hypo-EP operator if and only if R(T) is closed and R(T)⊆R(T * ). It is also shown that T is a hypo-EP operator if and only if T † T 2 T † � TT † . e hypo-EP operator is our focus of attention in this paper, and it has been studied in [10,11]. e EP operator can be applied to the solution of operator equations, see Section 3 of this article. e properties of hypo-EP and EP operators can find applications also in reverse order law [12] and core partial order [13] and will be useful in some other applied fields [14,15]. In this note, T ∈ L(H, K) is said to be regular if there is an operator It is easy to prove that T is regular if and only if R(T) is closed. e 1 { }-inverse of T is not unique in general.
In this paper, we use the generalized inverse to the generalized Schur complement as defined in [16]. Suppose M ∈ L(H ⊕ K) is a modular operator matrix of the form where where A − is an inner inverse of A. Similarly, the generalized where D − is an inner inverse of D. e formulas (2) and (3) have previously appeared in papers dealing with generalized inverses of partitioned matrices (cf. [17][18][19]).
ese equations imply that T † will be uniquely determined if it exists, and T † T and TT † are both orthogonal projections. Moreover, R(T † ) � R(T † T), R(T) � R (TT † ), N(T) � N(T † T), and N(T † ) � N(TT † ). Clearly, the Moore-Penrose inverse T † of T exists if and only if R(T) is closed; T is Moore-Penrose invertible if and only if T * is Moore-Penrose invertible, and in this case, (T * ) † � (T † ) * . Obviously, the Moore-Penrose inverse T † of T is one of inner inverses of T.
From Lemma 2, we can obtain the following corollary.
Proof. e proof is similar to that in [22], Corollary 12, for Hilbert space operators. □ Definition 2 (see [4]). Let H be a Hilbert A-module. An Obviously, the range of an EP or a hypo-EP operator on Hilbert C * -modules is not necessarily closed, and we further have the following properties.
Proposition 1 (see [4]). Let H be a Hilbert A-module and T ∈ L(H) with closed range. en, the following conditions are equivalent: Let H be a Hilbert A-module and T ∈ L(H) with closed range. en, the following conditions are equivalent: e class of all hypo-EP operators contains the class of all EP operators on Hilbert A-modules. Meanwhile, the EP operator with closed range is an extension of the invertible operator and the normal operator with closed range. In the case of finite dimensional situation, EP and hypo-EP are the same.

Main Results and Proofs
First, using generalized Schur complements, we study the hypo-EP property of matrices of adjointable operators on Hilbert C * -modules. Theorem 1. Let M be a modular operator matrix of the form (1) with where Obviously, L and R are invertible. By using Lemma 1 and by assumptions N(A) ⊆ N(C) and . By using Lemma 1 again, it is immediate that holds for every inner inverse P − of P. In particular, for we have from relation (7) that by Corollary 1. Using N(A * ) ⊆ N(B * ) and N((M/ A) * ) ⊆ N(C * ), by Lemma 1, MM † is described as Similarly, by using N(A) ⊆ N(C), N(M/A) ⊆ N(B), and Lemma 1, it is given that en, Since A and M/A are hypo-EP operators with closed range, us, M † M 2 M † � MM † . erefore, M is a hypo-EP operator matrix with closed range. e following conclusion is a natural extension of [10], eorem 3.1, on Hilbert C * -modules. Next, using the properties of generalized inverses, we study upper triangular hypo-EP matrices of adjointable operators on Hilbert C * -modules. Proof. Let M be a hypo-EP operator matrix with closed range. We write Obviously, L is invertible. By Lemma 1 and assumption N(D) ⊆ N(B), it is clear that M can be decomposed as M � LP. Hence, N(M) � N(P). Since M is a hypo-EP operator matrix with closed range, N(P) � N(M) ⊆ N (M * ). By Lemma 1, it is immediate that M * � M * P † P, where P † is given by is gives Since N(A * ) ⊆ N(B * ), by Lemma 1, MM † is described as Similarly, by Lemma 1, N(D) ⊆ N(B) leads to en, Since A and D are hypo-EP operators with closed range, erefore M is a hypo-EP operator matrix with closed range. Complexity Hence, A * � AA † A * and D * � DD † D * imply N(A * ) ⊆ N(A) and N(D * ) ⊆ N(D), respectively.
Conversely, let A and D be EP operators. Since R(A)and R(D) are closed, N(D) ⊆ N(B), and N erefore, M is an EP operator matrix with closed range. e Hilbert space version of the preceding four conclusions is given by [10], and the conditions of closed range can be naturally omitted there. Moreover, the alternative proofs of the conclusions in Hilbert space setting can be found in section 3 of [10]. In addition, these results originated from the research of the EP property of block matrices, according to Hartwig [24].
Finally, the following are devoted to investigating the hypo-EP property of antitriangular block matrices of adjointable operators on Hilbert C * -modules.
and We write X: By Definition 1, M † � X as desired. Necessity: since Using N(C) ⊆ N(A) and N(B * ) ⊆ N(A * ), by Lemma 1, we have us, M is an EP operator matrix with closed range. □ Proof. e sufficiency is clear by Lemma 5. Now, we suppose that M is a hypo-EP operator matrix with closed range. We write In the similar way as in the proof of eorem 3, we have M � LP, and hence, N(P) � N(M) ⊆ N(M * ), since M is a hypo-EP with closed range. is means M * � M * P † P by Lemma 1, i.e., Hence, C * � C * B † B, which together with N(C) � N (B), implies N(C) ⊆ N(C * ). us, C is a hypo-EP operator. Similarly, it follows from B * � B * C † C and N(B) � N(C) that N(B) ⊆ N(B * ), and therefore, B is a hypo-EP operator. Proof. By Corollary 7, we only need to show the necessity, which can be easily verified according to the proofs of Corollary 4 and eorem 4.