# Hypo-EP Matrices of Adjointable Operators on HilbertC∗-Modules

Authors: Xiaopeng Li; Junjie Huang; Alatancang Chen
Journal: Complexity
 (2021)

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## Abstract

This paper introduces and studies hypo-EP matrices of adjointable operators on HilbertC∗-modules, based on the generalized Schur complement. The necessary and sufficient conditions for some modular operator matrices to be hypo-EP are given, and some special circumstances are also analyzed. Furthermore, an application of the EP operator in operator equations is given.


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## Body

## 1. Introduction and Preliminaries

The EP matrix, as an extension of the normal matrix, was proposed by Schwerdtfeger; a square matrixT over the complex field ℂ is said to be an EP matrix if T and T∗ share the same range [1, 2]. The 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 ℬH the collection of all bounded linear operators on H. Let T∈ℬH. Recall that T is called an EP operator if its range, ℛT, is closed, and ℛT=ℛT∗ [3]. It is well known that ℛT 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. This 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 ℛT is closed and ℛT⊆ℛT∗. It is also shown that T is a hypo-EP operator if and only if T†T2T†=TT†. The hypo-EP operator is our focus of attention in this paper, and it has been studied in [10, 11]. The EP operator can be applied to the solution of operator equations, see Section 3 of this article. The 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, andC∗-algebras can all be regarded as Hilbert C∗-modules, one can study hypo-EP modular 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. Throughout 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 that ℋ is complete with respect to the induced norm ⋅=⋅,⋅1/2. Suppose that ℋ and K are Hilbert A-modules, and let ℒℋ,K be the set of all maps T:ℋ⟶K for which there is a map T∗:K⟶ℋ such that Tx,y=x,T∗y for x∈ℋ and y∈K. It is well known that an arbitrary element T of ℒℋ,K must be a bounded linear operator, which is also A-linear in the sense of Txa=Txa for any x∈ℋ and a∈A. We call ℒℋ,K the set of adjointable operators from ℋ to K. We use ℒℋ to denote the C∗-algebra ℒℋ,ℋ. Let ℒℋsa be the set of Hermitian elements of ℒℋ. For T∈ℒℋ,K, the range and the null space of T are denoted by ℛT and NT, respectively. An operator T∈ℒℋ,K is said to be regular if there is an operator T−∈ℒℋ,K satisfying TT−T=T; T− is called an inner inverse (or 1-inverse) of T. It is easy to prove that T is regular if and only if ℛT is closed. The 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∈ℒℋ⊕K is a modular operator matrix of the form(1)M=ABCD,where A∈ℒℋ,B∈ℒK,ℋ,C∈ℒℋ,K,D∈ℒK. Then, the generalized Schur complement of A in M is(2)MA=D−CA−B,where A− is an inner inverse of A. Similarly, the generalized Schur complement of D in M is(3)MD=A−BD−C,where D− is an inner inverse of D. The formulas (2) and (3) have previously appeared in papers dealing with generalized inverses of partitioned matrices (cf. [17–19]).Definition 1.
(see [20]). Let T∈ℒℋ,K. The Moore–Penrose inverse T† of T (if exists) is an element in ℒK,ℋ which satisfies(a)
TT†T=T(b)
T†TT†=T†(c)
TT†∗=TT†(d)
T†T∗=T†T
These equations imply thatT† will be uniquely determined if it exists, and T†T and TT† are both orthogonal projections. Moreover, ℛT†=ℛT†T, ℛT=ℛTT†, NT=NT†T, and NT†=NTT†. Clearly, the Moore–Penrose inverse T† of T exists if and only if ℛT 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.
Similar to [21], Lemma 2.2.4, and [22], Lemma 2.2, we have the following conclusions on Hilbert C∗-modules.Lemma 1.
LetA∈ℒℋ,B∈ℒK,ℋ and C∈ℒℋ,K. If A has an inner inverse A−, then(i)
NA⊆NC if and only if C=CA−A(ii)
NA∗⊆NB∗ if and only if B=AA−BLemma 2.
LetM be a modular operator matrix of form (1) with NA⊆NC and NA∗⊆NB∗. If A has an inner inverse A−, then M is regular if and only if M/A is regular, where M/A=D−CA−B. In this case, the inner inverse of M is given by(4)M−=A−+A−BMA−CA−−A−BMA−−MA−CA−MA−.From Lemma2, we can obtain the following corollary.Corollary 1.
LetM be a modular operator matrix of form (1) with NA⊆NC, NA∗⊆NB∗, NM/A⊆NB, and NM/A∗⊆NC∗. IfℛAandℛM/A are closed, then the Moore–Penrose inverse M† of M can be expressed as(5)M†=A†+A†BMA†CA†−A†BMA†−MA†CA†MA†.Remark 1.
The preceding result given in [17], Theorem 1, was proved for finite matrices.Lemma 3 (see [23]).
LetM=AB0D∈ℒℋ⊕K, where A∈ℒℋ,B∈ℒK,ℋ, and D∈ℒK. If ℛA and ℛD are closed, then M†=A†−A†BD†0D† if and only if ND⊆NB and NA∗⊆NB∗.Proof.
The 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 operator T∈ℒℋ is called EP if ℛT¯=ℛT∗¯.Definition 3.
Letℋ be a Hilbert A-module. An operator T∈ℒℋ is called hypo-EP if ℛT¯⊆ℛT∗¯.
Obviously, the range of an EP or a hypo-EP operator on HilbertC∗-modules is not necessarily closed, and we further have the following properties.Proposition 1 (see [4]).
Letℋ be a Hilbert A-module and T∈ℒℋ with closed range. Then, the following conditions are equivalent:(i)
T is an EP operator(ii)
NT=NT∗(iii)
T is Moore–Penrose invertible and T†T=TT†Proposition 2.
Letℋ be a Hilbert A-module and T∈ℒℋ with closed range. Then, the following conditions are equivalent:(i)
T is a hypo-EP operator(ii)
NT⊆NT∗(iii)
T is Moore–Penrose invertible and T†T2T†=TT†Remark 2.
The class of all hypo-EP operators contains the class of all EP operators on HilbertA-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.


## 2. Main Results and Proofs

First, using generalized Schur complements, we study the hypo-EP property of matrices of adjointable operators on HilbertC∗-modules.Theorem 1.
LetM be a modular operator matrix of the form (1) with NA⊆NC, NA∗⊆NB∗, NM/A⊆NB, and NM/A∗⊆NC∗. Suppose that ℛA and ℛM/A are closed. Then, the following conditions are equivalent:(i)
M is a hypo-EP operator matrix with closed range(ii)
A and M/A are hypo-EP operatorsProof.
LetM be a hypo-EP operator matrix with closed range. Since ℛA and ℛM/A are closed, let us consider the operator matrices(6)L=I0CA−I,R=IBMA−0I,P=A00MA,where M/A=D−CA−B. Obviously, L and R are invertible. By using Lemma 1 and by assumptions NA⊆NC and NM/A⊆NB, it is clear that M can be factorized as M=LRP. Hence, NP=NM. Since M is a hypo-EP operator matrix with closed range, NP=NM⊆NM∗. By using Lemma 1 again, it is immediate that(7)M∗=M∗P−Pholds for every inner inverse P− of P. In particular, for(8)P−=A−00MA−,we have from relation (7) that(9)M∗=A∗C∗B∗D∗=A∗C∗B∗D∗A−A00MA−MA,=A∗A−AC∗MA−MAB∗A−AD∗MA−MA.
Then,A∗=A∗A−A implies NA⊆NA∗. Hence, A is a hypo-EP operator. Since C∗=C∗M/A−M/A, substituting D=M/A+CA−B into(10)D∗=D∗M/A−M/Ayields M/A∗=M/A∗M/A−M/A. This implies NM/A⊆NM/A∗. Thus, M/A is a hypo-EP operator.
Conversely, according to the assumptionsNA⊆NC, NM/A⊆NB, NA∗⊆NB∗, and NM/A∗⊆NC∗, the Moore–Penrose inverse M† of M exists, and M† is given by(11)M†=A†+A†BMA†CA†−A†BMA†−MA†CA†MA†,by Corollary 1. Using NA∗⊆NB∗ and NM/A∗⊆NC∗, by Lemma 1, MM† is described as(12)MM†=AA†00MAMA†.
Similarly, by usingNA⊆NC, NM/A⊆NB, and Lemma 1, it is given that(13)M†M=A†A00MA†MA.
Then,(14)M†M2M†=M†MMM†=A†A2A†00MA†MA2MA†.
SinceA and M/A are hypo-EP operators with closed range,(15)A†A2A†=AA†,MA†MA2MA†=MAMA†.
Thus,M†M2M†=MM†. Therefore, M is a hypo-EP operator matrix with closed range.
The following conclusion is a natural extension of [10], Theorem 3.1, on Hilbert C∗-modules.Corollary 2.
LetA,X∈ℒℋ and M=AAXX∗AX∗AX∈ℒℋ⊕ℋ. If ℛA is closed, then M is a hypo-EP operator matrix with closed range if and only if A is a hypo-EP operator.If using the generalized Schur complementM/D=A−BD−C of D in M, similar to Theorem 1, one can get the following results.Theorem 2.
LetM be a modular operator matrix of form (1) with ND⊆NB, ND∗⊆NC∗, NM/D⊆NC, and NM/D∗⊆NB∗. Suppose that ℛD and ℛM/D are closed. Then, the following conditions are equivalent:(i)
M is a hypo-EP operator matrix with closed range(ii)
D and M/D are hypo-EP operatorsCorollary 3.
LetD,X∈ℒℋ and M=X∗DXX∗DDXD∈ℒℋ⊕ℋ. If ℛD is closed, then M is a hypo-EP operator matrix with closed range if and only if D is a hypo-EP operator.Next, using the properties of generalized inverses, we study upper triangular hypo-EP matrices of adjointable operators on HilbertC∗-modules.Theorem 3.
LetM=AB0D∈ℒℋ⊕K with ND⊆NB and NA∗⊆NB∗, where A∈ℒℋ,B∈ℒK,ℋ, and D∈ℒK. If ℛA and ℛD are closed, then M is a hypo-EP operator matrix with closed range if and only if A and D are hypo-EP operators.Proof.
LetM be a hypo-EP operator matrix with closed range. We write(16)L≔IBD†0I,P≔A00D.
Obviously,L is invertible. By Lemma 1 and assumption ND⊆NB, it is clear that M can be decomposed as M=LP. Hence, NM=NP. Since M is a hypo-EP operator matrix with closed range, NP=NM⊆NM∗. By Lemma 1, it is immediate that M∗=M∗P†P, where P† is given by(17)P†=A†00D†.
This gives(18)M∗=A∗0B∗D∗=A∗0B∗D∗A†A00D†D,=A∗A†A0B∗A†AD∗D†D.
Hence,A∗=A∗A†A implies NA⊆NA∗. Thus, A is a hypo-EP operator. From D∗=D∗D†D, it follows that ND⊆ND∗. Therefore, D is a hypo-EP operator.
Conversely, supposeA and D are hypo-EP operators. Since ℛA and ℛD are closed and ND⊆NB and NA∗⊆NB∗, by Lemma 3, the Moore–Penrose inverse M† of M exists and(19)M†=A†−A†BD†0D†.
SinceNA∗⊆NB∗, by Lemma 1, MM† is described as(20)MM†=AA†00DD†.
Similarly, by Lemma1, ND⊆NB leads to(21)M†M=A†A00D†D.
Then,(22)M†M2M†=M†MMM†=A†A2A†00D†D2D†.
SinceA and D are hypo-EP operators with closed range, A†A2A†=AA† and D†D2D†​​=​​DD†. Thus, M†M2M†=MM†. Therefore M is a hypo-EP operator matrix with closed range.Corollary 4.
LetM=AB0D∈ℒℋ⊕K with ND⊆NB and NA∗⊆NB∗, where A∈ℒℋ, B∈ℒK,ℋ, and D∈ℒK. If ℛA and ℛD are closed, then M is an EP operator matrix with closed range if and only if A and D are EP operators.Proof.
LetM be an EP operator matrix with closed range. In view of Theorem 3, to prove the necessity, it is enough to show NA∗⊆NA and ND∗⊆ND. Since M is an EP operator matrix with closed range, by the proof of Theorem 3, NP=NM=NM∗. Applying Lemma 1, we have P∗=MM†P∗, i.e.,(23)P∗=A∗00D∗=AA†00DD†A∗00D∗,=AA†A∗00DD†D∗.
Hence,A∗=AA†A∗ and D∗=DD†D∗ imply NA∗⊆NA and ND∗⊆ND, respectively.
Conversely, letA and D be EP operators. Since ℛAand ℛD are closed, ND⊆NB, and NA∗⊆NB∗, we get M†=A†−A†BD†0D†. Thus,(24)M†M−MM†=A†A−AA†00D†D−DD†=0000.
Therefore,M is an EP operator matrix with closed range.Corollary 5.
LetA,X,D∈ℒℋ and M=AAXD0D∈ℒℋ⊕ℋ. If ℛA and ℛD are closed, then M is a hypo-EP operator matrix with closed range if and only if A and D are hypo-EP operators.Corollary 6.
LetA∈ℒℋ and M=AA0A∈ℒℋ⊕ℋ. If ℛA is closed, then M is a hypo-EP operator matrix with closed range if and only if A is a hypo-EP operator.Remark 3.
The 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 HilbertC∗-modules.Lemma 4.
LetM=ABC0∈ℒℋ⊕ℋ. If ℛB and ℛC are closed, then M†=0C†B†−B†AC† if and only if NC⊆NA and NB∗⊆NA∗.Proof.
Sufficiency: sinceℛB and ℛC are closed, B and C are Moore–Penrose invertible. From NC⊆NA and NB∗⊆NA∗, it follows that AC†C=A and BB†A=A. We write X:=0C†B†−B†AC†. A direct calculation shows that(25)MXM=M,XMX=X,MX∗=MX,XM∗=XM.
By Definition1, M†=X as desired.
Necessity: since(26)MM†=BB†AC†−BB†AC†0CC†,M†M=C†C0B†A−B†AC†CB†Bare self-adjoint, we have AC†−BB†AC†=0 and B†A−B†AC†C=0. From MM†M=M, we get BB†A=A=AC†C. Therefore, NC⊆NA and NB∗⊆NA∗.Lemma 5.
LetM=ABC0∈ℒℋ⊕ℋ with NC⊆NA and NB∗⊆NA∗. If B and C are hypo-EP operators with closed ranges, then M is a hypo-EP operator matrix with closed range.Proof.
SinceℛB and ℛC are closed, NC⊆NA, and NB∗⊆NA∗, by Lemma 4, the Moore–Penrose inverse M† of M is given by(27)M†=0C†B†−B†AC†.
UsingNC⊆NA and NB∗⊆NA∗, by Lemma 1, we have(28)MM†=BB†00CC†,M†M=C†C00B†B.
By Definition1, we have ℛMM†=ℛM and ℛM†M=ℛM†=ℛM∗. Since B and C are hypo-EP operators with closed ranges, ℛBB†=ℛB⊆ℛB∗=ℛB†B and ℛCC†=ℛC⊆ℛC∗=ℛC†C. Then,(29)ℛM=ℛMM†=ℛBB†⊕ℛCC†⊆ℛB†B⊕ℛC†C=ℛM†M=ℛM∗.
Therefore,M is a hypo-EP operator with closed range.Corollary 7.
LetM=ABC0∈ℒℋ⊕ℋ with NC⊆NA and NB∗⊆NA∗. If B and C are EP operators with closed ranges, then M is an EP operator matrix with closed range.Proof.
According to the assumption, as with Lemma5, we have equation (28). Since B and C are EP operators with closed ranges, ℛBB†=ℛB=ℛB∗=ℛB†B and ℛCC†=ℛC=ℛC∗=ℛC†C. Then,(30)ℛM=ℛMM†=ℛBB†⊕ℛCC†=ℛB†B⊕ℛC†C=ℛM†M=ℛM∗.
Thus,M is an EP operator matrix with closed range.Theorem 4.
LetM=ABC0∈ℒℋ⊕ℋ with NB=NC⊆NA and NB∗⊆NA∗. If ℛB and ℛCare closed, then M is a hypo-EP operator matrix with closed range if and only if B and C are hypo-EP operators.Proof.
The sufficiency is clear by Lemma5. Now, we suppose that M is a hypo-EP operator matrix with closed range. We write(31)L≔IAC†0I,P≔0BC0.
In the similar way as in the proof of Theorem3,
we haveM=LP, and hence, NP=NM⊆NM∗, since M is a hypo-EP with closed range. This means M∗=M∗P†P by Lemma 1, i.e.,(32)A∗C∗B∗0=M∗=M∗P†P=A∗C†CC∗B†BB∗C†C0.
Hence,C∗=C∗B†B, which together with NC=NB, implies NC⊆NC∗. Thus, C is a hypo-EP operator. Similarly, it follows from B∗=B∗C†C and NB=NC that NB⊆NB∗, and therefore, B is a hypo-EP operator.Corollary 8.
LetB,X,C∈ℒℋ and M=BXCBC0∈ℒℋ⊕ℋ with NB=NC. If ℛB and ℛC are closed, then M is a hypo-EP operator matrix with closed range if and only if B and C are hypo-EP operators.Corollary 9.
LetM=ABC0∈ℒℋ⊕ℋ with NB=NC⊆NA and NB∗⊆NA∗. If ℛB and ℛCare closed, then M is an EP operator matrix with closed range if and only if B and C are EP operators.Proof.
By Corollary7, we only need to show the necessity, which can be easily verified according to the proofs of Corollary 4 and Theorem 4.Corollary 10.
LetB,X,C∈ℒℋ and M=BXCBC0∈ℒℋ⊕ℋ with NB=NC. If ℛB and ℛC are closed, then M is an EP operator matrix with closed range if and only if B and C are EP operators.Remark 4.
In Hilbert space case, the conditions of closed range in Theorem4 and Corollary 9 can be naturally omitted in Theorem 3.8 and Theorem 3.9 of [10], and the alternative proofs of Theorem 4 and Corollary 9 can be, respectively, found in Theorem 3.8 and Theorem 3.9 of [10].


## 3. The Application of EP Operators

In this section, letℋ,K, and G be Hilbert spaces. We establish the solvability conditions and the general expression for the EP solution to the operator equations(33)AX=C,XB=D,where A,C∈ℬℋ,K, B,D∈ℬG,ℋ, and X∈ℬℋ.Lemma 6 (see [25]).
LetT∈ℬℋ with closed range. Then, the operator T is EP if and only if there exist Hilbert spaces ℋ1 and ℋ2, U∈ℬℋ1⊕ℋ2,ℋ unitary, and T1∈ℬℋ1 isomorphism such that(34)T=UT1⊕0U∗,where ℋ1⊕ℋ2=ℋ.Lemma 7 (see [22]).
LetA,C∈ℬℋ,K and B,D∈ℬG,ℋ. Suppose that A and B have closed ranges. Then, equation (33) has a common solution X∈ℬℋ if and only if(35)NA∗⊆NC∗,NB⊆ND,AD=CB.In this case, the general common solution is given by(36)X=A−C+DB−−A−ADB−+Iℋ−A−AYIℋ−BB−,where Y∈ℬℋ is arbitrary.Now, we consider the EP solution to equation (33). By the Lemma 6, for the unitary operator U∈ℬℋ1⊕ℋ2,ℋ, the solution has the following factorization:(37)X=UX1000U∗.LetℛA, ℛB be closed, and(38)AU=A1A2,CU=C1C2,U∗B=B1B2,U∗D=D1D2,where A1,C1∈ℬℋ1,K, A2,C2∈ℬℋ2,K, B1,D1∈ℬG1,ℋ, B2,D2∈ℬG2,ℋ, and ℛA1, ℛB1 are closed. Then, equation (33) has an EP solution if and only if operator equations(39)A1X1=C1,X1B1=D1,C2=0,D2=0have a common solution. By Lemma 7, we have the following theorem.Theorem 5.
LetA,C∈ℬℋ,K and B,D∈ℬG,ℋ, and let ℛA, ℛB be closed. Suppose that U∈ℬℋ1⊕ℋ2,ℋ is unitary such that(40)AU=A1A2,CU=C1C2,U∗B=B1B2,U∗D=D1D2,where A1,C1∈ℬℋ1,K, A2,C2∈ℬℋ2,K, B1,D1∈ℬG1,ℋ, B2,D2∈ℬG2,ℋ, and ℛA1 and ℛB1 are closed. Then, equation (33) has an EP solution X∈ℬℋ if and only if(41)NA1∗⊆NC1∗,NB1⊆ND1,A1D1=C1B1,C2=D2=0.In this case, the general EP solution of (33) is given by(42)X=UA1−C1+D1B1−−A1−A1D1B1−+Iℋ1−A1−A1Y1Iℋ1−B1B1−000U∗,where Y1∈ℬℋ1 is arbitrary.


## 4. Concluding Remarks

In this work, we have characterized hypo-EP and EP matrices of adjointable operators on HilbertC∗-modules, based on the generalized Schur complement, and an application of EP operator in operator equations is presented. In addition, the properties of hypo-EP and EP operators may have potential applications in some fields involving mathematics and its applications. In our opinion, it is worth establishing the hypo-EP and EP matrices of bounded linear operators on Krein space and of the adjointable operators on Krein C∗-modules.



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