- Email: [email protected]
A note on partial orders of Hartwig, Mitsch, and Šemrl
Applied Mathematics and Computation 270 (2015) 711–713
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A note on partial orders of Hartwig, Mitsch, and Šemrl Gregor Dolinar a, Bojan Kuzma b,c, Janko Marovt d,∗ a
Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, Ljubljana SI-1000, Slovenia University of Primorska, Glagoljaška 8, Koper SI-6000, Slovenia IMFM, Jadranska 19, Ljubljana SI-1000, Slovenia d Faculty of Economics and Business, University of Maribor, Razlagova 14, Maribor SI-2000, Slovenia b c
a r t i c l e
i n f o
Keywords: Rickart ring Minus partial order Bounded linear operator
a b s t r a c t We show that on Rickart rings the partial orders of Mitsch and Šemrl are equivalent. In particular, these orders are equivalent on B(H), the algebra of all bounded linear operators on a Hilbert space H. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Let S be a semigroup and let E (S) = {e ∈ S : e = e2 } be the set of all idempotent elements in S. The set E(S) is partially ordered in the following way:
e≤ f
e = ef = fe.
if
(1)
There are many examples of orders on semigroups which, whe n restricted from S to E(S), coincide with the partial order (1). For example, on regular semigroups such an order, now known as the minus partial order, was introduced by Hartwig [3]. Recall that an element a in a semigroup S is called regular when there exists an element b ∈ S such that a = aba, and that a regular semigroup is a semigroup in which every element is regular. Hartwig defined the minus partial order in the following way: For a regular semigroup S and a, b ∈ S we write
a <− b if
a a = a b and aa = ba
(2)
for some a ∈ V (a) = {x ∈ S : a = axa, x = xax}. There are many other equivalent definitions of the minus partial order. For example, Nambooripad introduced in a separate paper [9] the partial order for which Mitsch [8] later proved that it is equivalent to Hartwig’s orders. Moreover, Mitsch further generalized the minus partial order to arbitrary semigroups. The definition is as follows [8, Theorem 3]: Suppose a, b are two elements of an arbitrary semigroup S. Then
a ≤M b if S1
a = xb = by and xa = a
(3)
S1
where denotes the set S, if S has the identity, and the set S with the identity adjoined otherwise. Mitsch for some x, y ∈ proved that this is indeed a partial order for any semigroup S and that the following statement is equivalent to the original definition (3): a = xb = by and xa = ay = a for some x, y ∈ S1 (see [8, corollary to Theorem 4]). Mitsch also showed that on regular semigroups definition (3) is equivalent to the following one which is due to Jones (see for example [4]):
a ≤J b if ∗
a = pb = bq
Corresponding author. Tel.: +38622290305. E-mail addresses: [email protected] (G. Dolinar), [email protected] (B. Kuzma), [email protected] (J. Marovt).
http://dx.doi.org/10.1016/j.amc.2015.08.066 0096-3003/© 2015 Elsevier Inc. All rights reserved.
(4)
712
G. Dolinar et al. / Applied Mathematics and Computation 270 (2015) 711–713
for some p, q ∈ E(S1 ). In fact Jones introduced this relation in an arbitrary semigroup. Clearly, even in this setting ≤ J is a reflexive relation, but it is not necessarily a partial order. However if S is a regular semigroup, then the definitions (2), (3), and (4) are equivalent (see [8]). Note also that in the case when S is a group-bound semigroup the orders (3) and (4) are again equivalent (see [5] for definitions and details). The set Mn (F) of all n × n matrices over a field F is a regular semigroup and the minus partial order was extensively studied on Mn (F). The motivation to study the minus partial order on matrices comes for example from applications in statistics and in the study of shorted operators which are closely connected with the electrical network theory (see [7] and references therein). Although B(H), the algebra of all bounded linear operators on a (real or complex) Hilbert space H, is equal to Mn (R) or Mn (C), if H is a finite-dimensional space, in general B(H) is not a regular semigroup. In fact (see for example [10]), an operator A ∈ B(H) has an inner generalized inverse A− , i.e., an element such that A = AA− A, if and only if its image is closed. Without using generalized inner inverses Šemrl provided in [11] another equivalent definition of the minus partial order on Mn (C) and then extended this definition to B(H). Namely, for A, B ∈ B(H) we write A ≤ S B if there exist idempotent operators P, Q ∈ B(H) such that Im P = Im A, Ker Q = Ker A,PA = PB, and AQ = BQ. Here Im A denotes the image of A and Ker A the kernel of A. Šemrl proved in [11] that this is indeed a partial order on B(H). Since Im P = Im A, we have PA = A, hence on B(H) it follows that A ≤S B implies A ≤M B. It is the aim of this paper to show that the other implication is also true. Actually we will prove that Šemrl’s order and Mitsch’s order are equivalent even in a more general setting of Rickart rings. 2. Partial orders in Rickart rings A ring A is called a Rickart ring if for every a ∈ A there exist some idempotent elements p, q ∈ A such that for the right annihilator a◦ = {x ∈ A : ax = 0} of a and for the left annihilator ◦ a = {x ∈ A : xa = 0} of a it holds a◦ = p · A and ◦ a = A · q. Note that every Rickart ring A has the unit element (see [1] or [6]). Let us mention that the class of Rickart rings includes von Neumann algebras and rings with no proper zero divisors (see [1]). In [2] authors found an equivalent definition of Šemrl’s partial order ≤ S on B(H) which allowed them to consider an algebraic version of this order. Namely, the following order, called the minus order on a ring A with the unit 1, was introduced: For a, b ∈ A we write ◦
a ≤S b if
a = A · (1 − p), a◦ = (1 − q) · A, pa = pb, and aq = bq
(5)
for some idempotents p, q ∈ E (A). If A is a Rickart ring, then it was proved in [2] that this is a partial order and that it coincides with Šemrl’s order if A = B(H ). In what follows we will prove that on Rickart rings the partial order ≤S from definition (5) is equivalent to the partial order ≤M from definition (3). First let us write some observations. Suppose p and q are idempotent elements in A. Then any x ∈ A can be represented in the following form:
x = pxq + px(1 − q) + (1 − p)xq + (1 − p)x(1 − q) =
x1,1
x1,2
x2,1
x2,2
. p×q
Here x1,1 = pxq, x1,2 = px(1 − q), x2,1 = (1 − p)xq, x2,2 = (1 − p)x(1 − q). If r ∈ A is an idempotent, x = (xi, j ) p×q and y = (yi, j )q×r , then xy = ( 2k=1 xi,k yk, j ) p×r . Thus, if we have idempotents in A, then the usual algebraic operations in A can be interpreted as operations between appropriate 2 × 2 matrices over A. Let us also introduce two sets of idempotents in a ring A with the unit 1:
LP(a) = p ∈ A : p = p2 , ◦ a = A · (1 − p) , RP(a) = q ∈ A : q = q2 , a◦ = (1 − q) · A . Note that for an idempotent p ∈ A the following holds: A · (1 − p) = °p and (1 − p) · A = p◦ . Lemma 1. Let A be a Rickart ring and a, b ∈ A. Then a ≤S b if and only if there exist p ∈ LP(a) and q ∈ RP(a) with a = pb = bq. Proof. Let a, b ∈ A with a ≤S b. Then pa = pb for some p ∈ LP(a) hence (1 − p) = 1 · (1 − p) ∈ A · (1 − p) = ◦ a, thus a = pa = pb. Similarly, a = aq = bq. The proof of the reverse implication is trivial. Lemma 2. Let A be a Rickart ring and a, b ∈ A. Then a ≤ M b if and only if there exist p ∈ LP(a) and q ∈ RP(a) with a = pb = bq. Proof. Let a ≤M b for some a, b ∈ A. Then there exist x, y ∈ A such that a = xb = by and xa = a = ay. Since A is a Rickart ring, there exist idempotents r, s ∈ A, such that ◦ a = A(1 − r) and a◦ = (1 − s)A, so r ∈ LP(a) and s ∈ RP(a). It follows that
a=
a1
0
0
0
x
, r×s
x
Decompose x = x1 x2 3 4
a = xa =
x1 a1 x3 a1
where a1 = ras = a.
y
y
, y = y1 y2 3 4 r×r
0 0
r×s
b
b
, and b = b1 b2 3 4 s×s
. From r×s
G. Dolinar et al. / Applied Mathematics and Computation 270 (2015) 711–713
713
we have x1 a1 = a1 and x3 a1 = 0. So, x3 ∈ ◦ a1 =◦ a = A · (1 − r) = ◦ r. It follows that x3 r = 0. Note that x3 = (1 − r)xr = (1 − r)xr2 = x3 r = 0. From x1 a1 = a1 we have (1 − x1 ) ∈ ◦ a1 = ◦ a = ◦ r which yields (1 − x1 )r = 0. We conclude that x1 = rxr = rxr2 = x1 r = r. So,
r x2 x= 0 x4
. r×r
The equation a = xb implies
a1
0
0
0
=
b 1 + x2 b 3
b 2 + x2 b 4
x4 b 3
x4 b 4
r×s
, r×s
hence a1 = b1 + x2 b3 and 0 = b2 + x2 b4 . Let
p=
r
x2
0
0
. r×r
Then, by an easy exercise, p ∈ LP(a). Also,
r
x2
0
0
r×r
b1
b2
b3
b4
=
b 1 + x2 b 3
b 2 + x2 b 4
0
0
r×s
= r×s
a1
0
0
0
, r×s
hence pb = a. Similarly, from a = ay we have a1 y1 = a1 and a1 y2 = 0. Observe that a◦1 = a◦ = s◦ , so (1 − y1 ) ∈ s◦ and y2 ∈ s°. It follows that s(1 − y1 ) = 0 and sy2 = 0. So, y1 = sys = s2 ys = sy1 = s and y2 = sy(1 − s) = s2 y(1 − s) = sy2 = 0. The equation a = by implies b1 + b2 y3 = a1 and b3 + b4 y3 = 0. Let
q=
s
0
y3
0
. s×s
Analogously, q ∈ RP(a). Also,
b1
b2
b3
b4
r×s
s
0
y3
0
= s×s
b 1 + b 2 y3
0
b 3 + b 4 y3
0
= r×s
a1
0
0
0
, r×s
and thus bq = a. The reverse implication follows easily since a = pb implies pa = p2 b = pb = a for a, b ∈ A and any idempotent p ∈ A. As a direct corollary of Lemmas 1 and 2 we may conclude that on Rickart rings orders defined by Mitsch and Šemrl are the same. Theorem 3. Let A be a Rickart ring and a, b ∈ A. Then
a ≤S b if and only if a ≤M b. Since B(H) is a Rickart ring (see for example [1]), we also have the following. Corollary 4. For every A, B ∈ B(H),
A ≤S B if and only if A ≤M B. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
S.K. Berberian, Baer ∗ -rings, Springer-Verlag, New York, 1972. ´ D.S. Rakic, ´ J. Marovt, Minus partial order in Rickart rings, Publicationes Mathematicae Debrecen, 2015. (in press). D.S. Djordjevic, R.E. Hartwig, How to partially order regular elements, Math. Japon. 25 (1980) 1–13. P.M. Higgins, Techniques of the Semigroup Theory, Oxford University Press, Oxford, 1992. P.M. Higgins, The Mitsch order in a semigroup, Semigroup Forum 49 (1994) 261–266. I. Kaplansky, Rings of Operators, Benjamin, New York, 1968. S.K. Mitra, P. Bhimasankaram, S.B. Malik, Matrix Partial Orders, Shorted Operators and Applications, Word Scientific, London, 2010. H. Mitsch, A natural partial order for semigroups, Proc. Am. Math. Soc. 97 (No. 3) (1986) 384–388. K.S.S. Nambooripad, The natural partial order on a regular semigroup, Proc. Edinb. Math. Soc. 23 (1980) 249–260. M.Z. Nashed (Ed.), Generalized Inverses and Applications, Academic Press, New York-London, 1976. P. Šemrl, Automorphisms of B(H) with respect to minus partial order, J. Math. Anal. Appl. 369 (2010) 205–213.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A note on partial orders of Hartwig, Mitsch, and Šemrl Gregor Dolinar a, Bojan Kuzma b,c, Janko Marovt d,∗ a
Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, Ljubljana SI-1000, Slovenia University of Primorska, Glagoljaška 8, Koper SI-6000, Slovenia IMFM, Jadranska 19, Ljubljana SI-1000, Slovenia d Faculty of Economics and Business, University of Maribor, Razlagova 14, Maribor SI-2000, Slovenia b c
a r t i c l e
i n f o
Keywords: Rickart ring Minus partial order Bounded linear operator
a b s t r a c t We show that on Rickart rings the partial orders of Mitsch and Šemrl are equivalent. In particular, these orders are equivalent on B(H), the algebra of all bounded linear operators on a Hilbert space H. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Let S be a semigroup and let E (S) = {e ∈ S : e = e2 } be the set of all idempotent elements in S. The set E(S) is partially ordered in the following way:
e≤ f
e = ef = fe.
if
(1)
There are many examples of orders on semigroups which, whe n restricted from S to E(S), coincide with the partial order (1). For example, on regular semigroups such an order, now known as the minus partial order, was introduced by Hartwig [3]. Recall that an element a in a semigroup S is called regular when there exists an element b ∈ S such that a = aba, and that a regular semigroup is a semigroup in which every element is regular. Hartwig defined the minus partial order in the following way: For a regular semigroup S and a, b ∈ S we write
a <− b if
a a = a b and aa = ba
(2)
for some a ∈ V (a) = {x ∈ S : a = axa, x = xax}. There are many other equivalent definitions of the minus partial order. For example, Nambooripad introduced in a separate paper [9] the partial order for which Mitsch [8] later proved that it is equivalent to Hartwig’s orders. Moreover, Mitsch further generalized the minus partial order to arbitrary semigroups. The definition is as follows [8, Theorem 3]: Suppose a, b are two elements of an arbitrary semigroup S. Then
a ≤M b if S1
a = xb = by and xa = a
(3)
S1
where denotes the set S, if S has the identity, and the set S with the identity adjoined otherwise. Mitsch for some x, y ∈ proved that this is indeed a partial order for any semigroup S and that the following statement is equivalent to the original definition (3): a = xb = by and xa = ay = a for some x, y ∈ S1 (see [8, corollary to Theorem 4]). Mitsch also showed that on regular semigroups definition (3) is equivalent to the following one which is due to Jones (see for example [4]):
a ≤J b if ∗
a = pb = bq
Corresponding author. Tel.: +38622290305. E-mail addresses: [email protected] (G. Dolinar), [email protected] (B. Kuzma), [email protected] (J. Marovt).
http://dx.doi.org/10.1016/j.amc.2015.08.066 0096-3003/© 2015 Elsevier Inc. All rights reserved.
(4)
712
G. Dolinar et al. / Applied Mathematics and Computation 270 (2015) 711–713
for some p, q ∈ E(S1 ). In fact Jones introduced this relation in an arbitrary semigroup. Clearly, even in this setting ≤ J is a reflexive relation, but it is not necessarily a partial order. However if S is a regular semigroup, then the definitions (2), (3), and (4) are equivalent (see [8]). Note also that in the case when S is a group-bound semigroup the orders (3) and (4) are again equivalent (see [5] for definitions and details). The set Mn (F) of all n × n matrices over a field F is a regular semigroup and the minus partial order was extensively studied on Mn (F). The motivation to study the minus partial order on matrices comes for example from applications in statistics and in the study of shorted operators which are closely connected with the electrical network theory (see [7] and references therein). Although B(H), the algebra of all bounded linear operators on a (real or complex) Hilbert space H, is equal to Mn (R) or Mn (C), if H is a finite-dimensional space, in general B(H) is not a regular semigroup. In fact (see for example [10]), an operator A ∈ B(H) has an inner generalized inverse A− , i.e., an element such that A = AA− A, if and only if its image is closed. Without using generalized inner inverses Šemrl provided in [11] another equivalent definition of the minus partial order on Mn (C) and then extended this definition to B(H). Namely, for A, B ∈ B(H) we write A ≤ S B if there exist idempotent operators P, Q ∈ B(H) such that Im P = Im A, Ker Q = Ker A,PA = PB, and AQ = BQ. Here Im A denotes the image of A and Ker A the kernel of A. Šemrl proved in [11] that this is indeed a partial order on B(H). Since Im P = Im A, we have PA = A, hence on B(H) it follows that A ≤S B implies A ≤M B. It is the aim of this paper to show that the other implication is also true. Actually we will prove that Šemrl’s order and Mitsch’s order are equivalent even in a more general setting of Rickart rings. 2. Partial orders in Rickart rings A ring A is called a Rickart ring if for every a ∈ A there exist some idempotent elements p, q ∈ A such that for the right annihilator a◦ = {x ∈ A : ax = 0} of a and for the left annihilator ◦ a = {x ∈ A : xa = 0} of a it holds a◦ = p · A and ◦ a = A · q. Note that every Rickart ring A has the unit element (see [1] or [6]). Let us mention that the class of Rickart rings includes von Neumann algebras and rings with no proper zero divisors (see [1]). In [2] authors found an equivalent definition of Šemrl’s partial order ≤ S on B(H) which allowed them to consider an algebraic version of this order. Namely, the following order, called the minus order on a ring A with the unit 1, was introduced: For a, b ∈ A we write ◦
a ≤S b if
a = A · (1 − p), a◦ = (1 − q) · A, pa = pb, and aq = bq
(5)
for some idempotents p, q ∈ E (A). If A is a Rickart ring, then it was proved in [2] that this is a partial order and that it coincides with Šemrl’s order if A = B(H ). In what follows we will prove that on Rickart rings the partial order ≤S from definition (5) is equivalent to the partial order ≤M from definition (3). First let us write some observations. Suppose p and q are idempotent elements in A. Then any x ∈ A can be represented in the following form:
x = pxq + px(1 − q) + (1 − p)xq + (1 − p)x(1 − q) =
x1,1
x1,2
x2,1
x2,2
. p×q
Here x1,1 = pxq, x1,2 = px(1 − q), x2,1 = (1 − p)xq, x2,2 = (1 − p)x(1 − q). If r ∈ A is an idempotent, x = (xi, j ) p×q and y = (yi, j )q×r , then xy = ( 2k=1 xi,k yk, j ) p×r . Thus, if we have idempotents in A, then the usual algebraic operations in A can be interpreted as operations between appropriate 2 × 2 matrices over A. Let us also introduce two sets of idempotents in a ring A with the unit 1:
LP(a) = p ∈ A : p = p2 , ◦ a = A · (1 − p) , RP(a) = q ∈ A : q = q2 , a◦ = (1 − q) · A . Note that for an idempotent p ∈ A the following holds: A · (1 − p) = °p and (1 − p) · A = p◦ . Lemma 1. Let A be a Rickart ring and a, b ∈ A. Then a ≤S b if and only if there exist p ∈ LP(a) and q ∈ RP(a) with a = pb = bq. Proof. Let a, b ∈ A with a ≤S b. Then pa = pb for some p ∈ LP(a) hence (1 − p) = 1 · (1 − p) ∈ A · (1 − p) = ◦ a, thus a = pa = pb. Similarly, a = aq = bq. The proof of the reverse implication is trivial. Lemma 2. Let A be a Rickart ring and a, b ∈ A. Then a ≤ M b if and only if there exist p ∈ LP(a) and q ∈ RP(a) with a = pb = bq. Proof. Let a ≤M b for some a, b ∈ A. Then there exist x, y ∈ A such that a = xb = by and xa = a = ay. Since A is a Rickart ring, there exist idempotents r, s ∈ A, such that ◦ a = A(1 − r) and a◦ = (1 − s)A, so r ∈ LP(a) and s ∈ RP(a). It follows that
a=
a1
0
0
0
x
, r×s
x
Decompose x = x1 x2 3 4
a = xa =
x1 a1 x3 a1
where a1 = ras = a.
y
y
, y = y1 y2 3 4 r×r
0 0
r×s
b
b
, and b = b1 b2 3 4 s×s
. From r×s
G. Dolinar et al. / Applied Mathematics and Computation 270 (2015) 711–713
713
we have x1 a1 = a1 and x3 a1 = 0. So, x3 ∈ ◦ a1 =◦ a = A · (1 − r) = ◦ r. It follows that x3 r = 0. Note that x3 = (1 − r)xr = (1 − r)xr2 = x3 r = 0. From x1 a1 = a1 we have (1 − x1 ) ∈ ◦ a1 = ◦ a = ◦ r which yields (1 − x1 )r = 0. We conclude that x1 = rxr = rxr2 = x1 r = r. So,
r x2 x= 0 x4
. r×r
The equation a = xb implies
a1
0
0
0
=
b 1 + x2 b 3
b 2 + x2 b 4
x4 b 3
x4 b 4
r×s
, r×s
hence a1 = b1 + x2 b3 and 0 = b2 + x2 b4 . Let
p=
r
x2
0
0
. r×r
Then, by an easy exercise, p ∈ LP(a). Also,
r
x2
0
0
r×r
b1
b2
b3
b4
=
b 1 + x2 b 3
b 2 + x2 b 4
0
0
r×s
= r×s
a1
0
0
0
, r×s
hence pb = a. Similarly, from a = ay we have a1 y1 = a1 and a1 y2 = 0. Observe that a◦1 = a◦ = s◦ , so (1 − y1 ) ∈ s◦ and y2 ∈ s°. It follows that s(1 − y1 ) = 0 and sy2 = 0. So, y1 = sys = s2 ys = sy1 = s and y2 = sy(1 − s) = s2 y(1 − s) = sy2 = 0. The equation a = by implies b1 + b2 y3 = a1 and b3 + b4 y3 = 0. Let
q=
s
0
y3
0
. s×s
Analogously, q ∈ RP(a). Also,
b1
b2
b3
b4
r×s
s
0
y3
0
= s×s
b 1 + b 2 y3
0
b 3 + b 4 y3
0
= r×s
a1
0
0
0
, r×s
and thus bq = a. The reverse implication follows easily since a = pb implies pa = p2 b = pb = a for a, b ∈ A and any idempotent p ∈ A. As a direct corollary of Lemmas 1 and 2 we may conclude that on Rickart rings orders defined by Mitsch and Šemrl are the same. Theorem 3. Let A be a Rickart ring and a, b ∈ A. Then
a ≤S b if and only if a ≤M b. Since B(H) is a Rickart ring (see for example [1]), we also have the following. Corollary 4. For every A, B ∈ B(H),
A ≤S B if and only if A ≤M B. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
S.K. Berberian, Baer ∗ -rings, Springer-Verlag, New York, 1972. ´ D.S. Rakic, ´ J. Marovt, Minus partial order in Rickart rings, Publicationes Mathematicae Debrecen, 2015. (in press). D.S. Djordjevic, R.E. Hartwig, How to partially order regular elements, Math. Japon. 25 (1980) 1–13. P.M. Higgins, Techniques of the Semigroup Theory, Oxford University Press, Oxford, 1992. P.M. Higgins, The Mitsch order in a semigroup, Semigroup Forum 49 (1994) 261–266. I. Kaplansky, Rings of Operators, Benjamin, New York, 1968. S.K. Mitra, P. Bhimasankaram, S.B. Malik, Matrix Partial Orders, Shorted Operators and Applications, Word Scientific, London, 2010. H. Mitsch, A natural partial order for semigroups, Proc. Am. Math. Soc. 97 (No. 3) (1986) 384–388. K.S.S. Nambooripad, The natural partial order on a regular semigroup, Proc. Edinb. Math. Soc. 23 (1980) 249–260. M.Z. Nashed (Ed.), Generalized Inverses and Applications, Academic Press, New York-London, 1976. P. Šemrl, Automorphisms of B(H) with respect to minus partial order, J. Math. Anal. Appl. 369 (2010) 205–213.