Aequat. Math. 85 (2013), 465–470c© Springer Basel AG 20120001-9054/13/030465-6published online August 30, 2012DOI 10.1007/s00010-012-0155-9 Aequationes Mathematicae

Reverse order law for the weighted Moore–Penrose inversein C∗-algebras

Dijana Mosic

Abstract. In this paper we give a positive solution to a conjecture on the reverse order lawfor the weighted Moore–Penrose inverse in C∗-algebras (Mosic and Djordjevic in Electron.J. Linear Algebra 22:92–111, 2011).

Mathematics Subject Classification (2010). 16W10, 15A09, 46L05.

Keywords. Weighted Moore–Penrose inverse, reverse order law, weighted-EP elements,

C∗-algebras.

1. Introduction

Let A be a unital C∗-algebra. An element a ∈ A is regular if there exists someb ∈ A satisfying aba = a. The set of all regular elements of A will be denotedby A−. An element a ∈ A is self-adjoint if a∗ = a. An element x ∈ A is positiveif x = x∗ and σ(x) ⊆ [0,+∞), where the spectrum of element x is denoted byσ(x). If x ∈ A, then x∗x is a positive element.

An element a ∈ A is group invertible if there exists a# ∈ A such that

aa#a = a, a#aa# = a#, aa# = a#a.

Recall that a# is uniquely determined by these equations.The Moore–Penrose inverse (or MP-inverse) of a ∈ A is the element b ∈ A

satisfying the following equations [11]:

(1) aba = a, (2) bab = b, (3) (ab)∗ = ab, (4) (ba)∗ = ba.

There is at most one such element (see [11]) denoted by a†. If a is invertible,then a† coincides with the ordinary inverse of a.

The author is supported by the Ministry of Science, Republic of Serbia, grant no. 174007.

466 D. Mosic AEM

Theorem 1.1. [6] In a unital C∗-algebra A, a ∈ A is MP-invertible if and onlyif a is regular.

Let e, f be invertible positive elements in A. An element a ∈ A is said tobe weighted Moore–Penrose invertible with weights e, f , if there exists b ∈ Asuch that

(1) aba = a, (2) bab = b, (3e) (eba)∗ = eba, (4f) (fab)∗ = fab.

A weighted MP inverse of a with weights e, f is unique if it exists (see [8]) andwill be denoted by a†

e,f .

Theorem 1.2. [8] Let A be a unital C∗-algebra and let e, f be two positiveinvertible elements of A. If a ∈ A is regular, then the unique weighted MP-inverse a†

e,f exists.

Define the mapping a �→ a∗e,f = e−1a∗f , for all a ∈ A. Notice that (∗, e, f) :A → A is not an involution, because in general (ab)∗e,f �= b∗e,fa∗e,f . Now, westate some basic properties of weighted MP-inverse which are used in the restof the paper.

Lemma 1.1. Let a ∈ A− and let e, f be two invertible positive elements in A.Then(a) a∗e,f = a†

e,faa∗e,f = a∗e,faa†e,f ;

(b) a†e,f = a†

e,f (a†e,f )∗f,ea∗e,f = a∗e,f (a†

e,f )∗f,ea†e,f ;

(c) a†e,fA = a†

e,faA = a∗e,faA = a∗e,fA;(d) aA = aa†

e,fA = aa∗e,fA.

Proof. Since a∗e,f = e−1a∗f and (a†e,f )∗f,e = f−1(a†

e,f )∗e, by the definition ofweighted MP inverse, we can verify the statements (a)–(d). �

We recall the definitions of EP elements and weighted-EP elements.

Definition 1.1. An element a ∈ A− is EP if aa† = a†a.

Lemma 1.2. [7] An element a ∈ A is EP, if a ∈ A− and aA = a∗A (or,equivalently, if a ∈ A− and a◦ = (a∗)◦, where a◦ = {x ∈ A : ax = 0}).

Definition 1.2. [10] Let e, f ∈ A be invertible positive elements. An elementa ∈ A is (e, f) − weighted EP if both ea and af−1 are EP; that is, if a ∈ A−,eaA = (ea)∗A and af−1A = (af−1)∗A.

Lemma 1.3. Let f ∈ A be an invertible positive element. Then a ∈ A is (f, f)-weighted EP if a ∈ A− and aA = a∗f,fA.

Proof. The proof depends on the fact that fA = A = f−1A and uA = vA ⇔fuA = fvA. �

Vol. 85 (2013) Reverse order law for the weighted Moore–Penrose inverse 467

Weighted EP elements have been studied in detail in [10]. They areimportant since they are characterized as the elements commuting with theirweighted MP inverses. Further, they are exactly those elements for which thegroup and the weighted MP inverse exist and coincide.

The weighted Moore–Penrose inverse has several important applicationsin theoretical research and numerical computations [2]. The weighted Moore–Penrose inverse of the product is applied to the generalized least squares prob-lem, optimization problems, the weighted perturbation theory of the singularmatrix, as well as in approximation methods in general Hilbert spaces. There-fore, we investigate the reverse order rule in C∗-algebras.

Greville [5] proved that the reverse order rule (ab)† = b†a† holds for com-plex matrices if and only if a†a commutes with bb∗ and bb† commutes withaa∗. Tian [12] studied a group of rank equalities related to the Moore–Penroseinverse of products of two matrices, which implies necessary and sufficientconditions for (ab)† = b†a†. The extensions of these results to the weightedMoore–Penrose inverse have been considered, too. The operator analogues ofthese results for the Moore–Penrose inverse were proved in [3,4] for boundedlinear operators on Hilbert spaces. In [9] the reverse order law for the Moore–Penrose inverse and the weighted Moore–Penrose inverse in C∗-algebras wereinvestigated, extending the results from [3,4,12].

E. Arghiriade proved that (ab)† = b†a† holds for matrices if and only ifa∗abb∗ is EP (see [1,2]). In [9], the authors posed the following question: Isthere a similar result involving the weighted MP inverse and weighted EPelements? In this paper we answer this question in the positive.

The following results will be used in the next section.

Lemma 1.4. [9, Corollary 3.1] Let A be a unital C∗-algebra and let e, f , h bepositive invertible elements of A. If a, b ∈ A are regular, then the followingconditions are equivalent:

(a) abb†e,fa†

f,hab = ab;(b) b†

e,fa†f,habb†

e,fa†f,h = b†

e,fa†f,h;

(c) a†f,habb†

e,f = bb†e,fa†

f,ha.

Lemma 1.5. [9, Corollary 3.4] Let A be a unital C∗-algebra and let e, f , h bepositive invertible elements of A. If a, b, ab ∈ A are regular, then the followingconditions are equivalent:

(a) (ab)†e,h = b†

e,fa†f,h;

(b) a∗f,hab = bb†e,fa∗f,hab and abb∗e,f = abb∗e,fa†

f,ha;(c) a∗f,habb†

e,f = bb†e,fa∗f,ha and bb∗e,fa†

f,ha = a†f,habb∗e,f .

468 D. Mosic AEM

2. Results

In the following theorem we present a positive solution to a conjecture onthe reverse order law for the weighted MP-inverse stated in [9]. Precisely, wewill show that the reverse order law (ab)†

e,h = b†e,fa†

f,h holds for elements ofC∗-algebra if and only if a∗f,habb∗e,f is (f ,f)-weighted EP.

Theorem 2.1. Let A be a unital C∗-algebra and let e, f , h be positive invert-ible elements of A. If a, b, ab ∈ A are regular, then the following conditions areequivalent:

(a) (ab)†e,h = b†

e,fa†f,h;

(b) a∗f,habb∗e,f is (f, f)-weighted EP;(c) a∗f,habA ⊆ bA and bb∗e,fa∗f,hA ⊆ a∗f,hA.

Proof. (a) ⇒ (b): Let

w = a∗f,habb∗e,f , u = (b†e,f )∗f,eb†

e,fa†f,h(a†

f,h)∗h,f .

Then

wuw = a∗f,habb†e,fa†

f,habb∗e,f by Lemma 1.1 (b)

= a∗f,habb∗e,f = w by hypothesis (a),

which shows that w is regular. Further,

a∗f,habb†e,f = bb†

e,fa∗f,ha by Lemma 1.5 (c), (A)

a†f,habb†

e,f = bb†e,fa†

f,ha by Lemma 1.4 (c), (B)

bb∗e,fa†f,ha = a†

f,habb∗e,f by Lemma 1.5 (c). (C)

Then

a∗f,habb∗e,fA = a∗f,habb†e,fA by Lemma 1.1 (d)

= bb†e,fa∗f,haA by (A)

= bb†e,fa†

f,haA by Lemma 1.1 (c)

= a†f,habb†

e,fA by (B)

= a†f,habb∗e,fA by Lemma 1.1 (d)

= bb∗e,fa†f,haA by (C)

= bb∗e,fa∗f,haA by Lemma 1.1 (c),

that is,

wA = bb∗e,fa∗f,haA. (D)

Vol. 85 (2013) Reverse order law for the weighted Moore–Penrose inverse 469

The equality

w∗f,f = f−1(f−1a∗habe−1b∗f)∗f = be−1b∗a∗ha = bb∗e,fa∗f,ha (E)

combined with (D) gives

w∗f,fA = wA.

By Lemma 1.3, w is (f ,f)-weighted EP.(b) ⇒ (c): If w = a∗f,habb∗e,f is (f ,f)-weighted EP, then w is regular

and w∗f,fA = wA. Then wA = bb∗e,fa∗f,haA, by (E). By Lemma 1.1, bA =bb∗e,fA and a∗f,hA = a∗f,haA. Hence we obtain

a∗f,habA = wA = bb∗e,fa∗f,haA ⊆ bAand

bb∗e,fa∗f,hA = bb∗e,fa∗f,haA = wA ⊆ a∗f,hA.

So, condition (c) is satisfied.(c) ⇒ (a): From a∗f,habA ⊆ bA and bb∗e,fa∗f,hA ⊆ a∗f,hA, we have

a∗f,hab = bx, for some x ∈ A, and bb∗e,fa∗f,h = a∗f,hy, for some y ∈ A.The equalities

bb†e,fa∗f,hab = bb†

e,f bx = bx = a∗f,hab

and

abb∗e,fa†f,ha = abe−1b∗fa†

f,ha = (fa†f,habb∗e,fa∗f,hh−1)∗

= (fa†f,haa∗f,hyh−1)∗ = (fa∗f,hyh−1)∗ = (fbb∗e,fa∗f,hh−1)∗

= (fbe−1b∗a∗)∗ = abb∗e,f

imply (ab)†e,h = b†

e,fa†f,h, by Lemma 1.5. �

Applying Theorem 2.1, we obtain the following result.

Corollary 2.1. Let A be a unital C∗-algebra and let e, f , h be positive invertibleelements of A. If a, b, ab ∈ A− and (ab)†

e,h = b†e,fa†

f,h, then

(a∗f,habb∗e,f )# = (b†e,f )∗f,eb†

e,fa†f,h(a†

f,h)∗h,f = (bb∗e,f )†f,f (a∗f,ha)†

f,f .

Proof. Since (ab)†e,h = b†

e,fa†f,h implies that a∗f,habb∗e,f is (f ,f)-weighted

EP, by Theorem 2.1, (a∗f,habb∗e,f )# = (a∗f,habb∗e,f )†f,f , by [10]. Then

(a∗f,habb∗e,f )†f,f = (b†

e,f )∗f,eb†e,fa†

f,h(a†f,h)∗h,f . �

If e = f = 1 in Theorem 2.1, we obtain the following well-known result.The equivalences (a) and (c) of the following corollary appear in [10].

Corollary 2.2. Let A be a unital C∗-algebra. If a, b, ab ∈ A are regular, thenthe following conditions are equivalent:

470 D. Mosic AEM

(a) (ab)† = b†a†;(b) a∗abb∗ is EP;(c) a∗abA ⊆ bA and bb∗a∗A ⊆ a∗A.

Acknowledgements

The author would like to thank the anonymous referees for their useful sug-gestions, which helped to improve the original version of this paper.

References

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Dijana MosicFaculty of Sciences and MathematicsUniversity of NisVisegradska 33P. O. Box 22418000 NisSerbiae-mail: dijana@pmf.ni.ac.rs

Received: April 5, 2012

Revised: July 20, 2012