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When do a module map and its adjoint have the same range?

K. Sharifi Shahrood University of Technology, sharifi[email protected]

Abstract A closed range module map is called EP if the map and its adjoint have the same range. We study first EP module maps on Hilbert C*-modules and then we provide necessary and sufficient conditions for the product of two EP modular operators to be EP. These enable us to improve some of older results for the product of two EP elements in C*-algebras.

Keywords: Hilbert C*-modules, C*-algebras, EP operators, Moore-Penrose inverse, C*-algebra of compact

operators.

Mathematics Subject Classification (2010): 47A05, 46L08, 46L05, 15A09.

1 Introduction

A bounded linear operator T with closed range on a complex Hilbert H is called an EP operator if

T and T ∗ have the same range. This was introduced for matrices by Schwerdtfeger and has been studied

in detail by several authors, see e.g. [3, 6] and references therein. A problem that has been open for over

twenty-five years is when the product of two EP matrices is again EP. Hartwig and Katz [1], and Koliha [2]

gave necessary and sufficient conditions for a product of two n n complex EP matrices to be EP. Djordjević × provided a generalization of the result for EP operators on Hilbert . In this note we investigate about

the EP operators on Hilbert C*-modules over an arbitrary C*-algebra of coefficients, and then we reformulate

some results of [2, 3] for the product of EP modular operators.

Since the finite-dimensional spaces, Hilbert spaces and C*-algebras can all be regarded as Hilbert C*-

modules, one can study EP modular operators in a unified way in the framework of Hilbert C*-modules.

Indeed, a Hilbert C*-module is an object like a except that the inner product is not scalar-

valued, but takes its values in a C*-algebra of coefficients. Since the geometry of these modules emerges from

the C*-valued inner product, some basic properties of Hilbert spaces like Pythagoras’ equality, self-duality,

and decomposition into orthogonal complements must be given up. These modules play an important role

in the modern theory of C*-algebras and the study of locally compact quantum groups. A (right) pre-

Hilbert C*-module over a C*-algebra is a right -module X endowed with an -valued inner product A A A , : X X , (x, y) x, y which is linear in the second variable y (and conjugate-linear in x), ⟨· ·⟩ × → A 7→ ⟨ ⟩ 338 www.SID.ir Archive of SID

satisfying the conditions

x, y = y, x ∗, x, ya = x, y a for all a , ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ∈ A x, x 0 with equality if and only if x = 0. ⟨ ⟩ ≥

A pre-Hilbert -module X is called a Hilbert -module if X is a with respect to the A A norm x = x, x 1/2. If X, Y are two Hilbert -modules then the set of all ordered pairs of elements ∥ ∥ ∥⟨ ⟩∥ A X Y from X and Y is a Hilbert -module with respect to the -valued inner product (x , y ), (x , y ) = ⊕ A A ⟨ 1 1 2 2 ⟩ x , x + y , y . It is called the direct orthogonal sum of X and Y . If V is a (possibly non-closed) ⟨ 1 2⟩X ⟨ 1 2⟩Y -submodule of X, then V := y X : x, y = 0 for all x V is a closed -submodule of X and A ⊥ { ∈ ⟨ ⟩ ∈ } A V V . A Hilbert -submodule V of a Hilbert -module X is orthogonally complemented if V and ⊆ ⊥ ⊥ A A its orthogonal complement V yield X = V V , in this case, V and its biorthogonal complement V ⊥ ⊕ ⊥ ⊥ ⊥ coincide. For the basic theory of Hilbert C*-modules we refer to the books [4].

Throughout the present paper we assume to be an arbitrary C*-algebra (i.e. not necessarily unital). A We use the notations Ker( ) and Ran( ) for and range of operators, respectively. We denote by · · (X,Y ) the Banach space of all bounded adjointable operators between X and Y , i.e., all bounded -linear L A maps T : X Y such that there exists T : Y X with the property T x, y = x, T y for all x X, → ∗ → ⟨ ⟩ ⟨ ∗ ⟩ ∈ y Y . The C*-algebra (X,X) is abbreviated by (X). ∈ L L In this paper we first briefly investigate some basic facts about EP modular operators with closed ranges

and then we give some factorizations and characterizations of such operators. If T,S and TS are EP modular

operators with closed ranges then Ran(TS) = Ran(T ) Ran(S). If, in addition, Ker(T ) + Ker(S) is dense ∩ in its biorthogonal complement then we obtain Ker(TS) = Ker(T ) + Ker(S). Some special cases for

EP elements of C*-algebras and C*-algebras of compact operators are considered. The following results

express when the product of two modular operators with closed range again has closed range. Suppose

T,S (X) are bounded adjointable operators with closed range. Then TS has closed range, if and only ∈ L if Ker(T ) + Ran(S) is an orthogonal summand in X if an only if Ker(S∗) + Ran(T ∗) is an orthogonal

summand in X. For the proof of the results and historical notes about the problem we refer to [5] and

references therein.

Let T (X), then a bounded adjointable operator T (X) is called the Moore-Penrose inverse of ∈ L † ∈ L T if

TT †T = T,T †TT † = T †, (TT †)∗ = TT † and (T †T )∗ = T †T. (1.1)

The notation T † is reserved to denote the Moore-Penrose inverse of T . These properties imply that T †

is unique and T †T and TT † are orthogonal projections. Moreover, Ran(T †) = Ran(T †T ), Ran(T ) =

Ran(TT ), Ker(T ) = Ker(T T ) and Ker(T ) = Ker(TT ) which lead us to X = Ker(T T ) Ran(T T ) = † † † † † ⊕ †

339 www.SID.ir Archive of SID

1 Ker(T ) Ran(T †) and X = Ker(T †) Ran(T ). If T † exists then T † = limω 0+ (ω1 + T ∗T )− T ∗ = ⊕ ⊕ → 1 limω 0+ T (ω1 + T ∗T )− . → Xu and Sheng in have shown that a bounded adjointable operator between two Hilbert C*-modules admits

a bounded Moore-Penrose inverse if and only if the operator has closed range. The reader should be aware

of the fact that a bounded adjointable operator may admit an as its Moore-Penrose.

In the Hilbert C*-module context, one needs to add the extra condition, closeness of the range, in order to

get a reasonably good theory. This ensures that an EP operator has a bounded adjointable Moore-Penrose

inverse. Like in the general theory of Hilbert spaces one can easily see that the following conditions are

equivalent:

T is EP with closed range, •

T and T have the same kernel, • ∗

T is Moore-Penrose invertible and TT = T T , • † †

Ran(T ) is orthogonally complemented in X, with complement Ker(T ). •

Proposition 1.1. Let X be a Hilbert -module and T (X) have a closed range. Then the following A ∈ L conditions are equivalent:

(i) T is EP with closed range,

(ii) there exists an V (X) such that T = VT , ∈ L ∗

(iii) there exists an isomorphism V (X) such that T = VT = TV . ∈ L † 2 Main Results

In this section we try to generalize some results of Koliha [2, 3] to the framework of Hilbert C*-modules.

Some special cases for EP elements of C*-algebras and the C*-algebra of compact operators are also obtained.

Lemma 2.1. Suppose X is a Hilbert -module. Let T (X) have closed range and S (X) be an A ∈ L ∈ L arbitrary operator which commutes with T . Then S commutes with T †.

Proposition 2.2. Suppose X is a Hilbert -module. Let T,S (X) be EP operators with closed range A ∈ L and TS = ST . Then TS is an EP operator with closed range.

Proposition 2.3. Let X be a Hilbert -module. If T, S, T S (X) are EP operators then T (Ran(S)) A ∈ L ⊆ Ran(S) and S (Ran(T )) Ran(T ). ∗ ⊆

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Example 2.4. let be unital C*-algebra and H be the standard Hilbert -module which is countably A A A generated by orthonormal basis ξj = (0, ..., 0, 1, 0, ..., 0), j N. Let W = span ξ2j : j N and S be ∈ { ∈ } the orthogonal projection onto the closed submodule W . We define T0 by T0(ξ1) = ξ2, T0(ξ2j) = ξ2j+2

1 1 and T0(ξ2j+1) = ξ2j 1, for all j N. The inverse of T0 is defined by T0− (ξ2) = ξ1, T0− (ξ2j) = ξ2j 2 and − ∈ − 1 1 1 1 T0− (ξ2j+1) = ξ2j+3. Then T0 and T0− can be extended uniquely to T and T − on H which satisfy T ∗ = T − . A One can easily see that T (Ran(S)) Ran(S) and S (Ran(T )) = Ran(S ) = Ran(S) = W Ran(T ). ⊆ ∗ ∗ ⊆ However, TS is not EP since ξ2 is orthogonal to Ran(TS) and to Ker(TS).

Lemma 2.5. Let X and Y be Hilbert -modules and T (X,Y ) have a closed range. If -submodule A ∈ L A W is orthogonally complemented in Y then the operator T has a matrix representation with respect to the

orthogonal sums X = Ran(T ) Ker(T ) and Y = W W as follows: ∗ ⊕ ⊕ ⊥

T 0 Ran(T ) W T = 1 : ∗ . (2.1) T2 0 Ker(T ) → W ⊥ [ ] [ ] [ ] In this case, A = T T + T T : Ran(T ) Ran(T ) is invertible. Moreover, 1∗ 1 2 2∗ ∗ → ∗

1 1 A− T1∗ A− T2∗ T † = . (2.2) 0 0 [ ] Theorem 2.6. Let X be a Hilbert -module and T,S (X) be EP operators with closed range. Among A ∈ L the following four properties of T , S and TS, the implication (i) (iii) holds. Moreover, (i) and (ii) are → equivalent to (iii) and (iv).

(i) TS is an EP operator with closed range.

(ii) Ker(T ) + Ker(S) is dense in its biorthogonal complement.

(iii) Ran(TS) = Ran(T ) Ran(S). ∩

(iv) Ker(TS) = Ker(T ) + Ker(S).

Recall that a C*-algebra of compact operators is a c - of elementary C*-algebras (H ) of all 0 K i compact operators acting on Hilbert spaces Hi, i I, i.e. = c0- i I (Hi). Suppose is an arbitrary ∈ A ⊕ ∈ K A C*-algebra of compact operators. It is well known that every norm closed submodule of every Hilbert - A module is automatically an orthogonal summand. We can reformulate Theorem 2.6 in terms of bounded

-linear maps on Hilbert C*-modules over C*-algebras of compact operators. A

Corollary 2.7. Suppose is an arbitrary C*-algebra of compact operators, X is a Hilbert -module and A A T,S (X) are EP operators with closed range. Then TS is an EP operator with closed range if and only ∈ L if Ran(TS) = Ran(T ) Ran(S) and Ker(TS) = Ker(T ) + Ker(S). ∩ 341 www.SID.ir Archive of SID

Koliha in [3] gave necessary and sufficient conditions for elements of C*-algebras which commute with

their Moore-Penrose inverse. He also studied conditions which ensure that the property is preserved under

multiplication. As a special case of our results we recover some parts of Theorem 4.3 of [3].

Corollary 2.8. Suppose is an arbitrary C*-algebra and a, b and ab commute with their Moore-Penrose A inverse. Then ab = a b . A A ∩ A

Corollary 2.9. Suppose is an arbitrary C*-algebra of compact operators and a and b commute with their A Moore-Penrose inverse. Then ab commutes with its Moore-Penrose inverse if and only if ab = a b A A ∩ A and (ab) 1 = a 1(0) + a 1(0), in which a 1(0) = x : xa = 0 . − − − − { ∈ A }

Recall that every C*-algebra is an -module on its own and define the bounded operators L : , A a A → A L (x) = ax then a 1(0) = Ker(L ) and Ran(L ) = a . The above facts follows from Theorem 2.6 and a − a a A Corollary 2.7.

References

[1] R. Hartwig and I. J. Katz, On products of EP matrices, Appl. 252 (1997), 339-345.

[2] J. J. Koliha, A simple proof of the product theorem for EP matrices, Linear Algebra Appl. 294 (1999), no. 1-3, 213-215.

[3] J. J. Koliha, Elements of C*-algebras commuting with their Moore-Penrose inverse, Studia Math. 139 (2000), no. 1, 81-90.

[4] E. C. Lance, Hilbert C*-Modules, LMS Lecture Note Series 210, Cambridge Univ. Press, 1995.

[5] K. Sharifi, The product of operators with closed range in Hilbert C*-modules, Linear Algebra Appl. 435 (2011), 1122-1130.

[6] K. Sharifi, EP modular operators and their products, J. Math. Anal. Appl. 419 (2014), 870-877.

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