The product of operators with closed range in Hilbert C*-modules
aa r X i v : . [ m a t h . OA ] F e b THE PRODUCT OF OPERATORS WITH CLOSED RANGE IN HILBERTC*-MODULES
K. SHARIFI
Abstract.
Suppose T and S are bounded adjointable operators with close range betweenHilbert C*-modules, then T S has closed range if and only if
Ker ( T ) + Ran ( S ) is an orthog-onal summand, if and only if Ker ( S ∗ ) + Ran ( T ∗ ) is an orthogonal summand. Moreover,if the Dixmier (or minimal) angle between Ran ( S ) and Ker ( T ) ∩ [ Ker ( T ) ∩ Ran ( S )] ⊥ ispositive and Ker ( S ∗ ) + Ran ( T ∗ ) is an orthogonal summand then T S has closed range. Introduction.
The closeness of range of operators is an attractive and important problem which appearsin operator theory, especially, in the theory of Fredholm operators and generalized inverses.In this paper we will investigate when the product of two operators with closed range againhas closed range. This problem was first studied by Bouldin for bounded operators betweenHilbert spaces in [3, 4]. Indeed, for Hilbert space operators
T, S whose ranges are closed,he proved that the range of
T S is closed if and only if the Dixmier (or minimal) anglebetween
Ran ( S ) and Ker ( T ) ∩ [ Ker ( T ) ∩ Ran ( S )] ⊥ is positive, where the Dixmier anglebetween subspaces M and N of a certain Hilbert space is the angle α ( M, N ) in [0 , π/ c ( M, N ) = sup {kh x, y ik : x ∈ M, k x k ≤ , y ∈ N, k y k ≤ } . Nikaido [24, 25] also gave topological characterizations of the problem for the Banach spaceoperators. Recently (Dixmier and Friedrichs) angles between linear subspaces have beenstudied systematically by Deutsch [7], he also has reconsidered the closeness of range of theproduct of two operators with closed range. In this note we use C*-algebras techniques toreformulate some results of Bouldin and Deutsch in the framework of Hilbert C*-modules.Some further characterizations of modular operators with closed range are obtained.Hilbert C*-modules are essentially objects like Hilbert spaces, except that the inner prod-uct, instead of being complex-valued, takes its values in a C*-algebra. Since the geometry ofthese modules emerges from the C*-valued inner product, some basic properties of Hilbert
Mathematics Subject Classification.
Primary 47A05; Secondary 15A09, 46L08, 46L05.
Key words and phrases.
Bounded adjointable operator, Moore-Penrose inverse, closed range, Hilbert C*-module, C*-algebra, Dixmier angle.This research was in part supported by a grant from IPM (No. 89460018). spaces like Pythagoras’ equality, self-duality, and decomposition into orthogonal comple-ments do not hold. The theory of Hilbert C*-modules, together with adjointable operatorsforms an infrastructure for some of the most important research topics in operator algebras,in Kasparov’s KK-theory and in noncommutative geometry.A (left) pre-Hilbert C*-module over a C*-algebra A is a left A -module E equipped withan A -valued inner product h· , ·i : E × E → A , ( x, y )
7→ h x, y i , which is A -linear in the firstvariable x (and conjugate-linear in y ) and has the properties: h x, y i = h y, x i ∗ , h ax, y i = a h x, y i for all a in A , h x, x i ≥ x = 0 . A pre-Hilbert A -module E is called a Hilbert A -module if E is a Banach space with respectto the norm k x k = kh x, x ik / . A Hilbert A -submodule E of a Hilbert A -module F is anorthogonal summand if F = E ⊕ E ⊥ , where E ⊥ := { y ∈ F : h x, y i = 0 for all x ∈ E } denotes the orthogonal complement of E in F . The papers [9, 10] and the books [19, 22] areused as standard sources of reference.Throughout the present paper we assume A to be an arbitrary C*-algebra (i.e. notnecessarily unital). We use the notations Ker ( · ) and Ran ( · ) for kernel and range of operators,respectively. We denote by L ( E, F ) the Banach space of all bounded adjointable operatorsbetween E and F , i.e., all bounded A -linear maps T : E → F such that there exists T ∗ : F → E with the property h T x, y i = h x, T ∗ y i for all x ∈ E , y ∈ F . The C*-algebra L ( E, E ) is abbreviated by L ( E ).In this paper we first briefly investigate some basic facts about Moore-Penrose inverses ofbounded adjointable operators on Hilbert C*-modules and then we give some necessary andsufficient conditions for closeness of the range of the product of two orthogonal projections.These lead us to our main results. Indeed, for adjointable module maps T, S whose rangesare closed we show that the operator
T S has closed range if and only if
Ker ( T ) + Ran ( S )is an orthogonal summand, if and only if Ker ( S ∗ ) + Ran ( T ∗ ) is an orthogonal summand.The Dixmier angle between submodules M and N of a Hilbert C*-module E is the angle α ( M, N ) in [0 , π/
2] whose cosine is defined by c ( M, N ) = sup {kh x, y ik : x ∈ M, k x k ≤ , y ∈ N, k y k ≤ } . If the Dixmier angle between
Ran ( S ) and Ker ( T ) ∩ [ Ker ( T ) ∩ Ran ( S )] ⊥ is positive and Ker ( S ∗ ) + Ran ( T ∗ ) is an orthogonal summand then T S has closed range. Since every C*-algebra is a Hilbert C*-module over itself, our results are also remarkable in the case ofbounded adjointable operators on C*-algebras.
HE PRODUCT OF OPERATORS WITH CLOSED RANGE 3 Preliminaries
Closed submodules of Hilbert modules need not to be orthogonally complemented at all,but Lance states in [19, Theorem 3.2] under which conditions closed submodules may beorthogonally complemented (see also [22, Theorem 2.3.3]). Let E , F be two Hilbert A -modules and suppose that an operator T in L ( E, F ) has closed range, then one has: • Ker ( T ) is orthogonally complemented in E , with complement Ran ( T ∗ ), • Ran ( T ) is orthogonally complemented in F , with complement Ker ( T ∗ ), • the map T ∗ ∈ L ( F, E ) has closed range, too.
Lemma 2.1.
Suppose T ∈ L ( E, F ) . The operator T has closed range if and only if T T ∗ has closed range. In this case, Ran ( T ) = Ran ( T T ∗ ) .Proof. Suppose T has closed range, the proof of Theorem 3.2 of [19] indicates that Ran ( T T ∗ )is closed and Ran ( T ) = Ran ( T T ∗ ).Conversely, if T T ∗ has closed range then F = Ran ( T T ∗ ) ⊕ Ker ( T T ∗ ) = Ran ( T T ∗ ) ⊕ Ker ( T ∗ ) ⊂ Ran ( T ) ⊕ Ker ( T ∗ ) ⊂ F which implies T has closed range. (cid:3) Let T ∈ L ( E, F ), then a bounded adjointable operator S ∈ L ( F, E ) is called an innerinverse of T if T ST = T . If T ∈ L ( E, F ) has an inner inverse S then the bounded adjointableoperator T × = ST S in L ( F, E ) satisfies(2.1)
T T × T = T and T × T T × = T. The bounded adjointable operator T × which satisfies (2.1) is called generalized inverse of T .It is known that a bounded adjointable operator T has a generalized inverse if and only if Ran ( T ) is closed, see e.g. [5, 31].Let T ∈ L ( E, F ), then a bounded adjointable operator T † ∈ L ( F, E ) is called the
Moore-Penrose inverse of T if(2.2) T T † T = T, T † T T † = T † , ( T T † ) ∗ = T T † and ( T † T ) ∗ = T † T. The notation T † is reserved to denote the Moore-Penrose inverse of T . These propertiesimply that T † is unique and T † T and T T † are orthogonal projections. Moreover, Ran ( T † ) = Ran ( T † T ), Ran ( T ) = Ran ( T T † ), Ker ( T ) = Ker ( T † T ) and Ker ( T † ) = Ker ( T T † ) whichlead us to E = Ker ( T † T ) ⊕ Ran ( T † T ) = Ker ( T ) ⊕ Ran ( T † ) and F = Ker ( T † ) ⊕ Ran ( T ) . Xu and Sheng in [30] have shown that a bounded adjointable operator between two HilbertC*-modules admits a bounded Moore-Penrose inverse if and only if the operator has closedrange. The reader should be aware of the fact that a bounded adjointable operator may admitan unbounded operator as its Moore-Penrose, see [13, 28, 29] for more detailed information.
K. SHARIFI
Proposition 2.2.
Suppose
E, F, G are Hilbert A -modules and S ∈ L ( E, F ) and T ∈ L ( F, G ) are bounded adjointable operators with closed ranges. Then T S has a generalized inverse ifand only if T † T SS † has. In particular, T S has closed range if and only if T † T SS † has.Proof. Suppose first that V is a generalized inverse of T S . Then T † T SS † ( SV T ) T † T SS † = T † T ( SS † S ) V ( T T † T ) SS † = T † T S V T SS † = T † T SS † . Similarly,
SV T ( T † T SS † ) SV T = SV T and so
SV T is a generalized inverse of T † T SS † .Conversely, suppose that U ∈ L ( F ) is a generalized inverse of T † T SS † . Let P = SS † and Q = T † T are orthogonal projections onto Ran ( S ) and Ker ( T ) ⊥ , respectively, then QP U QP = QP . We set W = P U Q , then
P W Q = W and QW P = QP . The later equalityimplies that Q (1 − W ) P = 0, that is, 1 − W maps Ran ( P ) = Ran ( S ) into Ker ( Q ) = Ker ( T ).Consequently, T (1 − W ) S = 0. Hence, T S ( S † W T † ) T S = T P W QS = T W S = T S.
On the other hand, S † W T † = S † P U QT † = S † SS † U T † T T † = S † U T † which shows that( S † W T † ) T S ( S † W T † ) = S † U T † = S † W T † , i.e. S † W T † is a generalized inverse of T S . Inparticular,
T S has closed range if and only if T † T SS † has. (cid:3) Lemma 2.3.
Let T ∈ L ( E, F ) , then T has closed range if and only if Ker ( T ) is orthogonallycomplemented in E and T is bounded below on Ker ( T ) ⊥ , i.e. k T x k ≥ c k x k , for all x ∈ Ker ( T ) ⊥ for a certain positive constant c . The statement directly follows from Proposition 1.3 of [12].
Lemma 2.4.
Let T be a non-zero bounded adjointable operator in L ( E, F ) , then T has closedrange if and only if Ker ( T ) is orthogonally complemented in E and γ ( T ) = inf {k T x k : x ∈ Ker ( T ) ⊥ and k x k = 1 } > . In this case, γ ( T ) = k T † k − and γ ( T ) = γ ( T ∗ ) .Proof. The first assertion follows directly from Lemma 2.3. To prove the first equality,suppose T has closed range, x ∈ Ker ( T ) ⊥ = Ran ( T † T ) and k x k = 1, then 1 = k x k = k T † T x k ≤ k T † k k T x k , consequently, k T † k − ≤ γ ( T ) . Suppose x ∈ Ker ( T ) ⊥ then γ ( T ) k x k ≤k T x k . Suppose w ∈ F and x = T † w then x ∈ Ran ( T † ) = Ker ( T ) ⊥ , hence, γ ( T ) k T † w k ≤ k T T † w k ≤ k T T † k k w k ≤ k w k . HE PRODUCT OF OPERATORS WITH CLOSED RANGE 5
We therefore have γ ( T ) ≤ k T † k − . To establish the second equality just recall that T hasclosed range if and only if T ∗ has. It now follows from the first equality and the fact k T ∗ † k = k T † ∗ k = k T † k . (cid:3) Closeness of the range of the products
Suppose F is a Hilbert A -module and T be a bounded adjointable operator in the unitalC*-algebra L ( F ), then σ ( T ) and acc σ ( T ) denote the spectrum and the set of all accumu-lation points of σ ( T ), respectively. According to [17, Theorem 2.4] and [30, Theorem 2.2],a bounded adjointable operator T in L ( F ) has closed range if and only if T has a Moore-Penrose inverse, if and only if 0 / ∈ acc σ ( T T ∗ ), if and only if 0 / ∈ acc σ ( T ∗ T ). In particular,if T is selfadjoint then T has closed range if and only if 0 / ∈ acc σ ( T ). We use these facts inthe proof of the following results. Lemma 3.1.
Suppose F is a Hilbert A -module and P, Q are orthogonal projections in L ( F ) .Then P − Q has closed range if and only if P + Q has closed range.Proof. Following the argument of Koliha and Rakoˇcevi´c [18], for every λ ∈ C we have(3.1) ( λ − P )( λ − ( P − Q ))( λ − Q ) = λ ( λ − P Q ) , (3.2) ( λ − P )( λ − ( P + Q ))( λ − Q ) = λ (( λ − − P Q ) . Using the above equations and the facts that σ ( P ) ⊂ { , } and σ ( Q ) ⊂ { , } , we obtainthat Ran ( P − Q ) is closed if and only if 0 / ∈ acc σ ( P − Q ), if and only if 1 / ∈ acc σ ( P Q ), ifand only if 0 / ∈ acc σ ( P + Q ), if and only if Ran ( P + Q ) is closed. (cid:3) Lemma 3.2.
Suppose F is a Hilbert A -module and P, Q are orthogonal projections in L ( F ) .Then the following conditions are equivalent:(i) P Q has closed range,(ii) − P − Q has closed range,(iii) − P + Q has closed range,(iv) − Q + P has closed range.Proof. Suppose λ ∈ C \ { , } . In view of the equation (3.2), we conclude that λ ∈ σ ( P + Q )if and only if ( λ − ∈ σ ( P Q ).The above fact together with Remark 1.2.1 of [23] imply that
P Q has closed range if andonly if 0 / ∈ acc σ ( P QP ), if and only if 0 / ∈ acc σ ( P Q ), if and only if 1 / ∈ acc σ ( P + Q ), ifand only if 0 / ∈ acc σ (1 − P − Q ), if and only if 1 − P − Q has closed range. This proves K. SHARIFI the equivalence of (i) and (ii). The statements (ii), (iii) and (iv) are equivalent by Lemma3.1. (cid:3)
Remark . Suppose E , F are two Hilbert A -modules then the set of all ordered pairs ofelements E ⊕ F from E and F is a Hilbert A -module with respect to the A -valued innerproduct h ( x , y ) , ( x , y ) i = h x , x i E + h y , y i F , cf. [26, Example 2.14]. In particular, it canbe easily seen that L is a closed submodule of F if and only if L ⊕ { } is a closed submoduleof F ⊕ F . Lemma 3.4.
Suppose P and Q are orthogonal projections on a Hilbert A -module F thenthe following conditions are equivalent:(i) P Q has closed range,(ii)
Ker ( P ) + Ran ( Q ) is an orthogonal summand,(iii) Ker ( Q ) + Ran ( P ) is an orthogonal summand.Proof. Suppose T = − P Q ! ∈ L ( F ⊕ F ) . Then
Ran ( T ) = ( Ran (1 − P ) + Ran ( Q )) ⊕ { } and Ran ( T T ∗ ) = Ran (1 − P + Q ) ⊕ { } .Using Lemmata 2.1, 3.2 and Remark 3.3, we infer that P Q has closed range if and only if1 − P + Q has closed range, if and only if Ran ( T T ∗ ) = Ran (1 − P + Q ) ⊕ { } is closed, if andonly if Ran ( T ) = ( Ran (1 − P ) + Ran ( Q )) ⊕ { } is closed, if and only if Ran (1 − P ) + Ran ( Q )is closed. In particular, Ran (1 − P + Q ) = Ran (1 − P ) + Ran ( Q ) is an orthogonal summand.This proves that the conditions (i) and (ii) are equivalent. Now, consider the matrix operator˜ T = − Q P ! ∈ L ( F ⊕ F ) . A similar argument shows that
P Q has closed range if and only if
Ran (1 − Q + P ) = Ran (1 − Q ) + Ran ( P ) is closed which shows that conditions (i) and (iii) are equivalent. (cid:3) Suppose M and N are closed submodule of a Hilbert A -module E and P M and P N areorthogonal projection onto M and N , respectively. Then P M P N = P M if and only if P N P M = P M , if and only if M ⊂ N . Beside these, the following statements are equivalent • P M and P N commute, i.e. P M P N = P N P M , • P M P N = P M ∩ N , • P M P N is an orthogonal projection, • P M ⊥ and P N commute, HE PRODUCT OF OPERATORS WITH CLOSED RANGE 7 • P N ⊥ and P M commute, • P M ⊥ and P N ⊥ commute, • M = M ∩ N + M ∩ N ⊥ . Proposition 3.5.
Suppose P and Q are orthogonal projections on a Hilbert A -module F and Ker ( Q ) + Ran ( P ) is an orthogonal summand in F . If R is the orthogonal projectiononto the closed submodule Ker ( Q ) + Ran ( P ) and P Q = 0 then (3.3) γ ( P Q ) + k (1 − P ) QR k ≥ . Proof.
The inclusion
Ker ( Q ) ⊂ Ker ( Q ) + Ran ( P ) implies that the orthogonal projection1 − Q onto Ker ( Q ) satisfies (1 − Q ) R = R (1 − Q ) = 1 − Q , consequently, QR is an orthogonalprojection and Ran ( QR ) is orthogonally complemented in F . Since Ran ( QP ) ⊂ Ran ( QR ) ⊂ Ran ( QP ) , we have Ran ( QP ) = Ran ( QR ) and so Ran ( QP ) is orthogonally complemented. Therefore, Ker ( P Q ) ⊥ = Ran ( QR ) . Suppose x ∈ Ker ( P Q ) ⊥ ⊂ Ran ( Q ) and k x k = 1. Then, since x = QR x = Qx , we have k P Q x k + k (1 − P ) QR k ≥ k P Q x k + k (1 − P ) Q x k ≥ kh P Q x, P Q x i + h (1 − P ) Q x, (1 − P ) Q x ik = kh Qx, Qx ik = k Qx k = 1 . By definition, the infimum of k P Q x k is γ ( P Q ). Therefore, γ ( P Q ) + k (1 − P ) QR k ≥ (cid:3) Note that as we set A = C i.e. if we take F to be a Hilbert space, the inequality (3.3)changes to an equality. In view of this notification, the following problem arises in theframework of Hilbert C*-modules. Problem 3.6.
Suppose P and Q are orthogonal projections on a Hilbert A -module F and Ker ( Q ) + Ran ( P ) is an orthogonal summand in F . If R is the orthogonal projection ontothe closed submodule Ker ( Q ) + Ran ( P ) and P Q = 0 then characterize those C*-algebras A for which the following equality holds:(3.4) γ ( P Q ) + k (1 − P ) QR k = 1 . To solve the problem, it might be useful to know that γ ( P Q ) ≤ k P Q x k for all x ∈ Ker ( P Q ) ⊥ ⊂ Ran ( Q ) of norm k x k = 1, therefore γ ( P Q ) + k (1 − P ) Q x k ≤ k P Q x k + k (1 − P ) Q x k = k P x k + k (1 − P ) x k . K. SHARIFI
Corollary 3.7.
Suppose P and Q are orthogonal projections on a Hilbert A -module F . If δ = k (1 − P ) QR k < and R is the orthogonal projection onto the orthogonal summand Ker ( Q ) + Ran ( P ) then P Q has closed range.Proof.
Suppose
P Q = 0 (in the case P Q = 0 the result is clear). According to Proposition3.5 and its proof,
Ker ( P Q ) ⊥ = Ran ( QR ) is orthogonally complimented and γ ( P Q ) ≥ − δ >
0. Therefore,
P Q has closed range by Lemma 2.4. (cid:3)
Two different concepts of angle between subspaces of a Hilbert space was first introducedby Dixmier and Friedrichs, see [8, 14, 1] and the excellent survey by Deutsch [7] for morehistorical notes and information. We generalized Dixmier’s definition for the angle betweentwo submodules of a Hilbert C*-module.
Definition 3.8.
The Dixmier (or minimal) angle between submodules M and N of a HilbertC*-module E is the angle α ( M, N ) in [0 , π/
2] whose cosine is defined by c ( M, N ) = sup {kh x, y ik : x ∈ M, k x k ≤ , y ∈ N, k y k ≤ } . Suppose M and N are submodule of a Hilbert C*-module E , then ( M + N ) ⊥ = M ⊥ ∩ N ⊥ .In particular, if M + N is orthogonally complemented in E then( M ⊥ ∩ N ⊥ ) ⊥ = ( M + N ) ⊥ ⊥ = M + N .
Theorem 3.9.
Suppose S ∈ L ( E, F ) and T ∈ L ( F, G ) are bounded adjointable operatorswith closed range. Then the following three conditions are equivalent:(i) T S has closed range,(ii)
Ker ( T ) + Ran ( S ) is an orthogonal summand in F ,(iii) Ker ( S ∗ ) + Ran ( T ∗ ) is an orthogonal summand in F .Furthermore, if c ( Ran ( S ) , Ker ( T ) ∩ [ Ker ( T ) ∩ Ran ( S )] ⊥ ) < and Ker ( S ∗ ) + Ran ( T ∗ ) isan orthogonal summand then T S has closed range.Proof.
Taking P = T † T and Q = SS † , then Ker ( P ) = Ker ( T ) , Ran ( P ) = Ran ( T † ) = Ran ( T ∗ ) ,Ker ( Q ) = Ker ( S † ) = Ker ( S ∗ ) , and Ran ( Q ) = Ran ( S ) . The equivalence of (i), (ii) and (iii) directly follows from the above equalities and Lemma3.4. To establish the statement of the second part suppose R is the orthogonal projection HE PRODUCT OF OPERATORS WITH CLOSED RANGE 9 onto the orthogonal summand
Ker ( Q ) + Ran ( P ) then (1 − P ) R is the projection onto M = Ker ( P ) ∩ [ Ran ( P ) + Ker ( Q ) ] = Ker ( T ) ∩ [ Ran ( T ∗ ) + Ker ( S ∗ ) ]= Ker ( T ) ∩ [ Ran ( T ∗ ) ⊥ ∩ Ker ( S ∗ ) ⊥ ] ⊥ = Ker ( T ) ∩ [ Ker ( T ) ∩ Ran ( S )] ⊥ . If neither M nor Ran ( S ) is { } , by commutativity of R with P and Q , we obtain k (1 − P ) QR k = k RQ (1 − P ) k = k Q (1 − P ) R k = sup {kh Q (1 − P ) Rx, y ik : x, y ∈ F and k x k ≤ , k y k ≤ } = sup {kh (1 − P ) Rx, Qy ik : x, y ∈ F and k x k ≤ , k y k ≤ } = sup {kh x, y ik : x ∈ M, y ∈ Ran ( S ) and k x k ≤ , k y k ≤ } = c ( M, Ran ( S )) . The statement is now derived from the above argument and Corollary 3.7. (cid:3)
Recall that a bounded adjointable operator between Hilbert C*-modules admits a boundedadjointable Moore-Penrose inverse if and only if the operator has closed range. This lead usto the following results.
Corollary 3.10.
Suppose S ∈ L ( E, F ) and T ∈ L ( F, G ) possess bounded adjointable Moore-Penrose inverses S † and T † . Then ( T S ) † is bounded if and only if Ker ( T ) + Ran ( S ) isan orthogonal summand, if and only if Ker ( S ∗ ) + Ran ( T ∗ ) is an orthogonal summand.Moreover, if the Dixmier angle between Ran ( S ) and Ker ( T ) ∩ [ Ker ( T ) ∩ Ran ( S )] ⊥ is positiveand Ker ( S ∗ ) + Ran ( T ∗ ) is an orthogonal summand then ( T S ) † is bounded. Now, it is natural to ask for the reverse order law, that is, if S ∈ L ( E, F ) and T ∈ L ( F, G )possess bounded adjointable Moore-Penrose inverses S † and T † , when does the equation( T S ) † = S † T † hold? We will answer this question elsewhere. Note that the above conditionsdo not ensure the equality.Recall that a C*-algebra of compact operators is a c -direct sum of elementary C*-algebras K ( H i ) of all compact operators acting on Hilbert spaces H i , i ∈ I , i.e. A = c - ⊕ i ∈ I K ( H i ),cf. [2, Theorem 1.4.5]. Suppose A is an arbitrary C*-algebra of compact operators. Magajnaand Schweizer have shown, respectively, that every norm closed (coinciding with its biorthog-onal complement, respectively) submodule of every Hilbert A -module is automatically an orthogonal summand, cf. [21, 27]. In this situation, every bounded A -linear map T : E → F is automatically adjointable. Recently further generic properties of the category of HilbertC*-modules over C*-algebras which characterize precisely the C*-algebras of compact oper-ators have been found in [11, 12, 13]. We close the paper with the observation that we canreformulate Theorem 3.9 in terms of bounded A -linear maps on Hilbert C*-modules overC*-algebras of compact operators. Corollary 3.11.
Suppose A is an arbitrary C*-algebra of compact operators, E, F, G areHilbert A -modules and S : E → F and T : F → G are bounded A -linear maps with closerange. Then the following conditions are equivalent:(i) T S has closed range,(ii)
Ker ( T ) + Ran ( S ) is closed,(iii) Ker ( S ∗ ) + Ran ( T ∗ ) is closed.Furthermore, if c ( Ran ( S ) , Ker ( T ) ∩ [ Ker ( T ) ∩ Ran ( S )] ⊥ ) < then T S has closed range.
In view of Corollary 3.11, one may ask about the converse of the last conclusion. To finda solution, one way reader has is to solve Problem 3.6.
Acknowledgement : The author would like to thank the referee for his/her careful read-ing and useful comments.
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Kamran Sharifi,Department of Mathematics, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, IranSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box:19395-5746, Tehran, Iran
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