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Moore-Penrose inverses in rings and weighted partial isometries in C*− algebras
Applied Mathematics and Computation 395 (2021) 125832
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Moore-Penrose inverses in rings and weighted partial isometries in C ∗ −algebras Ruju Zhao a,c,1,∗, Hua Yao b, Junchao Wei c a
College of Science, Beibu Gulf University, Qinzhou, Guangxi 535011, PR China School of Mathematics and Statistics, Huanghuai University, Zhumadian, Henan 463000, PR China c School of Mathematical Sciences, Yangzhou University, Yangzhou, Jiangsu 225002, PR China b
a r t i c l e
i n f o
Article history: Received 26 May 2018 Revised 7 June 2020 Accepted 22 November 2020
a b s t r a c t We explore the existence of MP-inverse in rings with involution. In addition, we give some equivalent conditions for an element of a C ∗ −algebra to be a weighted-EP element and weighted partial isometry. © 2020 Elsevier Inc. All rights reserved.
2010 MSC: 15A09 16U99 16W10 ∗−Regularity Moore-Penrose inverse Weighted-EP element Weighted partial isometry C ∗ −algebra,
1. Introduction Let R be an associative ring with unit 1R . As usual, denote by J (R ) the Jacobson radical of R, U (R ) the set of all invertible elements of R and E (R ) the set of all idempotents of R. An element a ∈ R is said to be regular if there exists b ∈ R such that aba = a. We denote the set of all regular elements of R by R− . For any element a ∈ R, we define the commutant of a by comm(a ) = {x ∈ R|ax = xa}. An element a ∈ R is said to be group invertible if there exists a# ∈ R such that aa# a = a, a# aa# = a# , aa# = a# a. Note that if a# exists, then it is unique by the above conditions [1]. We denote the set of all group invertible elements of R by R# . An element a ∈ R is group invertible if and only if a ∈ a2 R ∩ Ra2 [4,17]. An involution in R is an anti-isomorphism ∗ : R → R, a → a∗ of degree 2, that is, (a∗ )∗ = a, (a + b)∗ = a∗ + b∗ , (ab)∗ = b∗ a∗ .
∗
Corresponding author at: College of Science, Beibu Gulf University, Qinzhou, Guangxi 535011, PR China. E-mail address: [email protected] (R. Zhao). 1 The research has been supported by the National Science Foundation of China under Grant Nos. 11471282, 11661014, and 11701499 and the High-level Scientific Research Foundation for the introduction of talent of Beibu Gulf University in 2020 under Grant No. 2020KYQD08.
https://doi.org/10.1016/j.amc.2020.125832 0 096-30 03/© 2020 Elsevier Inc. All rights reserved.
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832
Definition 1.1. Let R be a ring and a, b ∈ R. a is said to be right b−regular, if there exists x∗ = x ∈ comm(ba ), such that a = axba. Analogously, an element a is said to be left c−regular, if there is an y∗ = y ∈ comm(ac ) satisfying a = acya. If a is both left c−regular and right b−regular, then it is said to be (c, b)−regular. An element a ∈ R satisfying a∗ = a is said to be symmetric (or Hermitian) and an element p ∈ R is called a projection if 2 p = p = p∗ . Definition 1.2. An element a of a ring R is said to be well-supported if there exists a projection p such that ap = a, a∗ a + 1 − p ∈ U ( R ). The idempotent p is called the support of a. An element a is said to be left ∗−cancellable if a∗ ax = a∗ ay yields ax = ay, right ∗−cancellable if xaa∗ = yaa∗ yields xa = ya, and ∗−cancellable if it is both left and right ∗−cancellable. Note that a is left ∗−cancellable if and only if a∗ is right ∗−cancellable. The Moore-Penrose inverse (or MP-inverse) [16] of a ∈ R is the element a† ∈ R satisfying aa† a = a, a† aa† = a† , (aa† )∗ = aa† , (a† a )∗ = a† a. There is at most one a† satisfying the above equations [2,7,8]. Denote by R† the set of all MP-invertible elements of R. Actually, a is MP-invertible if and only if a is ∗−cancellable and a∗ aa∗ is regular [9]. An element a ∈ R† satisfying aa† = a† a is said to be EP. Denote by REP the set of all EP elements of R. An element a ∈ REP if and only if a ∈ R# ∩ R† and a# = a† . The Drazin inverse of a ∈ R is the element aD ∈ R satisfying aD aaD = aD , aaD = aD a, ak+1 aD = ak for some nonnegative integer k. The least such k is called the index of a, denoted by ind (a ). We denote the set of all Drazin invertible elements of R by RD . If ind (a ) = 1, then the Drazin inverse aD is the group inverse a# . Let A be a unital C ∗ −algebra with unite 1A . Actually, the unital C ∗ −algebra A is an associative ring with the addition and the multiplication of A. Thus, the definitions and conclusions mentioned above still work for A. Particularly, in C ∗ −algebra, every element is ∗−cancellable [2]. An element x ∈ A is said to be positive if there exists y ∈ A satisfying x = y∗ y. Alternatively, x ∈ A is said to be positive if x is Hermitian and σ (x ) ⊆ [0, +∞ ), where σ (x ) is the spectrum of x. Definition 1.3. [2,3] Let A be a unital C ∗ −algebra and let e, f be positive invertible elements in A. An element a ∈ A is said to be weighted MP-invertible with weights e, f if there exists b ∈ A such that aba = a, bab = b, (eab)∗ = eab, ( f ba )∗ = f ba. †
The unique such b is called weighted MP-inverse of a with weights e and f, and denoted by ae, f if it exists. Denote by †
Ae, f the set of all weighted MP-invertible elements of A with weights e and f .
Assume that e, f ∈ A are positive and invertible. Define the mapping (∗, e ) : A −→ A, x −→ x∗e = e−1 x∗ e. It is easy to 1
1
check that the mapping (∗, e ) is an involution of A. Moreover, define x e = e 2 xe− 2 , for any x ∈ A. Then · e is a norm of A. We can show that Ae = (A, (∗, e ), · e ) is a unital C ∗ −algebra with the involution (∗, e ) and the norm · e . Consequently, the conditions of Definition 1.3 can be rewritten as aba = a, bab = b, (ab)∗e = ab, (ba )∗ f = ba. If f = λe, where 0 = λ ∈ C, then x∗e = x∗ f for any x ∈ A, where C is the complex field. In this case, the weighted MP† inverse ae, f is a special MP-inverse a† with the involution (∗, e ). In fact, mapping (∗, e ) is an involution and x∗e = x∗ f for †
any x ∈ A. Let ∗ = (∗, e ) and b = ae, f . From the definition of weighted MP-inverse, we get aba = a, bab = b, (ab)∗ = ab, and
(ba )∗
†
= ba. From the definition of MP-inverse, b = ae, f is the MP-inverse of a.
Define the mapping (∗, e, f ): A −→ A, x −→ x∗e, f = e−1 x∗ f, for all x ∈ A. In general, (∗, e, f ) is not an involution. But if f = e, then (∗, e, f ) is an involution, in this case x∗e, f = x∗ f,e for any x ∈ A. Theorem 1.4. [2] Let A be a unital C ∗ −algebra and let e, f be positive invertible elements of A. If a ∈ A− , then the unique † weighted MP-inverse ae, f exists and 1
1
1
1
ae, f = f − 2 (e 2 a f − 2 )† e 2 . †
Definition 1.5. [11] Let A be a unital C ∗ −algebra and let e, f be positive invertible elements in A. The element a ∈ A− is said to be weighted-EP with respect to elements e, f (or weighted-EP w.r.t. (e, f )) if both ea and a f −1 are EP (which is † † equivalent to aae, f = ae, f a). †
An element a ∈ A− satisfying a∗ f,e = ae, f is called a weighted partial isometry with respect to elements e, f (or a
weighted partial isometry w.r.t. (e, f )). The Moore-Penrose inverse in rings with involution and the weighted Moore-Penrose inverse in C ∗ −algebras have several important applications in matrices over an integral domain, bounded linear operators on Hilbert spaces and numerical computations [1,5,6,15]. 2
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832
Various characterizations of Moore-Penrose inverse in rings with involution were given in [10,18]. Weighted-EP matrix was defined and a number of conclusions about it were presented using various formulas for a rank of a complex matrix in [19]. The reverse order law for the Moore-Penrose inverse and the weighted Moore-Penrose inverse in C ∗ −algebras were investigated [6,14]. Many achievements have been made in weighted-EP elements and weighted partial isometries of C ∗ −algebras [11–13]. In this paper, we study the Moore-Penrose inverse in rings with involution, weighted-EP element and weighted partial isometry in C ∗ −algebras, extending the results from [2,11,12]. In Section 2, we prove that an element a of a ring R is right a∗ −regular if and only if a is well-supported or MP-invertible. In addition, we also present other equivalent conditions for an element to be MP-invertible. Under the condition that a∗ a = a2 , a is EP if and only if a is group invertible, if and only if a is Drazin invertible and left ∗−cancellable. In Section 3, letting A be a C ∗ −algebra, a ∈ A− and e, f be positive invertible † † elements of A, we give an example to show that the equality (a∗e, f ) f,e = a(a∗ f,e a ) f, f is not true in general. And then we present some equivalent conditions for an element a ∈ A to be a weighted-EP element w.r.t (e, f ) and weighted partial isometry w.r.t (e, f ) by means of E (A ), J (A ), and the solutions of equation x = xaa∗ f,e . 2. ∗− Regularities and MP-inverses In this section, we work in a ring with involution unless otherwise stated. We first present some equivalent conditions for an element a of a ring R to be well-supported, left ∗−cancellable and right a∗ −regular. Denote r (a ) = {x ∈ R|ax = 0}, for any a ∈ R. For convenience, here we set 1R = 1. Lemma 2.1. Let R be a ring and a ∈ R. If a is well-supported, then a is left ∗−cancellable. Proof. If there is an y ∈ R such that a∗ ay = 0, then we have (a∗ a + 1 − p) py = 0, where p is the support of a, because ap = a and p2 = p. From the hypothesis a∗ a + 1 − p ∈ U (R ), we obtain py = 0. Consequently, ay = apy = 0. Theorem 2.2. Let R be a ring and a ∈ R. Then a is right a∗ −regular if and only if a is well-supported. Proof. “⇒” By assumption, we know that there exists x∗ = x ∈ comm(a∗ a ) such that a = axa∗ a. Taking p = xa∗ a, we obtain p∗ = a∗ ax∗ = a∗ ax = xa∗ a = p, a = axa∗ a = ap, p2 = xa∗ axa∗ a = xa∗ a = p, pa∗ = (ap)∗ = a∗ . It is easy to check that (a∗ a + 1 − p)( px + 1 − p) = a∗ apx + a∗ a(1 − p) + 1 − p = a∗ ax + 1 − p = p + 1 − p = 1. The equality px(1 − p) = xa∗ a(x − x2 a∗ a ) = xa∗ a(x − xa∗ ax ) = xa∗ a(1 − xa∗ a )x = p(1 − p)x = 0 implies ( px + 1 − p)(a∗ a + 1 − p) = pxa∗ a + px(1 − p) + (1 − p)a∗ a + 1 − p = p + 1 − p = 1. That is, a∗ a + 1 − p ∈ U (R ). From the definition of well-supported element, the proof is completed. “⇐” According to the hypothesis, there exists p ∈ R such that a = ap, p∗ = p = p2 , and a∗ a + 1 − p = u ∈ U (R ). Applying involution to the above equalities, we get a∗ = p∗ a∗ = pa∗ and u∗ = (a∗ a + 1 − p)∗ = a∗ a + 1 − p = u. This gives (u−1 )∗ = u−1 . Furthermore, it is easy to check that ua∗ a = (a∗ a + 1 − p)a∗ a = a∗ aa∗ a = a∗ a(a∗ a + 1 − p) = a∗ au, which implies u ∈ comm(a∗ a ). Thus, u−1 ∈ comm(a∗ a ). Finally, we only need to show that a = au−1 a∗ a. In fact, up = (a∗ a + 1 − p) p = a∗ ap + (1 − p) p = a∗ ap = a∗ a, which leads to p = u−1 a∗ a. From the proof of Theorem 2.2, we have the following corollary. Corollary 2.3. Let R be a ring and a ∈ R. Then a is right a∗ −regular if and only if there is a projection p such that a∗ a + 1 − p ∈ U (R ), and one of the following conditions holds: (1) r (a ) = r ( p); (2) Ra = Rp; (3) a∗ R = pR. Proof. From Theorem 2.2, we only need to prove that the equation ap = a is satisfied. (1) Since p is a projection, we have p∗ = p = p2 and p − 1 ∈ r ( p). From the hypothesis, we get a( p − 1 ) = 0, that is, ap = a. (2) From the equation Ra = Rp, there exists x ∈ R such that a = xp. Since p is a projection, we conclude that ap = xp2 = xp = a. (3) From the hypothesis, there exists y ∈ R such that a∗ = py. Since p is a projection, we have a = y∗ p∗ = y∗ p. It follows that ap = y∗ p2 = y∗ p = a. An element a of a ring R is said to be weakly left ∗−cancellable, if a∗ ax = 0 implies ax2 = 0 for any x ∈ R. Obviously, if a is left ∗−cancellable, then a is weakly left ∗−cancellable. But the converse of this statement is not true from the following example. 3
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832
Example 2.4. Let R = Z4 Z4 . Define a mapping ∗ : R −→ R, (x, y ) −→ (y, x ). It is easy to check that the mapping ∗ is an involution of R. Take a = (2, 3 ). Then a∗ = (3, 2 ) and a∗ a = (2, 2 ). If a∗ a(x, y ) = 0, then (x, y ) ∈ {(0, 0 ) = 0, (0, 2 ), (2, 0 ), (2, 2 )}, which gives a(x, y )2 = 0. But if (x, y ) = (0, 2 ), then a(x, y ) = (0, 2 ) = 0. That is, a is weakly left ∗−cancellable, but it is not left ∗−cancellable. In the following, we explore the existence of MP-inverse in rings by means of weakly left ∗−cancellation, ∗−regularity, Hermitian element, and idempotent. Theorem 2.5. Let R be a ring and a ∈ R. Then the following statements are equivalent: (1) a ∈ R† ; (2) R = a∗ aR r (a∗ a ) and a is weakly left ∗−cancellable; (3) R = a∗ aR + r (a∗ a ) and a is weakly left ∗−cancellable. Proof. (1) ⇒ (2) It follows from the hypothesis that a† exists and a is left ∗−cancellable, which yield a is weakly left ∗−cancellable and r (a ) = r (a∗ a ). On the other hand, from [20, Theorem 3.11(2)], we obtain R = a∗ aR r (a ). Thus, R = a∗ aR r ( a∗ a ). (2) ⇒ (3) It is straightforward. (3) ⇒ (1) According to the condition R = a∗ aR + r (a∗ a ), we have a∗ aR = a∗ aa∗ aR. Write a∗ a = a∗ aa∗ ab, where b ∈ R. Then ∗ a a(1 − a∗ ab) = 0. Since a is weakly left ∗−cancellable, we deduce that a(1 − a∗ ab)2 = 0, which leads to a = aa∗ a(2b) − aa∗ aba∗ ab ∈ aa∗ aR. Therefore, a ∈ R† by [20, Theorem 3.1(3)]. Theorem 2.6. Let R be a ring and a ∈ R. Then a ∈ R† if and only if a is (a∗ , a∗ )−regular. Proof. “⇐” Since a is (a∗ , a∗ )−regular, we conclude that there are y∗ = y ∈ comm(aa∗ ) and z∗ = z ∈ comm(a∗ a ) satisfying a = aa∗ ya = aza∗ a. Fix x = yaz. Then aa∗ xa∗ a = aa∗ yaza∗ a = aza∗ a = a. Let u, v ∈ R satisfy a∗ au = 0 = vaa∗ . Then au = aa∗ xa∗ au = 0 and va = vaa∗ xa∗ a = 0. Thus, a is ∗−cancellable. On the other hand, we have a∗ aa∗ = a∗ aa∗ xa∗ aa∗ , which yields a∗ aa∗ is regular. From [9, Theorem 5.4(xi)], we get a ∈ R† . “⇒” Using the assumption, we get a = aa† a. Write y = a† (a† )∗ and z = (a† )∗ a† . Then it is easy to verify that y∗ = y ∈ comm(a∗ a ), z∗ = z ∈ comm(aa∗ ), and a = aa∗ za = aya∗ a. That is, a is both left and right a∗ −regular. Therefore, a is (a∗ , a∗ )−regular. Theorem 2.7. Let R be a ring and a ∈ R. Then a ∈ R† if and only if a is right a∗ −regular. Proof. “⇒” It follows from a ∈ R† that a† exists. Write x = a† (a† )∗ . Then x∗ = x, xa∗ a = a† (a† )∗ a∗ a = a† (aa† )∗ a = a† a, a∗ ax = a∗ aa† (a† )∗ = a∗ (a† )∗ = (a† a )∗ = a† a, which show that x∗ = x ∈ comm(a∗ a ) and axa∗ a = aa† a = a. Hence, a is right a∗ −regular. “⇐” Let x∗ = x ∈ comm(a∗ a ) satisfy a = axa∗ a. Taking b = xa∗ , we get ab = axa∗ , ba = xa∗ a, (ab)∗ = ax∗ a∗ = axa∗ = ab, (ba )∗ = a∗ ax∗ = a∗ ax = xa∗ a = ba, aba = axa∗ a = a, bab = xa∗ axa∗ = x(ax∗ a∗ a )∗ = x(axa∗ a )∗ = xa∗ = b. Consequently, a ∈ R† and a† = xa∗ . Theorem 2.8. Let R be a ring and a ∈ R. Then the following conditions are equivalent: (1) a ∈ R† ; (2) there is an x ∈ R such that a = axa∗ a, and a∗ ax is Hermitian. Proof. (1) ⇒ (2) Write x = a† (a† )∗ . Then a∗ ax = a∗ aa† (a† )∗ = a∗ (a† )∗ = a† a, which proves that a∗ ax is Hermitian. Furthermore, we have axa∗ a = aa† (a† )∗ a∗ a = (a† )∗ a∗ a = aa† a = a. (2) ⇒ (1) Let y ∈ R satisfy a∗ ay = 0. Then ay = (axa∗ a )y = ax(a∗ ay ) = 0. That is, a is left ∗−cancellable. Taking the involution on a = axa∗ a, we obtain a∗ = a∗ ax∗ a∗ . Post-multiplying by a, we have a∗ a = a∗ ax∗ a∗ a, which gives a = ax∗ a∗ a = a(a∗ ax )∗ = aa∗ ax, because a is left ∗−cancellable and a∗ ax is Hermitian. Applying the involution to axa∗ a = a = ax∗ a∗ a, we deduce that a∗ ax∗ a∗ = a∗ = a∗ axa∗ , which leads to ax∗ a∗ = axa∗ . Moreover, taking b = x∗ a∗ , we get ab = ax∗ a∗ = axa∗ . Applying the involution to the above equality, we see that (ab)∗ = (axa∗ )∗ = ax∗ a∗ = ab. From the hypothesis, we conclude that ba = x∗ a∗ a = (a∗ ax )∗ = a∗ ax, which shows that (ba )∗ = ba. It is left to prove that aba = a and bab = b. Actually, it is easy to check that aba = axa∗ a = a and bab = a∗ axx∗ a∗ = (a∗ ax )∗ x∗ a∗ = x∗ a∗ ax∗ a∗ = x∗ (axa∗ a )∗ = x∗ a∗ = b. 4
R. Zhao, H. Yao and J. Wei
Therefore, a ∈ R† and a† = x∗ a∗ .
Applied Mathematics and Computation 395 (2021) 125832
Theorem 2.9. Let R be a ring and a ∈ R. Then the following conditions are equivalent: (1) a ∈ R† ; (2) there are g, f ∈ E (R ) such that aR = gR, R f = Ra, 1 + (g∗ − g)∗ (g∗ − g) ∈ U (R ), and 1 + ( f ∗ − f )∗ ( f ∗ − f ) ∈ U (R ). Proof. (1) ⇒ (2) Fix g = aa† and f = a† a. Using the equality a = aa† a, we have g∗ = g = g2 and f ∗ = f = f 2 . It is evident that aR = aa† aR = gaR ⊆ gR = aa† R ⊆ aR, Ra = Raa† a = Ra f ⊆ R f = Ra† a ⊆ Ra, 1 + (g∗ − g)∗ (g∗ − g) = 1 ∈ U (R ), 1 + ( f ∗ − f )∗ ( f ∗ − f ) = 1 ∈ U (R ). (2) ⇒ (1) Write s = 1 + (g∗ − g)∗ (g∗ − g) ∈ U (R ) and t = 1 + ( f ∗ − f )∗ ( f ∗ − f ) ∈ U (R ). Then we get sgg∗ = gg∗ gg∗ = gg∗ s, s∗ = s, sg = gg∗ g = gs, t f ∗ f = f ∗ f f ∗ f = f ∗ f t , t ∗ = t , t f = f f ∗ f = f t. Take p = gg∗ s−1 and q = f ∗ f t −1 . By the above, we obtain a series of precious equalities as follows: p2 = gg∗ s−1 gg∗ s−1 = (gg∗ gg∗ )s−1 s−1 = gg∗ s−1 = p, q2 = f ∗ f t −1 f ∗ f t −1 = ( f ∗ f f ∗ f )t −1 t −1 = f ∗ f t −1 = q, p∗ = (s∗ )−1 gg∗ = s−1 gg∗ = gg∗ s−1 = p, q∗ = (t ∗ )−1 f ∗ f = f ∗ f t −1 = q, pg = gg∗ s−1 g = (gg∗ g)s−1 = gss−1 = g, gp = g(gg∗ s−1 ) = gg∗ s−1 = p, f q = f f ∗ f t −1 = f t t −1 = f, q f = f ∗ f t −1 f = f ∗ f f t −1 = f ∗ f t −1 = q. Since aR = gR and R f = Ra, we infer that a = ga = a f . It is immediate that pa = p(ga ) = ( pg)a = ga = a and aq = (a f )q = a( f q ) = a f = a. Set g = ax and f = ya, where x, y ∈ R. Write c = qyaxp. Then we conclude that ac = aqyaxp = ayaxp = a f xp = axp = gp = p = p∗ = (ac )∗ , ca = qyaxpa = qyaxa = qyga = qya = q f = q = q∗ = (ca )∗ , aca = pa = a, cac = qc = q(qyaxp) = qyaxp = c. Consequently, a ∈ R† and a† = qyaxp. In general, an element a of a ring R is not EP, if only a ∈ R# . In the following theorem we assume that a∗ a = a2 . Then the EP element and the group invertible element are equivalent. Theorem 2.10. Let R be a ring and a ∈ R. If a∗ a = a2 , then the following conditions are equivalent: (1) a ∈ REP ; (2) a ∈ R# ; (3) a is left ∗−cancellable and a ∈ RD . Proof. (1) ⇒ (2) The proof is straightforward. (2) ⇒ (3) Since a ∈ R# , it follows that a ∈ RD , in this case ind (a ) = 1. Let y ∈ R satisfy a∗ ay = 0. Then a2 y = a∗ ay = 0, which gives ay = a# a2 y = 0. (3) ⇒ (1) Using the assumptions, we see that there is an nonnegative integer k such that ak+1 aD = ak . If k = 1, then a ∈ R# and a# = aD . If k ≥ 2, then a∗ ak aD = a∗ ak−1 , because a∗ a = a2 . By the hypothesis, we assert that ak aD = ak−1 . Repeating the above process, we have a2 aD = a, which yields a ∈ R# . Summarizing, we have a ∈ R# . Applying the involution to a = aa# a, we conclude that a∗ = a∗ (a# )∗ a∗ . Post-multiplying by a, we have a∗ a = a∗ (a# )∗ a∗ a, which implies that a2 = a∗ (a# )∗ a2 . Post-multiplying by (a# )2 , we get aa# = a∗ (a# )∗ aa# = (aa# )∗ aa# . This gives (aa# )∗ = aa# . Therefore, a ∈ R† and a† = a# . That is, a ∈ REP . At the end of this section, we give an equivalent condition for an element to be EP. Theorem 2.11. Let R be a ring and a ∈ R. Then the following conditions are equivalent: (1) a ∈ REP ; (2) there is an g ∈ E (R ) such that Ra = Rg, aR = gR, gg∗ = g∗ g and 1 + (g∗ − g)∗ (g∗ − g) ∈ U (R ). Proof. (1) ⇒ (2) The conclusion is proved by taking g = a† a, because a† a = aa† . (2) ⇒ (1) Write s = 1 + (g∗ − g)∗ (g∗ − g) ∈ U (R ). Then sg∗ g = sgg∗ = gg∗ s = g∗ gs, s = s∗ , and sg = gg∗ g = gs. Take p = gg∗ s−1 . Then we have p = g∗ gs−1 , p2 = p, p∗ = p, pg = g, and gp = p, which imply g∗ p = p and p = p∗ = (g∗ p)∗ = p∗ g = pg = g. Since Ra = Rg and aR = gR, it follows that a = ga = ag. Fix g = ax = ya, where x, y ∈ R. Then a = axa = aya. Set b = yax. Then we can verify that ab = ayax = ax = g = p = p∗ = (ab)∗ , ba = yaxa = ya = g = p = p∗ = (ba )∗ , which give ab = ba. Moreover, we have aba = ga = a and bab = gb = yayax = yax = b. Consequently, a ∈ R# ∩ R† and a# = b = a† . That is, a ∈ REP . 5
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832
3. Weighted-EP elements and weighted partial isometries In this section, we study some equivalent conditions for an element to be a weighted-EP element and weighted partial isometry in C ∗ −algebras. Lemma 3.1. [11] Let A be a unital C ∗ −algebra and let e, f be positive invertible elements of A. For each a ∈ A− , the following is satisfied: † † (1) (ae, f ) f,e = a; †
†
(2) (a∗ f,e ) f,e = (ae, f )∗e, f ; †
†
(3) a∗ f,e = ae, f aa∗ f,e = a∗ f,e aae, f ; (4) (5) (6) (7) (8)
† † a∗ f,e (ae, f )∗e, f = ae, f a; (a†e, f )∗e, f a∗ f,e = aa†e, f ; (a∗ f,e a )†f, f = a†e, f (a†e, f )∗e, f ; (aa∗ f,e )†e,e = (a†e, f )∗e, f a†e, f ; † † † ae, f = (a∗ f,e a ) f, f a∗ f,e = a∗ f,e (aa∗ f,e )e,e . †
†
In addition, there is seemingly still a equality (a∗e, f ) f,e = a(a∗ f,e a ) f, f . But, in fact, it is not true in general according to the following example. Example 3.2. It is well-known that the complex field C is a unital C ∗ −algebra with involution ∗, the conjugate of complex numbers, and norm · , the absolute value (or modulus) of complex numbers. Take a = 2 + i, b = 15 (2 − i ), e = 2 and f = 5 in C. Then aba = a and e, f are√positive invertible. It is obvious that √ 1 1 e∗ = 2, e 2 = 2, e− 2 = 22 , √ √ 1 1 f ∗ = 5, f 2 = 5, f − 2 = 55 . Using the existence of weighted MP-inverse and the definition of (∗, e, f ), we obtain a∗e, f = 52 (2 − i ), a∗ f,e = 25 (2 − i ), √ √ 10 10 5 ( 5 (2 +
†
ae, f =
i ).
i ))† =
√
√ 10 10 5 [ 10 (2 − i )]
Furthermore, we have † a(a∗ f,e a ) f, f = (2 + i ) 15 (2 − i ) 12 (2 + i ) = †
1 5 (2
=
1 2 (2
− i ) = b. †
†
+ i ), (a∗e, f ) f,e = [ 25 (2 − i )]5,2 =
√
√ 10 5 10 2 [ 4 (2
− i )] † =
√
√ 10 2 10 2 [ 125 (2
+ i )] =
2 25 (2 +
†
Consequently, (a∗e, f ) f,e = a(a∗ f,e a ) f, f .
Lemma 3.3. Let A be a unital C ∗ −algebra and let e, f be positive invertible elements of A. If a ∈ A− , then (a∗ f,e )†f,e = a(a∗ f,e a )†f, f = (aa∗ f,e )†e,e a. 1
1
1
1
Proof. Write x = e 2 a f − 2 . Since a is regular, there exists b ∈ A such that a = aba. Take y = f 2 be− 2 . Then xyx = 1
1
1
1
1
1
1
1
1
1
e 2 a f − 2 f 2 be− 2 e 2 a f − 2 = e 2 aba f − 2 = e 2 a f − 2 = x. It means that x is also regular. From Theorem 1.4, we deduce that 1
1
1
1
1
1
1
1
1
1
a(a∗ f,e a )†f, f = a f − 2 ( f − 2 a∗ e 2 e 2 a f − 2 )† f 2 = a f − 2 (x∗ x )† f 2 = e− 2 xx† (x∗ )† f 2 1
1
1
1
1
1
1
1
= e− 2 ( x† )∗ f 2 = e− 2 ( x∗ )† f 2 = e− 2 ( f − 2 a∗ e 2 )† f 2 , 1
1
1
1
1
1
1
1
1
1
(aa∗ f,e )†e,e a = e− 2 (e 2 a f − 2 f − 2 a∗ e 2 )† e 2 a = e− 2 (xx∗ )† e 2 a = e− 2 (x∗ )† x† x f 2 1
1
1
1
1
1
1
1
= e− 2 ( x† )∗ f 2 = e− 2 ( x∗ )† f 2 = e− 2 ( f − 2 a∗ e 2 )† f 2 , and
1
1
1
1
1
1
1
1
(a∗ f,e )†f,e = e− 2 ( f 2 a∗ f,e e− 2 )† f 2 = e− 2 ( f − 2 a∗ e 2 )† f 2 .
Let A be a unital C ∗ −algebra and let e, f be positive invertible elements in A. If a ∈ A# , then a ∈ A− , because a = aa# a. That is, if a ∈ A# , then the unique weighted MP-inverse of a also exists. In the following theorem, we give the conditions to ensure that a is weighted-EP w.r.t. (e, f ), extending the results from [11]. Theorem 3.4. Let a ∈ A# and let e, f be positive invertible elements in A. Then the following statements are equivalent: (1) a is weighted-EP w.r.t. (e, f ); † † (2) ae, f + a# = 2ae, f aa# ; †
†
†
†
(3) aae, f = (ae, f )2 a2 and ae, f a = a2 (ae, f )2 ; †
†
(4) an+1 (ae, f )n = a = (ae, f )m am+1 , for any/some integers m, n ≥ 1; 6
R. Zhao, H. Yao and J. Wei †
Applied Mathematics and Computation 395 (2021) 125832
†
†
(5) ae, f a2n ae, f = an (ae, f )2 an , for any/some integers n ≥ 1. †
Proof. If a is weighted-EP w.r.t. (e, f ), then a# = ae, f by [11, Theorem 2.2(xi)]. It is not difficult to prove that conditions (2)-(5) hold. † † Conversely, in order to conclude that a is weighted-EP w.r.t. (e, f ), we only need to show that the condition ae, f a = aae, f is satisfied by [11, Theorem 2.2(viii)]. † † † (2) From the equality ae, f + a# = 2ae, f aa# , pre-multiplying by a, we have aae, f = aa# . On the other hand, post†
†
†
multiplying by a, we get ae, f a = a# a. Hence, ae, f a = aae, f . (3) Post-multiplying the hypothesis
† aae, f
=
(a†e, f )2 a2
†
by a# ae, f , we obtain
a# a†e, f = a(a# )2 a†e, f = aa†e, f a(a# )2 a†e, f = aa†e, f a# a†e, f = (a†e, f )2 a2 a# a†e, f = (a†e, f )2 aa†e, f = (a†e, f )2 , †
†
which yields a# aae, f = a(ae, f )2 . Post-multiplying by a, we can verify that a# a =
† a# aae, f a
=
† a(ae, f )2 a
†
†
†
†
†
†
= (aae, f )ae, f a = (ae, f )2 a2 ae, f a = (ae, f )2 a2 = aae, f . †
†
Similarly, using the equality ae, f a = a2 (ae, f )2 , we deduce that †
†
†
†
†
†
†
a# a = a(ae, f )2 a = aae, f (ae, f a ) = aae, f a2 (ae, f )2 = a2 (ae, f )2 = ae, f a. † ae, f a
† Therefore, = a# a = aae, f . † † (4) The condition an+1 (ae, f )n = a = (ae, f )m am+1 (m, n ≥ 1) gives † † † † † † a# a = a# an+1 (ae, f )n = an (ae, f )n = an (ae, f )n aae, f = a# a2 ae, f = aae, f , † † † † † † # m m +1 # m m m m 2 # aa = (ae, f ) a a = (ae, f ) a = ae, f a(ae, f ) a = ae, f a a = ae, f a. † † Consequently, ae, f a = a# a = aae, f . † † † † † (5) Pre-multiplying and post-multiplying the equality ae, f a2n ae, f = an (ae, f )2 an by a, we have a2n ae, f = an+1 (ae, f )2 an and † † ae, f a2n = an (ae, f )2 an+1 . It follows that † † † † a2n−1 ae, f = a# a2n ae, f = a# an+1 (ae, f )2 an = an (ae, f )2 an , † † † † † † ae, f a2n−1 = ae, f a2n a# = an (ae, f )2 an+1 a# = an (ae, f )2 an , which lead to a2n−1 ae, f = ae, f a2n−1 . From [11, Theorem 2.2(lv)], we † † obtain ae, f a = aae, f .
Motivated by [12], we present some novel equivalent conditions for an element a ∈ A to be a weighted partial isometry w.r.t. (e, f ) by means of E (A ), J (A ) and the solutions of certain equations. Proposition 3.5. Let a ∈ A− and let e, f be positive invertible elements in A. Then the following statements are equivalent: (1) a is a weighted partial isometry w.r.t. (e, f ); (2) aa∗ f,e ∈ E (A ); (3) a∗ f,e a ∈ E (A ). †
Proof. If a is a weighted partial isometry w.r.t. (e, f ), then a∗ f,e = ae, f . It is easy to check that conditions (2) and (3) hold.
Conversely, in order to show that a is a weighted partial isometry w.r.t. (e, f ), we only need to prove that the condition a∗ f,e = a∗ f,e aa∗ f,e is satisfied by [12, Theorem 2.1(v)]. † (2) Since aa∗ f,e ∈ E (A ), it follows that aa∗ f,e = aa∗ f,e aa∗ f,e . Pre-multiplying by ae, f , we have †
†
a∗ f,e = ae, f aa∗ f,e = ae, f aa∗ f,e aa∗ f,e = a∗ f,e aa∗ f,e . †
(3) From the hypothesis a∗ f,e a ∈ E (A ), we get a∗ f,e a = a∗ f,e aa∗ f,e a. Post-multiplying by ae, f , we obtain †
†
a∗ f,e = a∗ f,e aae, f = a∗ f,e aa∗ f,e aae, f = a∗ f,e aa∗ f,e . An element w of C ∗ −algebra A is called a semi-idempotent, if w − w2 ∈ J (A ), where J (A ) is the Jacobson radical of A. Corollary 3.6. Let a ∈ A− and let e, f be positive invertible elements in A. Then a is a weighted partial isometry w.r.t. (e, f ) if and only if the following statements hold: (1) aa∗ f,e is semi-idempotent; † (2) ae, f − a∗ f,e ∈ E (A ). †
Proof. If a is a weighted partial isometry w.r.t. (e, f ), then a∗ f,e = ae, f . It is straightforward that aa∗ f,e − aa∗ f,e aa∗ f,e = 0 ∈ †
J (A ) and ae, f − a∗ f,e = 0 ∈ E (A ). 7
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832 †
Conversely, write aa∗ f,e − aa∗ f,e aa∗ f,e = x ∈ J (A ). Pre-multiplying by ae, f , we have a∗ f,e
− a∗ f,e aa∗ f,e
† ae, f x.
=
†
Pre-multiplying the above equality by (ae, f )∗e, f , we conclude that † aae, f
− aa∗ f,e =
by the hypothesis.
(a†e, f )∗e, f a†e, f x, which implies a†e, f − a∗ f,e = a†e, f (a†e, f )∗e, f a†e, f x † † Thus, ae, f − a∗ f,e ∈ E (A ) ∩ J (A ). Hence, ae, f = a∗ f,e .
†
∈ J (A ). On the other hand, ae, f − a∗ f,e ∈ E (A )
Corollary 3.7. Let a ∈ A− and let e, f be positive invertible elements in A. Then a is a weighted partial isometry w.r.t. (e, f ) if † and only if aae, f − aa∗ f,e ∈ E (A ). †
Proof. If a is a weighted partial isometry w.r.t. (e, f ), then a∗ f,e = ae, f , which leads to aa∗ f,e ∈ E (A ). It is easy to see that
(aa†e, f − aa∗ f,e )2 = aa†e, f − aa∗ f,e − aa∗ f,e + aa∗ f,e = aa†e, f − aa∗ f,e . Conversely, by the hypothesis, we have † † † aae, f − aa∗ f,e = (aae, f − aa∗ f,e )2 = aae, f − 2aa∗ f,e + aa∗ f,e aa∗ f,e , which shows that aa∗ f,e = aa∗ f,e aa∗ f,e . That is, aa∗ f,e ∈
E (A ). From Proposition 3.5, we deduce that a is a weighted partial isometry w.r.t. (e, f ).
In order to consider the relation between weighted partial isometry and the solutions of equation, we need to introduce † † (e, f ) a set χa = {a, a# , ae, f , a∗ f,e , (ae, f )∗e, f }. Corollary 3.8. Let a ∈ A# and let e, f be positive invertible elements in A. Then a is a weighted partial isometry w.r.t. (e, f ) if (e, f ) and only if the equation x = xaa∗ f,e has at least one solution in χa . †
†
Proof. “⇒” From the assumption ae, f = a∗ f,e , it is proved by writing x = ae, f . “⇐” (1) If x = a, then a = a2 a∗ f,e . Pre-multiplying by † ae, f aa#
=
† ae, f a# a2 a∗ f,e
=
† ae, f aa∗ f,e
=
† ae, f a# ,
we have
a∗ f,e . †
†
Post-multiplying the above equality by a, we get ae, f a = a∗ f,e a, which yields ae, f = a∗ f,e by [12, Theorem 2.1(iii)]. (2) If x = a# , then a# = a# aa∗ f,e . Pre-multiplying by † ae, f aa#
From (3) If (4) If
† = = ae, f aa∗ f,e = a∗ f,e . † the proof of (1), we get ae, f = a∗ f,e . † † † x = ae, f , then ae, f = ae, f aa∗ f,e = a∗ f,e . x = a∗ f,e , then a∗ f,e = a∗ f,e aa∗ f,e . It follows
† ae, f a,
we can verify that
† ae, f aa# aa∗ f,e
†
from [12, Theorem 2.1(v)] that ae, f = a∗ f,e .
† † † (5) If x = (ae, f )∗e, f , then (ae, f )∗e, f = (ae, f )∗e, f aa∗ f,e . Pre-multiplying the above † † † † a = a(ae, f a ) = aa∗ f,e (ae, f )∗e, f = aa∗ f,e (ae, f )∗e, f aa∗ f,e = aae, f a2 a∗ f,e = a2 a∗ f,e . † That ae, f = a∗ f,e follows from the proof of (1).
equality by aa∗ f,e , we see that
Example 3.9. The complex field C is a unital C ∗ −algebra with involution ∗, the conjugate of complex numbers, and norm
· , the absolute value (or modulus) of complex numbers. Take a = 2 + i, e = 2 and f = 5 in C. From Example 3.2, we have † a2,5 = 15 (2 − i ). It is easy to check that a# = 15 (2 − i ). Then a is weighted-EP w.r.t. (2,5). Example 3.10. The set of all matrices over complex field is 1 pose of matrices, and Frobenius norm · F . Take A = 1 0
a unital C ∗ −algebra with involution ⎛ ∗, the ⎞ conjugate trans 3 1 1 1 0 0 2 2 ⎠. It follows from 1 , M= 0 1 0 , and N = ⎝ 1 3 0 0 0 2 2 2
Proposition 3.5 that A is not a weighted partial isometry w.r.t. (M, N ). In fact, the SVD of A has the form as
⎛ A = UDV ∗ = ⎝
√1 2 √1 2
0
⎛ where U =
√1 2 ⎝ √1 2
0
√1 2 − √12
√1 2 − √1 2
0
0
1
⎞
0
0 2 0⎠ 0 0 1
⎞
⎛
0⎠ and V = ⎝
0 0 0
⎛ √1
2
⎝
√1 2
√1 2
√1 2
√1 2
− √1
√1 2
− √12
⎞ ⎠,
⎞ ⎠ are unitary matrices.
2
8
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832
On the other hand, M = U D1
U∗
and N = V D2
V ∗,
where D1 =
Then M and N are positive. Moreover, we get
1 2
1 2
M = U D1 U ∗ =
1 0 0
⎛√ N 2 = V D2 V ∗ = ⎝ √ 1 2
1
⎛
3 4
− 41
− 41
3 4
and N −1 = ⎝
† AM,N
1
=
1 4
1
⎞
0 1 0 2+1 2 2−1 2
0 − 12 ∗ − 12 √0 , M = U D1 U = 2 √
√
2−1 2 2+1 2
⎞
1
1 0 0
⎛√
⎠, N− 12 = V D2 V ∗ = ⎝ √ − 12
0 1 0
0 1 0
0 0
0 0 2
and D2 =
2 0
0 1
are diagonal matrices.
√1 2
2+2 4
√ 2−2 4
2−2 4
√ 2+2 4
,
⎞ ⎠,
⎠. By direct computation, we have 0
and A∗N,M =
1
1 0 0
0
1 2
1
1
0
1
1
0
.
†
Then AM,N = A∗N,M . Acknowledgements The authors thank the anonymous referees for their valuable comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, second ed., Springer, New York, 2003. J.J. Koliha, D.S. Djordjevic´ , D. Cvetkovic´ , Moore-Penrose inverse in rings with involution, Linear Algebra Appl. 426 (2007) 371–381. D.S. Djordjevic´ , V. Rakocˇ evic´ , Lectures on Generalized Inverses, Faculty of Sciences and Mathematics, University of Niš, 2008. M.P. Drazin, Pseudo-inverses in associative rings and semigroups, Am. Math. Month. 65 (1958) 506–514. D.S. Djordjevic´ , Further results on the reverse order law for generalized inverses, SIAM J. Matrix Anal. Appl. 29 (4) (2007) 1242–1246. D.S. Djordjevic´ , N.v. Dincˇ ic´ , Reverse order law for the Moore-Penrose inverse, J. Math. Anal. Appl. 361 (1) (2010) 252–261. R.E. Harte, M. Mbekhta, On generalized inverses in c∗ −algebras, Studia Math. 103 (1) (1992) 71–77. R.E. Hartwig, K. Spindelböck, Matrices for which a∗ and a† commute, Linear Multilinear Algebra 14 (3) (1983) 241–256. J.J. Koliha, P. Patrício, Elements of rings with equal spectral idempotents, J. Aust. Math. Soc. 72 (1) (2002) 137–152. J.J. Koliha, V. Rakocˇ evic´ , Range projections and the Moore-Penrose inverse in rings with involution, Linear Multilinear Algebra 55 (2) (2007) 103–112. D. Mosic´ , D.S. Djordjevic´ , Weighted-EP elements in c∗ −algebras, Electron. J. Linear Algebra 22 (2011) 912–930. D. Mosic´ , D.S. Djordjevic´ , Weighted partial isometries and weighted-EP elements in c∗ −algebras, Appl. Math. Comput. 265 (2015) 17–30. D. Mosic´ , D.S. Djordjevic´ , Factorization of weighted-EP elements in c∗ −algebras, Appl. Math. Comput. 218 (9) (2012) 5383–5390. D. Mosic´ , Reverse order law for the weighted moore-penrose inverse in c∗ −algebras, Aequationes Math. 85 (3) (2013) 465–470. K.M. Prasad, R.B. Bapat, The generalized Moore-Penrose inverse, Linear Algebra Appl. 165 (1992) 59–69. R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955) 406–413. P. Patrício, A.V.d. Costa, On the Drazin index of regular elements, Cent. Eur. J. Math. 7 (2) (2009) 200–205. Y.G. Tian, Using rank formulas to characterize equalities for Moore-Penrose inverses of matrix products, Appl. Math. Comput. 147 (2) (2004) 581–600. Y.G. Tian, H.X. Wang, Characterizations of EP matrices and weighted-EP matrices, Linear Algebra Appl. 434 (5) (2011) 1295–1318. S.Z. Xu, J.L. Chen, The Moore-enrose inverse in rings with involution, Filomat 33 (18) (2019) 5791–5802.
9
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Moore-Penrose inverses in rings and weighted partial isometries in C ∗ −algebras Ruju Zhao a,c,1,∗, Hua Yao b, Junchao Wei c a
College of Science, Beibu Gulf University, Qinzhou, Guangxi 535011, PR China School of Mathematics and Statistics, Huanghuai University, Zhumadian, Henan 463000, PR China c School of Mathematical Sciences, Yangzhou University, Yangzhou, Jiangsu 225002, PR China b
a r t i c l e
i n f o
Article history: Received 26 May 2018 Revised 7 June 2020 Accepted 22 November 2020
a b s t r a c t We explore the existence of MP-inverse in rings with involution. In addition, we give some equivalent conditions for an element of a C ∗ −algebra to be a weighted-EP element and weighted partial isometry. © 2020 Elsevier Inc. All rights reserved.
2010 MSC: 15A09 16U99 16W10 ∗−Regularity Moore-Penrose inverse Weighted-EP element Weighted partial isometry C ∗ −algebra,
1. Introduction Let R be an associative ring with unit 1R . As usual, denote by J (R ) the Jacobson radical of R, U (R ) the set of all invertible elements of R and E (R ) the set of all idempotents of R. An element a ∈ R is said to be regular if there exists b ∈ R such that aba = a. We denote the set of all regular elements of R by R− . For any element a ∈ R, we define the commutant of a by comm(a ) = {x ∈ R|ax = xa}. An element a ∈ R is said to be group invertible if there exists a# ∈ R such that aa# a = a, a# aa# = a# , aa# = a# a. Note that if a# exists, then it is unique by the above conditions [1]. We denote the set of all group invertible elements of R by R# . An element a ∈ R is group invertible if and only if a ∈ a2 R ∩ Ra2 [4,17]. An involution in R is an anti-isomorphism ∗ : R → R, a → a∗ of degree 2, that is, (a∗ )∗ = a, (a + b)∗ = a∗ + b∗ , (ab)∗ = b∗ a∗ .
∗
Corresponding author at: College of Science, Beibu Gulf University, Qinzhou, Guangxi 535011, PR China. E-mail address: [email protected] (R. Zhao). 1 The research has been supported by the National Science Foundation of China under Grant Nos. 11471282, 11661014, and 11701499 and the High-level Scientific Research Foundation for the introduction of talent of Beibu Gulf University in 2020 under Grant No. 2020KYQD08.
https://doi.org/10.1016/j.amc.2020.125832 0 096-30 03/© 2020 Elsevier Inc. All rights reserved.
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832
Definition 1.1. Let R be a ring and a, b ∈ R. a is said to be right b−regular, if there exists x∗ = x ∈ comm(ba ), such that a = axba. Analogously, an element a is said to be left c−regular, if there is an y∗ = y ∈ comm(ac ) satisfying a = acya. If a is both left c−regular and right b−regular, then it is said to be (c, b)−regular. An element a ∈ R satisfying a∗ = a is said to be symmetric (or Hermitian) and an element p ∈ R is called a projection if 2 p = p = p∗ . Definition 1.2. An element a of a ring R is said to be well-supported if there exists a projection p such that ap = a, a∗ a + 1 − p ∈ U ( R ). The idempotent p is called the support of a. An element a is said to be left ∗−cancellable if a∗ ax = a∗ ay yields ax = ay, right ∗−cancellable if xaa∗ = yaa∗ yields xa = ya, and ∗−cancellable if it is both left and right ∗−cancellable. Note that a is left ∗−cancellable if and only if a∗ is right ∗−cancellable. The Moore-Penrose inverse (or MP-inverse) [16] of a ∈ R is the element a† ∈ R satisfying aa† a = a, a† aa† = a† , (aa† )∗ = aa† , (a† a )∗ = a† a. There is at most one a† satisfying the above equations [2,7,8]. Denote by R† the set of all MP-invertible elements of R. Actually, a is MP-invertible if and only if a is ∗−cancellable and a∗ aa∗ is regular [9]. An element a ∈ R† satisfying aa† = a† a is said to be EP. Denote by REP the set of all EP elements of R. An element a ∈ REP if and only if a ∈ R# ∩ R† and a# = a† . The Drazin inverse of a ∈ R is the element aD ∈ R satisfying aD aaD = aD , aaD = aD a, ak+1 aD = ak for some nonnegative integer k. The least such k is called the index of a, denoted by ind (a ). We denote the set of all Drazin invertible elements of R by RD . If ind (a ) = 1, then the Drazin inverse aD is the group inverse a# . Let A be a unital C ∗ −algebra with unite 1A . Actually, the unital C ∗ −algebra A is an associative ring with the addition and the multiplication of A. Thus, the definitions and conclusions mentioned above still work for A. Particularly, in C ∗ −algebra, every element is ∗−cancellable [2]. An element x ∈ A is said to be positive if there exists y ∈ A satisfying x = y∗ y. Alternatively, x ∈ A is said to be positive if x is Hermitian and σ (x ) ⊆ [0, +∞ ), where σ (x ) is the spectrum of x. Definition 1.3. [2,3] Let A be a unital C ∗ −algebra and let e, f be positive invertible elements in A. An element a ∈ A is said to be weighted MP-invertible with weights e, f if there exists b ∈ A such that aba = a, bab = b, (eab)∗ = eab, ( f ba )∗ = f ba. †
The unique such b is called weighted MP-inverse of a with weights e and f, and denoted by ae, f if it exists. Denote by †
Ae, f the set of all weighted MP-invertible elements of A with weights e and f .
Assume that e, f ∈ A are positive and invertible. Define the mapping (∗, e ) : A −→ A, x −→ x∗e = e−1 x∗ e. It is easy to 1
1
check that the mapping (∗, e ) is an involution of A. Moreover, define x e = e 2 xe− 2 , for any x ∈ A. Then · e is a norm of A. We can show that Ae = (A, (∗, e ), · e ) is a unital C ∗ −algebra with the involution (∗, e ) and the norm · e . Consequently, the conditions of Definition 1.3 can be rewritten as aba = a, bab = b, (ab)∗e = ab, (ba )∗ f = ba. If f = λe, where 0 = λ ∈ C, then x∗e = x∗ f for any x ∈ A, where C is the complex field. In this case, the weighted MP† inverse ae, f is a special MP-inverse a† with the involution (∗, e ). In fact, mapping (∗, e ) is an involution and x∗e = x∗ f for †
any x ∈ A. Let ∗ = (∗, e ) and b = ae, f . From the definition of weighted MP-inverse, we get aba = a, bab = b, (ab)∗ = ab, and
(ba )∗
†
= ba. From the definition of MP-inverse, b = ae, f is the MP-inverse of a.
Define the mapping (∗, e, f ): A −→ A, x −→ x∗e, f = e−1 x∗ f, for all x ∈ A. In general, (∗, e, f ) is not an involution. But if f = e, then (∗, e, f ) is an involution, in this case x∗e, f = x∗ f,e for any x ∈ A. Theorem 1.4. [2] Let A be a unital C ∗ −algebra and let e, f be positive invertible elements of A. If a ∈ A− , then the unique † weighted MP-inverse ae, f exists and 1
1
1
1
ae, f = f − 2 (e 2 a f − 2 )† e 2 . †
Definition 1.5. [11] Let A be a unital C ∗ −algebra and let e, f be positive invertible elements in A. The element a ∈ A− is said to be weighted-EP with respect to elements e, f (or weighted-EP w.r.t. (e, f )) if both ea and a f −1 are EP (which is † † equivalent to aae, f = ae, f a). †
An element a ∈ A− satisfying a∗ f,e = ae, f is called a weighted partial isometry with respect to elements e, f (or a
weighted partial isometry w.r.t. (e, f )). The Moore-Penrose inverse in rings with involution and the weighted Moore-Penrose inverse in C ∗ −algebras have several important applications in matrices over an integral domain, bounded linear operators on Hilbert spaces and numerical computations [1,5,6,15]. 2
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832
Various characterizations of Moore-Penrose inverse in rings with involution were given in [10,18]. Weighted-EP matrix was defined and a number of conclusions about it were presented using various formulas for a rank of a complex matrix in [19]. The reverse order law for the Moore-Penrose inverse and the weighted Moore-Penrose inverse in C ∗ −algebras were investigated [6,14]. Many achievements have been made in weighted-EP elements and weighted partial isometries of C ∗ −algebras [11–13]. In this paper, we study the Moore-Penrose inverse in rings with involution, weighted-EP element and weighted partial isometry in C ∗ −algebras, extending the results from [2,11,12]. In Section 2, we prove that an element a of a ring R is right a∗ −regular if and only if a is well-supported or MP-invertible. In addition, we also present other equivalent conditions for an element to be MP-invertible. Under the condition that a∗ a = a2 , a is EP if and only if a is group invertible, if and only if a is Drazin invertible and left ∗−cancellable. In Section 3, letting A be a C ∗ −algebra, a ∈ A− and e, f be positive invertible † † elements of A, we give an example to show that the equality (a∗e, f ) f,e = a(a∗ f,e a ) f, f is not true in general. And then we present some equivalent conditions for an element a ∈ A to be a weighted-EP element w.r.t (e, f ) and weighted partial isometry w.r.t (e, f ) by means of E (A ), J (A ), and the solutions of equation x = xaa∗ f,e . 2. ∗− Regularities and MP-inverses In this section, we work in a ring with involution unless otherwise stated. We first present some equivalent conditions for an element a of a ring R to be well-supported, left ∗−cancellable and right a∗ −regular. Denote r (a ) = {x ∈ R|ax = 0}, for any a ∈ R. For convenience, here we set 1R = 1. Lemma 2.1. Let R be a ring and a ∈ R. If a is well-supported, then a is left ∗−cancellable. Proof. If there is an y ∈ R such that a∗ ay = 0, then we have (a∗ a + 1 − p) py = 0, where p is the support of a, because ap = a and p2 = p. From the hypothesis a∗ a + 1 − p ∈ U (R ), we obtain py = 0. Consequently, ay = apy = 0. Theorem 2.2. Let R be a ring and a ∈ R. Then a is right a∗ −regular if and only if a is well-supported. Proof. “⇒” By assumption, we know that there exists x∗ = x ∈ comm(a∗ a ) such that a = axa∗ a. Taking p = xa∗ a, we obtain p∗ = a∗ ax∗ = a∗ ax = xa∗ a = p, a = axa∗ a = ap, p2 = xa∗ axa∗ a = xa∗ a = p, pa∗ = (ap)∗ = a∗ . It is easy to check that (a∗ a + 1 − p)( px + 1 − p) = a∗ apx + a∗ a(1 − p) + 1 − p = a∗ ax + 1 − p = p + 1 − p = 1. The equality px(1 − p) = xa∗ a(x − x2 a∗ a ) = xa∗ a(x − xa∗ ax ) = xa∗ a(1 − xa∗ a )x = p(1 − p)x = 0 implies ( px + 1 − p)(a∗ a + 1 − p) = pxa∗ a + px(1 − p) + (1 − p)a∗ a + 1 − p = p + 1 − p = 1. That is, a∗ a + 1 − p ∈ U (R ). From the definition of well-supported element, the proof is completed. “⇐” According to the hypothesis, there exists p ∈ R such that a = ap, p∗ = p = p2 , and a∗ a + 1 − p = u ∈ U (R ). Applying involution to the above equalities, we get a∗ = p∗ a∗ = pa∗ and u∗ = (a∗ a + 1 − p)∗ = a∗ a + 1 − p = u. This gives (u−1 )∗ = u−1 . Furthermore, it is easy to check that ua∗ a = (a∗ a + 1 − p)a∗ a = a∗ aa∗ a = a∗ a(a∗ a + 1 − p) = a∗ au, which implies u ∈ comm(a∗ a ). Thus, u−1 ∈ comm(a∗ a ). Finally, we only need to show that a = au−1 a∗ a. In fact, up = (a∗ a + 1 − p) p = a∗ ap + (1 − p) p = a∗ ap = a∗ a, which leads to p = u−1 a∗ a. From the proof of Theorem 2.2, we have the following corollary. Corollary 2.3. Let R be a ring and a ∈ R. Then a is right a∗ −regular if and only if there is a projection p such that a∗ a + 1 − p ∈ U (R ), and one of the following conditions holds: (1) r (a ) = r ( p); (2) Ra = Rp; (3) a∗ R = pR. Proof. From Theorem 2.2, we only need to prove that the equation ap = a is satisfied. (1) Since p is a projection, we have p∗ = p = p2 and p − 1 ∈ r ( p). From the hypothesis, we get a( p − 1 ) = 0, that is, ap = a. (2) From the equation Ra = Rp, there exists x ∈ R such that a = xp. Since p is a projection, we conclude that ap = xp2 = xp = a. (3) From the hypothesis, there exists y ∈ R such that a∗ = py. Since p is a projection, we have a = y∗ p∗ = y∗ p. It follows that ap = y∗ p2 = y∗ p = a. An element a of a ring R is said to be weakly left ∗−cancellable, if a∗ ax = 0 implies ax2 = 0 for any x ∈ R. Obviously, if a is left ∗−cancellable, then a is weakly left ∗−cancellable. But the converse of this statement is not true from the following example. 3
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832
Example 2.4. Let R = Z4 Z4 . Define a mapping ∗ : R −→ R, (x, y ) −→ (y, x ). It is easy to check that the mapping ∗ is an involution of R. Take a = (2, 3 ). Then a∗ = (3, 2 ) and a∗ a = (2, 2 ). If a∗ a(x, y ) = 0, then (x, y ) ∈ {(0, 0 ) = 0, (0, 2 ), (2, 0 ), (2, 2 )}, which gives a(x, y )2 = 0. But if (x, y ) = (0, 2 ), then a(x, y ) = (0, 2 ) = 0. That is, a is weakly left ∗−cancellable, but it is not left ∗−cancellable. In the following, we explore the existence of MP-inverse in rings by means of weakly left ∗−cancellation, ∗−regularity, Hermitian element, and idempotent. Theorem 2.5. Let R be a ring and a ∈ R. Then the following statements are equivalent: (1) a ∈ R† ; (2) R = a∗ aR r (a∗ a ) and a is weakly left ∗−cancellable; (3) R = a∗ aR + r (a∗ a ) and a is weakly left ∗−cancellable. Proof. (1) ⇒ (2) It follows from the hypothesis that a† exists and a is left ∗−cancellable, which yield a is weakly left ∗−cancellable and r (a ) = r (a∗ a ). On the other hand, from [20, Theorem 3.11(2)], we obtain R = a∗ aR r (a ). Thus, R = a∗ aR r ( a∗ a ). (2) ⇒ (3) It is straightforward. (3) ⇒ (1) According to the condition R = a∗ aR + r (a∗ a ), we have a∗ aR = a∗ aa∗ aR. Write a∗ a = a∗ aa∗ ab, where b ∈ R. Then ∗ a a(1 − a∗ ab) = 0. Since a is weakly left ∗−cancellable, we deduce that a(1 − a∗ ab)2 = 0, which leads to a = aa∗ a(2b) − aa∗ aba∗ ab ∈ aa∗ aR. Therefore, a ∈ R† by [20, Theorem 3.1(3)]. Theorem 2.6. Let R be a ring and a ∈ R. Then a ∈ R† if and only if a is (a∗ , a∗ )−regular. Proof. “⇐” Since a is (a∗ , a∗ )−regular, we conclude that there are y∗ = y ∈ comm(aa∗ ) and z∗ = z ∈ comm(a∗ a ) satisfying a = aa∗ ya = aza∗ a. Fix x = yaz. Then aa∗ xa∗ a = aa∗ yaza∗ a = aza∗ a = a. Let u, v ∈ R satisfy a∗ au = 0 = vaa∗ . Then au = aa∗ xa∗ au = 0 and va = vaa∗ xa∗ a = 0. Thus, a is ∗−cancellable. On the other hand, we have a∗ aa∗ = a∗ aa∗ xa∗ aa∗ , which yields a∗ aa∗ is regular. From [9, Theorem 5.4(xi)], we get a ∈ R† . “⇒” Using the assumption, we get a = aa† a. Write y = a† (a† )∗ and z = (a† )∗ a† . Then it is easy to verify that y∗ = y ∈ comm(a∗ a ), z∗ = z ∈ comm(aa∗ ), and a = aa∗ za = aya∗ a. That is, a is both left and right a∗ −regular. Therefore, a is (a∗ , a∗ )−regular. Theorem 2.7. Let R be a ring and a ∈ R. Then a ∈ R† if and only if a is right a∗ −regular. Proof. “⇒” It follows from a ∈ R† that a† exists. Write x = a† (a† )∗ . Then x∗ = x, xa∗ a = a† (a† )∗ a∗ a = a† (aa† )∗ a = a† a, a∗ ax = a∗ aa† (a† )∗ = a∗ (a† )∗ = (a† a )∗ = a† a, which show that x∗ = x ∈ comm(a∗ a ) and axa∗ a = aa† a = a. Hence, a is right a∗ −regular. “⇐” Let x∗ = x ∈ comm(a∗ a ) satisfy a = axa∗ a. Taking b = xa∗ , we get ab = axa∗ , ba = xa∗ a, (ab)∗ = ax∗ a∗ = axa∗ = ab, (ba )∗ = a∗ ax∗ = a∗ ax = xa∗ a = ba, aba = axa∗ a = a, bab = xa∗ axa∗ = x(ax∗ a∗ a )∗ = x(axa∗ a )∗ = xa∗ = b. Consequently, a ∈ R† and a† = xa∗ . Theorem 2.8. Let R be a ring and a ∈ R. Then the following conditions are equivalent: (1) a ∈ R† ; (2) there is an x ∈ R such that a = axa∗ a, and a∗ ax is Hermitian. Proof. (1) ⇒ (2) Write x = a† (a† )∗ . Then a∗ ax = a∗ aa† (a† )∗ = a∗ (a† )∗ = a† a, which proves that a∗ ax is Hermitian. Furthermore, we have axa∗ a = aa† (a† )∗ a∗ a = (a† )∗ a∗ a = aa† a = a. (2) ⇒ (1) Let y ∈ R satisfy a∗ ay = 0. Then ay = (axa∗ a )y = ax(a∗ ay ) = 0. That is, a is left ∗−cancellable. Taking the involution on a = axa∗ a, we obtain a∗ = a∗ ax∗ a∗ . Post-multiplying by a, we have a∗ a = a∗ ax∗ a∗ a, which gives a = ax∗ a∗ a = a(a∗ ax )∗ = aa∗ ax, because a is left ∗−cancellable and a∗ ax is Hermitian. Applying the involution to axa∗ a = a = ax∗ a∗ a, we deduce that a∗ ax∗ a∗ = a∗ = a∗ axa∗ , which leads to ax∗ a∗ = axa∗ . Moreover, taking b = x∗ a∗ , we get ab = ax∗ a∗ = axa∗ . Applying the involution to the above equality, we see that (ab)∗ = (axa∗ )∗ = ax∗ a∗ = ab. From the hypothesis, we conclude that ba = x∗ a∗ a = (a∗ ax )∗ = a∗ ax, which shows that (ba )∗ = ba. It is left to prove that aba = a and bab = b. Actually, it is easy to check that aba = axa∗ a = a and bab = a∗ axx∗ a∗ = (a∗ ax )∗ x∗ a∗ = x∗ a∗ ax∗ a∗ = x∗ (axa∗ a )∗ = x∗ a∗ = b. 4
R. Zhao, H. Yao and J. Wei
Therefore, a ∈ R† and a† = x∗ a∗ .
Applied Mathematics and Computation 395 (2021) 125832
Theorem 2.9. Let R be a ring and a ∈ R. Then the following conditions are equivalent: (1) a ∈ R† ; (2) there are g, f ∈ E (R ) such that aR = gR, R f = Ra, 1 + (g∗ − g)∗ (g∗ − g) ∈ U (R ), and 1 + ( f ∗ − f )∗ ( f ∗ − f ) ∈ U (R ). Proof. (1) ⇒ (2) Fix g = aa† and f = a† a. Using the equality a = aa† a, we have g∗ = g = g2 and f ∗ = f = f 2 . It is evident that aR = aa† aR = gaR ⊆ gR = aa† R ⊆ aR, Ra = Raa† a = Ra f ⊆ R f = Ra† a ⊆ Ra, 1 + (g∗ − g)∗ (g∗ − g) = 1 ∈ U (R ), 1 + ( f ∗ − f )∗ ( f ∗ − f ) = 1 ∈ U (R ). (2) ⇒ (1) Write s = 1 + (g∗ − g)∗ (g∗ − g) ∈ U (R ) and t = 1 + ( f ∗ − f )∗ ( f ∗ − f ) ∈ U (R ). Then we get sgg∗ = gg∗ gg∗ = gg∗ s, s∗ = s, sg = gg∗ g = gs, t f ∗ f = f ∗ f f ∗ f = f ∗ f t , t ∗ = t , t f = f f ∗ f = f t. Take p = gg∗ s−1 and q = f ∗ f t −1 . By the above, we obtain a series of precious equalities as follows: p2 = gg∗ s−1 gg∗ s−1 = (gg∗ gg∗ )s−1 s−1 = gg∗ s−1 = p, q2 = f ∗ f t −1 f ∗ f t −1 = ( f ∗ f f ∗ f )t −1 t −1 = f ∗ f t −1 = q, p∗ = (s∗ )−1 gg∗ = s−1 gg∗ = gg∗ s−1 = p, q∗ = (t ∗ )−1 f ∗ f = f ∗ f t −1 = q, pg = gg∗ s−1 g = (gg∗ g)s−1 = gss−1 = g, gp = g(gg∗ s−1 ) = gg∗ s−1 = p, f q = f f ∗ f t −1 = f t t −1 = f, q f = f ∗ f t −1 f = f ∗ f f t −1 = f ∗ f t −1 = q. Since aR = gR and R f = Ra, we infer that a = ga = a f . It is immediate that pa = p(ga ) = ( pg)a = ga = a and aq = (a f )q = a( f q ) = a f = a. Set g = ax and f = ya, where x, y ∈ R. Write c = qyaxp. Then we conclude that ac = aqyaxp = ayaxp = a f xp = axp = gp = p = p∗ = (ac )∗ , ca = qyaxpa = qyaxa = qyga = qya = q f = q = q∗ = (ca )∗ , aca = pa = a, cac = qc = q(qyaxp) = qyaxp = c. Consequently, a ∈ R† and a† = qyaxp. In general, an element a of a ring R is not EP, if only a ∈ R# . In the following theorem we assume that a∗ a = a2 . Then the EP element and the group invertible element are equivalent. Theorem 2.10. Let R be a ring and a ∈ R. If a∗ a = a2 , then the following conditions are equivalent: (1) a ∈ REP ; (2) a ∈ R# ; (3) a is left ∗−cancellable and a ∈ RD . Proof. (1) ⇒ (2) The proof is straightforward. (2) ⇒ (3) Since a ∈ R# , it follows that a ∈ RD , in this case ind (a ) = 1. Let y ∈ R satisfy a∗ ay = 0. Then a2 y = a∗ ay = 0, which gives ay = a# a2 y = 0. (3) ⇒ (1) Using the assumptions, we see that there is an nonnegative integer k such that ak+1 aD = ak . If k = 1, then a ∈ R# and a# = aD . If k ≥ 2, then a∗ ak aD = a∗ ak−1 , because a∗ a = a2 . By the hypothesis, we assert that ak aD = ak−1 . Repeating the above process, we have a2 aD = a, which yields a ∈ R# . Summarizing, we have a ∈ R# . Applying the involution to a = aa# a, we conclude that a∗ = a∗ (a# )∗ a∗ . Post-multiplying by a, we have a∗ a = a∗ (a# )∗ a∗ a, which implies that a2 = a∗ (a# )∗ a2 . Post-multiplying by (a# )2 , we get aa# = a∗ (a# )∗ aa# = (aa# )∗ aa# . This gives (aa# )∗ = aa# . Therefore, a ∈ R† and a† = a# . That is, a ∈ REP . At the end of this section, we give an equivalent condition for an element to be EP. Theorem 2.11. Let R be a ring and a ∈ R. Then the following conditions are equivalent: (1) a ∈ REP ; (2) there is an g ∈ E (R ) such that Ra = Rg, aR = gR, gg∗ = g∗ g and 1 + (g∗ − g)∗ (g∗ − g) ∈ U (R ). Proof. (1) ⇒ (2) The conclusion is proved by taking g = a† a, because a† a = aa† . (2) ⇒ (1) Write s = 1 + (g∗ − g)∗ (g∗ − g) ∈ U (R ). Then sg∗ g = sgg∗ = gg∗ s = g∗ gs, s = s∗ , and sg = gg∗ g = gs. Take p = gg∗ s−1 . Then we have p = g∗ gs−1 , p2 = p, p∗ = p, pg = g, and gp = p, which imply g∗ p = p and p = p∗ = (g∗ p)∗ = p∗ g = pg = g. Since Ra = Rg and aR = gR, it follows that a = ga = ag. Fix g = ax = ya, where x, y ∈ R. Then a = axa = aya. Set b = yax. Then we can verify that ab = ayax = ax = g = p = p∗ = (ab)∗ , ba = yaxa = ya = g = p = p∗ = (ba )∗ , which give ab = ba. Moreover, we have aba = ga = a and bab = gb = yayax = yax = b. Consequently, a ∈ R# ∩ R† and a# = b = a† . That is, a ∈ REP . 5
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3. Weighted-EP elements and weighted partial isometries In this section, we study some equivalent conditions for an element to be a weighted-EP element and weighted partial isometry in C ∗ −algebras. Lemma 3.1. [11] Let A be a unital C ∗ −algebra and let e, f be positive invertible elements of A. For each a ∈ A− , the following is satisfied: † † (1) (ae, f ) f,e = a; †
†
(2) (a∗ f,e ) f,e = (ae, f )∗e, f ; †
†
(3) a∗ f,e = ae, f aa∗ f,e = a∗ f,e aae, f ; (4) (5) (6) (7) (8)
† † a∗ f,e (ae, f )∗e, f = ae, f a; (a†e, f )∗e, f a∗ f,e = aa†e, f ; (a∗ f,e a )†f, f = a†e, f (a†e, f )∗e, f ; (aa∗ f,e )†e,e = (a†e, f )∗e, f a†e, f ; † † † ae, f = (a∗ f,e a ) f, f a∗ f,e = a∗ f,e (aa∗ f,e )e,e . †
†
In addition, there is seemingly still a equality (a∗e, f ) f,e = a(a∗ f,e a ) f, f . But, in fact, it is not true in general according to the following example. Example 3.2. It is well-known that the complex field C is a unital C ∗ −algebra with involution ∗, the conjugate of complex numbers, and norm · , the absolute value (or modulus) of complex numbers. Take a = 2 + i, b = 15 (2 − i ), e = 2 and f = 5 in C. Then aba = a and e, f are√positive invertible. It is obvious that √ 1 1 e∗ = 2, e 2 = 2, e− 2 = 22 , √ √ 1 1 f ∗ = 5, f 2 = 5, f − 2 = 55 . Using the existence of weighted MP-inverse and the definition of (∗, e, f ), we obtain a∗e, f = 52 (2 − i ), a∗ f,e = 25 (2 − i ), √ √ 10 10 5 ( 5 (2 +
†
ae, f =
i ).
i ))† =
√
√ 10 10 5 [ 10 (2 − i )]
Furthermore, we have † a(a∗ f,e a ) f, f = (2 + i ) 15 (2 − i ) 12 (2 + i ) = †
1 5 (2
=
1 2 (2
− i ) = b. †
†
+ i ), (a∗e, f ) f,e = [ 25 (2 − i )]5,2 =
√
√ 10 5 10 2 [ 4 (2
− i )] † =
√
√ 10 2 10 2 [ 125 (2
+ i )] =
2 25 (2 +
†
Consequently, (a∗e, f ) f,e = a(a∗ f,e a ) f, f .
Lemma 3.3. Let A be a unital C ∗ −algebra and let e, f be positive invertible elements of A. If a ∈ A− , then (a∗ f,e )†f,e = a(a∗ f,e a )†f, f = (aa∗ f,e )†e,e a. 1
1
1
1
Proof. Write x = e 2 a f − 2 . Since a is regular, there exists b ∈ A such that a = aba. Take y = f 2 be− 2 . Then xyx = 1
1
1
1
1
1
1
1
1
1
e 2 a f − 2 f 2 be− 2 e 2 a f − 2 = e 2 aba f − 2 = e 2 a f − 2 = x. It means that x is also regular. From Theorem 1.4, we deduce that 1
1
1
1
1
1
1
1
1
1
a(a∗ f,e a )†f, f = a f − 2 ( f − 2 a∗ e 2 e 2 a f − 2 )† f 2 = a f − 2 (x∗ x )† f 2 = e− 2 xx† (x∗ )† f 2 1
1
1
1
1
1
1
1
= e− 2 ( x† )∗ f 2 = e− 2 ( x∗ )† f 2 = e− 2 ( f − 2 a∗ e 2 )† f 2 , 1
1
1
1
1
1
1
1
1
1
(aa∗ f,e )†e,e a = e− 2 (e 2 a f − 2 f − 2 a∗ e 2 )† e 2 a = e− 2 (xx∗ )† e 2 a = e− 2 (x∗ )† x† x f 2 1
1
1
1
1
1
1
1
= e− 2 ( x† )∗ f 2 = e− 2 ( x∗ )† f 2 = e− 2 ( f − 2 a∗ e 2 )† f 2 , and
1
1
1
1
1
1
1
1
(a∗ f,e )†f,e = e− 2 ( f 2 a∗ f,e e− 2 )† f 2 = e− 2 ( f − 2 a∗ e 2 )† f 2 .
Let A be a unital C ∗ −algebra and let e, f be positive invertible elements in A. If a ∈ A# , then a ∈ A− , because a = aa# a. That is, if a ∈ A# , then the unique weighted MP-inverse of a also exists. In the following theorem, we give the conditions to ensure that a is weighted-EP w.r.t. (e, f ), extending the results from [11]. Theorem 3.4. Let a ∈ A# and let e, f be positive invertible elements in A. Then the following statements are equivalent: (1) a is weighted-EP w.r.t. (e, f ); † † (2) ae, f + a# = 2ae, f aa# ; †
†
†
†
(3) aae, f = (ae, f )2 a2 and ae, f a = a2 (ae, f )2 ; †
†
(4) an+1 (ae, f )n = a = (ae, f )m am+1 , for any/some integers m, n ≥ 1; 6
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Applied Mathematics and Computation 395 (2021) 125832
†
†
(5) ae, f a2n ae, f = an (ae, f )2 an , for any/some integers n ≥ 1. †
Proof. If a is weighted-EP w.r.t. (e, f ), then a# = ae, f by [11, Theorem 2.2(xi)]. It is not difficult to prove that conditions (2)-(5) hold. † † Conversely, in order to conclude that a is weighted-EP w.r.t. (e, f ), we only need to show that the condition ae, f a = aae, f is satisfied by [11, Theorem 2.2(viii)]. † † † (2) From the equality ae, f + a# = 2ae, f aa# , pre-multiplying by a, we have aae, f = aa# . On the other hand, post†
†
†
multiplying by a, we get ae, f a = a# a. Hence, ae, f a = aae, f . (3) Post-multiplying the hypothesis
† aae, f
=
(a†e, f )2 a2
†
by a# ae, f , we obtain
a# a†e, f = a(a# )2 a†e, f = aa†e, f a(a# )2 a†e, f = aa†e, f a# a†e, f = (a†e, f )2 a2 a# a†e, f = (a†e, f )2 aa†e, f = (a†e, f )2 , †
†
which yields a# aae, f = a(ae, f )2 . Post-multiplying by a, we can verify that a# a =
† a# aae, f a
=
† a(ae, f )2 a
†
†
†
†
†
†
= (aae, f )ae, f a = (ae, f )2 a2 ae, f a = (ae, f )2 a2 = aae, f . †
†
Similarly, using the equality ae, f a = a2 (ae, f )2 , we deduce that †
†
†
†
†
†
†
a# a = a(ae, f )2 a = aae, f (ae, f a ) = aae, f a2 (ae, f )2 = a2 (ae, f )2 = ae, f a. † ae, f a
† Therefore, = a# a = aae, f . † † (4) The condition an+1 (ae, f )n = a = (ae, f )m am+1 (m, n ≥ 1) gives † † † † † † a# a = a# an+1 (ae, f )n = an (ae, f )n = an (ae, f )n aae, f = a# a2 ae, f = aae, f , † † † † † † # m m +1 # m m m m 2 # aa = (ae, f ) a a = (ae, f ) a = ae, f a(ae, f ) a = ae, f a a = ae, f a. † † Consequently, ae, f a = a# a = aae, f . † † † † † (5) Pre-multiplying and post-multiplying the equality ae, f a2n ae, f = an (ae, f )2 an by a, we have a2n ae, f = an+1 (ae, f )2 an and † † ae, f a2n = an (ae, f )2 an+1 . It follows that † † † † a2n−1 ae, f = a# a2n ae, f = a# an+1 (ae, f )2 an = an (ae, f )2 an , † † † † † † ae, f a2n−1 = ae, f a2n a# = an (ae, f )2 an+1 a# = an (ae, f )2 an , which lead to a2n−1 ae, f = ae, f a2n−1 . From [11, Theorem 2.2(lv)], we † † obtain ae, f a = aae, f .
Motivated by [12], we present some novel equivalent conditions for an element a ∈ A to be a weighted partial isometry w.r.t. (e, f ) by means of E (A ), J (A ) and the solutions of certain equations. Proposition 3.5. Let a ∈ A− and let e, f be positive invertible elements in A. Then the following statements are equivalent: (1) a is a weighted partial isometry w.r.t. (e, f ); (2) aa∗ f,e ∈ E (A ); (3) a∗ f,e a ∈ E (A ). †
Proof. If a is a weighted partial isometry w.r.t. (e, f ), then a∗ f,e = ae, f . It is easy to check that conditions (2) and (3) hold.
Conversely, in order to show that a is a weighted partial isometry w.r.t. (e, f ), we only need to prove that the condition a∗ f,e = a∗ f,e aa∗ f,e is satisfied by [12, Theorem 2.1(v)]. † (2) Since aa∗ f,e ∈ E (A ), it follows that aa∗ f,e = aa∗ f,e aa∗ f,e . Pre-multiplying by ae, f , we have †
†
a∗ f,e = ae, f aa∗ f,e = ae, f aa∗ f,e aa∗ f,e = a∗ f,e aa∗ f,e . †
(3) From the hypothesis a∗ f,e a ∈ E (A ), we get a∗ f,e a = a∗ f,e aa∗ f,e a. Post-multiplying by ae, f , we obtain †
†
a∗ f,e = a∗ f,e aae, f = a∗ f,e aa∗ f,e aae, f = a∗ f,e aa∗ f,e . An element w of C ∗ −algebra A is called a semi-idempotent, if w − w2 ∈ J (A ), where J (A ) is the Jacobson radical of A. Corollary 3.6. Let a ∈ A− and let e, f be positive invertible elements in A. Then a is a weighted partial isometry w.r.t. (e, f ) if and only if the following statements hold: (1) aa∗ f,e is semi-idempotent; † (2) ae, f − a∗ f,e ∈ E (A ). †
Proof. If a is a weighted partial isometry w.r.t. (e, f ), then a∗ f,e = ae, f . It is straightforward that aa∗ f,e − aa∗ f,e aa∗ f,e = 0 ∈ †
J (A ) and ae, f − a∗ f,e = 0 ∈ E (A ). 7
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832 †
Conversely, write aa∗ f,e − aa∗ f,e aa∗ f,e = x ∈ J (A ). Pre-multiplying by ae, f , we have a∗ f,e
− a∗ f,e aa∗ f,e
† ae, f x.
=
†
Pre-multiplying the above equality by (ae, f )∗e, f , we conclude that † aae, f
− aa∗ f,e =
by the hypothesis.
(a†e, f )∗e, f a†e, f x, which implies a†e, f − a∗ f,e = a†e, f (a†e, f )∗e, f a†e, f x † † Thus, ae, f − a∗ f,e ∈ E (A ) ∩ J (A ). Hence, ae, f = a∗ f,e .
†
∈ J (A ). On the other hand, ae, f − a∗ f,e ∈ E (A )
Corollary 3.7. Let a ∈ A− and let e, f be positive invertible elements in A. Then a is a weighted partial isometry w.r.t. (e, f ) if † and only if aae, f − aa∗ f,e ∈ E (A ). †
Proof. If a is a weighted partial isometry w.r.t. (e, f ), then a∗ f,e = ae, f , which leads to aa∗ f,e ∈ E (A ). It is easy to see that
(aa†e, f − aa∗ f,e )2 = aa†e, f − aa∗ f,e − aa∗ f,e + aa∗ f,e = aa†e, f − aa∗ f,e . Conversely, by the hypothesis, we have † † † aae, f − aa∗ f,e = (aae, f − aa∗ f,e )2 = aae, f − 2aa∗ f,e + aa∗ f,e aa∗ f,e , which shows that aa∗ f,e = aa∗ f,e aa∗ f,e . That is, aa∗ f,e ∈
E (A ). From Proposition 3.5, we deduce that a is a weighted partial isometry w.r.t. (e, f ).
In order to consider the relation between weighted partial isometry and the solutions of equation, we need to introduce † † (e, f ) a set χa = {a, a# , ae, f , a∗ f,e , (ae, f )∗e, f }. Corollary 3.8. Let a ∈ A# and let e, f be positive invertible elements in A. Then a is a weighted partial isometry w.r.t. (e, f ) if (e, f ) and only if the equation x = xaa∗ f,e has at least one solution in χa . †
†
Proof. “⇒” From the assumption ae, f = a∗ f,e , it is proved by writing x = ae, f . “⇐” (1) If x = a, then a = a2 a∗ f,e . Pre-multiplying by † ae, f aa#
=
† ae, f a# a2 a∗ f,e
=
† ae, f aa∗ f,e
=
† ae, f a# ,
we have
a∗ f,e . †
†
Post-multiplying the above equality by a, we get ae, f a = a∗ f,e a, which yields ae, f = a∗ f,e by [12, Theorem 2.1(iii)]. (2) If x = a# , then a# = a# aa∗ f,e . Pre-multiplying by † ae, f aa#
From (3) If (4) If
† = = ae, f aa∗ f,e = a∗ f,e . † the proof of (1), we get ae, f = a∗ f,e . † † † x = ae, f , then ae, f = ae, f aa∗ f,e = a∗ f,e . x = a∗ f,e , then a∗ f,e = a∗ f,e aa∗ f,e . It follows
† ae, f a,
we can verify that
† ae, f aa# aa∗ f,e
†
from [12, Theorem 2.1(v)] that ae, f = a∗ f,e .
† † † (5) If x = (ae, f )∗e, f , then (ae, f )∗e, f = (ae, f )∗e, f aa∗ f,e . Pre-multiplying the above † † † † a = a(ae, f a ) = aa∗ f,e (ae, f )∗e, f = aa∗ f,e (ae, f )∗e, f aa∗ f,e = aae, f a2 a∗ f,e = a2 a∗ f,e . † That ae, f = a∗ f,e follows from the proof of (1).
equality by aa∗ f,e , we see that
Example 3.9. The complex field C is a unital C ∗ −algebra with involution ∗, the conjugate of complex numbers, and norm
· , the absolute value (or modulus) of complex numbers. Take a = 2 + i, e = 2 and f = 5 in C. From Example 3.2, we have † a2,5 = 15 (2 − i ). It is easy to check that a# = 15 (2 − i ). Then a is weighted-EP w.r.t. (2,5). Example 3.10. The set of all matrices over complex field is 1 pose of matrices, and Frobenius norm · F . Take A = 1 0
a unital C ∗ −algebra with involution ⎛ ∗, the ⎞ conjugate trans 3 1 1 1 0 0 2 2 ⎠. It follows from 1 , M= 0 1 0 , and N = ⎝ 1 3 0 0 0 2 2 2
Proposition 3.5 that A is not a weighted partial isometry w.r.t. (M, N ). In fact, the SVD of A has the form as
⎛ A = UDV ∗ = ⎝
√1 2 √1 2
0
⎛ where U =
√1 2 ⎝ √1 2
0
√1 2 − √12
√1 2 − √1 2
0
0
1
⎞
0
0 2 0⎠ 0 0 1
⎞
⎛
0⎠ and V = ⎝
0 0 0
⎛ √1
2
⎝
√1 2
√1 2
√1 2
√1 2
− √1
√1 2
− √12
⎞ ⎠,
⎞ ⎠ are unitary matrices.
2
8
R. Zhao, H. Yao and J. Wei
Applied Mathematics and Computation 395 (2021) 125832
On the other hand, M = U D1
U∗
and N = V D2
V ∗,
where D1 =
Then M and N are positive. Moreover, we get
1 2
1 2
M = U D1 U ∗ =
1 0 0
⎛√ N 2 = V D2 V ∗ = ⎝ √ 1 2
1
⎛
3 4
− 41
− 41
3 4
and N −1 = ⎝
† AM,N
1
=
1 4
1
⎞
0 1 0 2+1 2 2−1 2
0 − 12 ∗ − 12 √0 , M = U D1 U = 2 √
√
2−1 2 2+1 2
⎞
1
1 0 0
⎛√
⎠, N− 12 = V D2 V ∗ = ⎝ √ − 12
0 1 0
0 1 0
0 0
0 0 2
and D2 =
2 0
0 1
are diagonal matrices.
√1 2
2+2 4
√ 2−2 4
2−2 4
√ 2+2 4
,
⎞ ⎠,
⎠. By direct computation, we have 0
and A∗N,M =
1
1 0 0
0
1 2
1
1
0
1
1
0
.
†
Then AM,N = A∗N,M . Acknowledgements The authors thank the anonymous referees for their valuable comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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