Some properties on the star order of bounded operators

J. Math. Anal. Appl. 423 (2015) 32–40

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Some properties on the star order of bounded operators Chunyuan Deng 1 School of Mathematics Science, South China Normal University, Guangzhou 510631, PR China

a r t i c l e

i n f o

Article history: Received 27 January 2014 Available online 2 October 2014 Submitted by K. Jarosz Keywords: Logic order ∗-Order Operator matrix

a b s t r a c t In this paper, we give various characterizations of star order and present some new relationships between the logic order and the star order. We present some necessary and sufficient conditions for which the infimum or upper bound exists with respect to given orders. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Let H and K be separable complex Hilbert spaces. We denote the set of all bounded linear operators from H into K by B(H, K) and by B(H) when H = K. For an operator A, we shall denote by N (A) and R(A) the null space and the range of A, respectively. An operator A ∈ B(H) is said to be injective if N (A) = {0}. R(A) is the closure of R(A). An operator A ∈ B(H) is said to be positive if (Ax, x) ≥ 0 for all x ∈ H. For self-adjoint operators A and B, A ≤ B means (Ax, x) ≤ (Bx, x) for all x ∈ H. Let S(H) be the set of all linear bounded self-adjoint operators on H, S + (H) be the set of all positive operators in S(H), and P (H) be the set of all orthogonal projections on H. PM is the orthogonal projection on closed subspace M ⊆ H. Each element in P (H) is said to be a quantum event, and each element in S(H) is said to be a bounded quantum observable corresponding to a quantum system. For A, B ∈ S(H), if there exists a C ∈ S + (H) such that A + C = B, then we say A ≤ B. The order ≤ is a partial order on S(H). The physical meaning of A ≤ B is that the expectation of A is not greater than the expectation of B for each state of the system. The order ≤ is said to be a numerical order of S(H). If there exists a C ∈ S(H) such that AC = 0 and A + C = B, then we write A  B. The order  is said to be a logic order of S(H) (see [4–8,11]). Let A, B ∈ B(H). We say that A is lower than or equal to B with ∗

respect to the ∗-order [3], which is denoted by A ≤ B, if A∗ A = A∗ B = B ∗ A,

1

AA∗ = BA∗ = AB ∗ .

E-mail addresses: [email protected], [email protected]. Supported by the National Natural Science Foundation of China under grant 11171222.

http://dx.doi.org/10.1016/j.jmaa.2014.09.077 0022-247X/© 2014 Elsevier Inc. All rights reserved.

C.Y. Deng / J. Math. Anal. Appl. 423 (2015) 32–40

33

The ∗-order is a natural generalization of the usual partial ordering of the projection elements in B(H) [2,5]. For a given order, the infimum (supremum) problem of bounded quantum observables is to find under what condition the infimum (supremum) exists for the given order and the detailed structure and properties when this infimum (supremum) exists. In this paper, we present a new necessary and sufficient condition for which the infimum or supremum exists with respect to the given order in a totally different form. We give various characterizations of ∗-order and present some new relationships between the logic order and the star order. Furthermore, the much simpler matrix representations with respect to ∗-order relations are given. 2. Main results First, we present some necessary and sufficient conditions for which the logic order or ∗-order holds. For an operator A ∈ B(H, K), if N (A) = {0} and R(A) = K, we say A is injective dense defined. Theorem 2.1. Let M, N ∈ B(H1 ⊕ K1 , H2 ⊕ K2 ) have the 2 × 2 operator matrix forms:  M=

M1 M3

M2 M4

∗



 ,

N=

N1 0

0 0

 .

∗

(i) If M ≤ N , then Mi = 0, i = 2, 3, 4 and M1 ≤ N1 ; ∗

(ii) If N ≤ M and N1 ∈ B(H1 , H2 ) is injective dense defined, then Mi = 0, i = 2, 3 and N1 = M1 . Proof. The proof follows straight from the definition of ∗-order and injective density of N1 . 2 Let A ∈ B(H). Then A, as an operator from R(A∗ ) ⊕ N (A) into R(A) ⊕ N (A∗ ), has the 2 × 2 diagonal matrix form A = A1 ⊕ 0, where A1 ∈ B(R(A∗ ), R(A)) is injective dense defined. By Theorem 2.1(ii) we have the following result. Theorem 2.2. Let A, B ∈ B(H). Then ∗

A≤B

if and only if

A = A1 ⊕ 0,

B = A1 ⊕ B1 ,

(1)

where A1 ∈ B(R(A∗ ), R(A)) is injective dense defined and B1 ∈ B(N (A), N (A∗ )). As for operators A, B ∈ B(H), a similar result can be obtained by Theorems 2.1 and 2.2. See also [1] and [13, Lemma 2.1] for more details. Corollary 2.1. Let A, B ∈ B(H). ∗

(i) If A ≤ B, then R(A) ⊆ R(B) and R(A∗ ) ⊆ R(B ∗ ); ∗

(ii) If A ≤ B and B is idempotent, then A is idempotent and AB = BA = A; ∗

(iii) A ≤ B if and only if there exists V ∈ B(H) such that B = A + PN (A∗ ) V PN (A) . Proof. The range relations in (i) follow by Theorem 2.2. If B 2 = B, then A and B, as operators on  1 A2    B(R(B) ⊕ N (B)), have matrix representations A = A and B = I0 00 , respectively. The result (ii) A3 A4 follows by Theorem 2.1(i). Analogously we have (iii) by Theorem 2.2. 2

C.Y. Deng / J. Math. Anal. Appl. 423 (2015) 32–40

34

∗

∗

From (1), we observe that, if A is invertible, then A ≤ B ⇐⇒ A = B. If A ≤ B, then BX = BY (resp., XB = Y B) implies that AX = AY (resp., XA = Y A) for arbitrary X, Y ∈ B(H). If A, B ∈ S(H), then ∗-order reduces as logic order. Corollary 2.2. (See [12, Lemma 2.2] or [4].) Let A and B be in S(H). If A  B, then R(A) ⊆ R(B). AB

⇐⇒

A = A1 ⊕ 0,

B = A1 ⊕ B1 ,

⇐⇒

R(A) reduces B

⇐⇒

AB = A2 = BA,

where A, B act on H = R(A) ⊕ N (A) and A1 is injective dense defined. ∗

If A, B ∈ S(H), A ≤ B ⇐⇒ A  B; If B ∈ S(H)+ , A  B implies A ≤ B; If A, B ∈ P (H), S. Gudder in [4] had proved A ≤ B if and only if A  B, and A, B have the same infimum and the supremum with ∗

respect to the orders  and ≤. We denote by A ∧ B, the infimum, equivalently, the greatest lower bound, of ∗ A and B over the ∗-order, if it exists. To be more precise, A ∧ B is an operator in B(H) uniquely determined ∗

∗

∗

∗

∗

by the following properties: A ∧ B ≤ A, A ∧ B ≤ B, and an operator D ∈ B(H) satisfies both D ≤ A and ∗

∗

∗

∗

D ≤ B, if and only if D ≤ A ∧ B. A ∨ B denotes the least upper bound of A and B over the ∗-order. In the same way A B denotes the infimum of A and B over the  order and A ∧ B denotes the infimum of A and B over the ≤ order. ∗

Corollary 2.3. If A, B ∈ P (H), A ∧ B = A B = A ∧ B = PR(A)∩R(B) . If A, B ∈ S(H) and C ∈ S + (H) ∗

∗

∗

such that A ≤ C and B ≤ C, then A ≤ B ⇐⇒ A  B ⇐⇒ A ≤ B. In general, ∗

A  B =⇒ A ≤ B ⇓ ↓ ∗

A2  B 2 =⇒ A2 ≤ B 2 , where =⇒ denotes the usual implication and → denotes the implication which is valid under the commutativity condition AB = BA. ∗

∗

∗

It is trivial that A ≤ B ⇐⇒ A∗ ≤ B ∗ . Note that A ≤ B and B = B ∗ don’t imply that A is self-adjoint ∗ ∗    1  (for example A = 10 10 and B = 11 −1 ). Neither of the relations A ≤ B and A2 ≤ B 2 implies the other. For ∗ ∗    1     0  example, A = 10 10 , B = 12 −2 , then A ≤ B does not imply A2 ≤ B 2 . Whereas A = 00 10 , B = 10 −2 ∗

∗

illustrate that A2 ≤ B 2 and A2  B 2 don’t imply A ≤ B and A  B (see [1]). If A and B are normal and ∗

A ≤ B, then A∗ B = A∗ A = AA∗ = BA∗ . It follows that AB = BA by Fuglede–Putnam Theorem [10] and ∗

∗

A2 ≤ B 2 by definition of ∗-order. In general, neither of the relations A ≤ B and AB = BA implies the ∗

∗

other for the ∗-order. In [9, Theorem 3.2], the authors had proved that A2 ≤ B 2 ⇐⇒ A ≤ B if A and B are ∗

normal and R(A) ⊆ R(B). By (1), if A ≤ B with R(A) = R(A∗ ) and R(B) = R(B ∗ ), then c1 An + c2 B n is invertible if and only if B is invertible, where c1 , c2 ∈ C and c1 +c2 = 0. Next, we say A and B have common ∗

∗

∗-upper bound if there exists an operator C ∈ B(H) such that A ≤ C and B ≤ C. By Corollary 2.1(iii) we have the following special case. ∗

Corollary 2.4. If N (A∗ ) = {0} or N (A) = {0}, then A, B have common ∗-upper bound if and only if B ≤ A. ∗

In this case A ∨ B = A.

C.Y. Deng / J. Math. Anal. Appl. 423 (2015) 32–40

35

In general, we get the following theorem. Theorem 2.3. Let A, B ∈ B(H), B1 = PR(A) BPR(A∗ ) and B2 = PN (A∗ ) BPN (A) . ∗

(i) If B1 = 0, B2 = 0 and A, B have common ∗-upper bound, then B = 0 and A ∨ B = A; ∗

(ii) If B1 = 0 and A, B have common ∗-upper bound, then A ∨ B = A + B; ∗

∗

(iii) If B2 = 0 and A, B have common ∗-upper bound, then B ≤ A and A ∨ B = A; (iv) In general, A, B have common ∗-upper bound C if and only if A, B and C, as operators from          R B1∗ ⊕ N (B1 )  N (A) ⊕ R B2∗ ⊕ N (B2 )  R A∗ into          R(B1 ) ⊕ N B1∗  N A∗ ⊕ R(B2 ) ⊕ N B2∗  R(A) , can be written as ⎛

A11 ⎜ 0 A=⎜ ⎝ 0 0

0 A22 0 0

0 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

⎛

0 0 0 0

B11 ⎜ 0 B=⎜ ⎝ B31 0

B13 0 B33 0

⎞ 0 0⎟ ⎟, 0⎠ 0

⎛

A11 ⎜ 0 C=⎜ ⎝ 0 0

0 A22 0 0

0 0 C33 0

⎞ 0 0 ⎟ ⎟, 0 ⎠ C44

(2)

where diagonal elements A11 , B11 , A22 and B33 are injective dense defined and ∗ ∗ (A11 − B11 )B11 = B13 B13 ,

∗ ∗ B11 (A11 − B11 ) = B31 B31 ,

∗ ∗ (C33 − B33 )B33 = B31 B31 ,

∗ ∗ B33 (C33 − B33 ) = B13 B13 ,

∗ ∗ ∗ ∗ = B11 B31 + B13 B33 = B13 C33 , A11 B31

∗ ∗ ∗ B31 C33 = B11 B13 + B31 B33 = A∗11 B13 .

(3)

Proof. Suppose that A and B have common ∗-upper bound C ∈ B(H). By Theorem 2.2, A and C, as operators from R(A∗ ) ⊕ N (A) into R(A) ⊕ N (A∗ ), have matrix forms as A = A011 ⊕ 0,

0 C = A011 ⊕ C22 ,

where A011 is injective dense defined. Let B have the corresponding matrix form with  B=

0 B11 0 B21

0 B12 0 B22



∗

≤



A011 0

0 0 C22



 .

Then B1 =

0 B11 0

0 0



 , B2 =

0 0

0 0 B22

 .

By the definition of ∗-order, 

0 A011 − B11 0 −B21

0 −B12 0 0 C22 − B22



0∗ B11 0∗ B12

0∗ B21 0∗ B22



 =

0 0∗ 0 0∗ (A011 − B11 )B11 − B12 B12 0 0 0∗ 0 0∗ (C22 − B22 )B12 − B21 B11

0 0∗ 0 0∗ (A011 − B11 )B21 − B12 B22 0 0 0∗ 0 0∗ (C22 − B22 )B22 − B21 B21

0∗ 0 0∗ 0 B11 (A011 − B11 ) − B21 B21 0∗ 0 0 0∗ 0 B12 (A11 − B11 ) − B22 B21

0∗ 0 0 0∗ 0 B21 (C22 − B22 ) − B11 B12 0∗ 0 0 0∗ 0 B22 (C22 − B22 ) − B12 B12

 =0

and 

0∗ B11 0∗ B12

0∗ B21 0∗ B22



0 A011 − B11 0 −B21

0 −B12 0 0 C22 − B22



 =

Comparing the two sides of the above equations, we have

 = 0.

C.Y. Deng / J. Math. Anal. Appl. 423 (2015) 32–40

36

⎧ (a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) ⎪ ⎪ ⎪ ⎪ ⎨ (c)

0∗ 0 0∗ 0 0∗ = B11 B11 + B12 B12 , A011 B11 0∗ 0 0∗ 0 0∗ 0 B11 A11 = B11 B11 + B21 B21 , 0 0∗ 0 0∗ 0 0∗ C22 B22 = B21 B21 + B22 B22 ,

(4)

0∗ 0 0∗ 0 0∗ 0 ⎪ (d) B22 C22 = B22 B22 + B12 B12 , ⎪ ⎪ ⎪  0 0∗ ∗ ⎪ ⎪ 0 0∗ 0 0∗ 0 0∗ ⎪ (e) A11 B21 = B11 B21 + B12 B22 = C22 B12 , ⎪ ⎪ ⎪   ⎩ 0∗ 0 0∗ 0 0∗ 0 0∗ 0 ∗ (f) B21 C22 = B21 B22 + B11 B12 = B12 A11 .

0 0 0 Case (i). If B1 = 0 and B2 = 0, then B11 = 0 and B22 = 0. By (a) and (c) in (4) we get B12 = 0 and ∗

∗

0 B21 = 0, i.e. B = 0. Hence A ∨ B = A ∨ 0 = A. 0 0 0 Case (ii). If B1 = 0, then B11 = 0. By (a) and (b) in (4) we get B12 = 0, B21 = 0. By (c) and (d) in (4) ∗

∗

0 0∗ 0∗ 0∗ 0 0 we get B22 (C22 − B22 ) = 0 and B22 (C22 − B22 ) = 0, respectively, i.e. B22 ≤ C22 . Hence A ∨ B = A + B. 0 0 0 = 0, B21 = 0. By (a) and (b) in (4) Case (iii). If B2 = 0, then B22 = 0. By (c) and (d) in (4) we get B12 ∗

∗

0 0∗ 0∗ 0 0 (A0∗ we get B11 11 − B11 ) = 0 and B11 (A11 − B11 ) = 0, respectively, i.e. B11 ≤ A11 . It follows that B ≤ A and ∗

A ∨ B = A. Case (iv). The implication (⇐) is clear. We consider the converse. Suppose B1 = 0 and B2 = 0. Note that    0∗  = R B1∗ , R B11

 0 R B11 = R(B1 ),

 0 = N (B1 )  N (A), N B11

     0∗  = N B1∗  N A∗ N B11

 0 R B22 = R(B2 ),

 0   N B22 = N (B2 )  R A∗ ,

 0∗    N B22 = N B2∗  R(A).

and  0∗    R B22 = R B2∗ , Then  0 B11

=

B11 0

0 0

B33 0

0 0

  :

R(B1∗ ) N (B1 )  N (A)



 →

R(B1 ) ∗ N (B1 )  N (A∗ )



and  0 B22 =

  :



R(B2∗ ) N (B2 )  R(A∗ )

 →



R(B2 ) N (B2∗ )  R(A)

,

where B11 , B33 are injective dense defined. B, C can be written correspondingly as ⎛

B11 ⎜ 0 B=⎜ ⎝ B31 B41

0 0 B32 B42

B13 B23 B33 0

⎞ B14 B24 ⎟ ⎟, 0 ⎠ 0

⎛

A11 ⎜ A21 C=⎜ ⎝ 0 0

A12 A22 0 0

0 0 C33 C43

⎞ 0 0 ⎟ ⎟ C34 ⎠ C44

0∗ 0∗ 0 By (a) and (b) in (4), we know A011 B11 and B11 A11 are selfadjoint, which implies that A12 = 0 and A21 = 0 since B11 is injective dense defined. Similarly, (c) and (d) imply that C34 = 0 and C43 = 0. Now, we get

C = A11 ⊕ A22 ⊕ C33 ⊕ C44 , where diagonal elements A11 and A22 are injective dense defined. From (a) we get 

∗ A11 B11 0

0 0



 =

∗ B11 B11 0

0 0



 +

∗ ∗ + B14 B14 B13 B13 ∗ ∗ B23 B13 + B24 B14

∗ ∗ B13 B23 + B14 B24 ∗ ∗ B23 B23 + B24 B24

 .

C.Y. Deng / J. Math. Anal. Appl. 423 (2015) 32–40

37

∗ ∗ We get B23 B23 + B24 B24 = 0, i.e. B23 = 0, B24 = 0. Hence ∗ ∗ ∗ ∗ A11 B11 = B11 B11 + B13 B13 + B14 B14 .

In a similar way, ∗ ∗ ∗ ∗ (b) implies that B32 = 0, B42 = 0 and B11 A11 = B11 B11 + B31 B31 + B41 B41 ; ∗ ∗ ∗ (c) implies that B41 = 0 and C33 B33 = B31 B31 + B33 B33 ; ∗ ∗ ∗ C33 = B13 B13 + B33 B33 ; (d) implies that B14 = 0 and B33 ∗ ∗ ∗ ∗ (e) implies that A11 B31 = B11 B31 + B13 B33 = B13 C33 ; ∗ ∗ ∗ (f) implies that B31 C33 = B11 B13 + B31 B33 = A∗11 B13 .

Hence (2) and (3) hold. 2 In Theorem 2.3, the cases (i)–(iii) can be treated as some particular cases of (iv). By (2) we know C = A11 ⊕ A22 ⊕ C33 ⊕ C44 if C is one common ∗-upper bound of A, B, where C44 is arbitrary. If C33   exists, then C33 is unique. In fact, suppose C  = A11 ⊕ A22 ⊕ C33 ⊕ C44 is another common ∗-upper bound ∗ ∗ ∗  ∗ of A and B. Then C33 B33 = B33 B33 + B31 B31 = C33 B33 by the fourth relation in (3). Since B33 is injective  dense defined, we get C33 = C33 . ∗

Corollary 2.5. A ∨ B exists if and only if A and B have common ∗-upper bound C. If A and B are written as (2), then ∗

A ∨ B = A11 ⊕ A22 ⊕ C33 ⊕ 0, where C33 is unique defined by the relations in (3). By (2) and (3), we have following further result. Theorem 2.4. If A and B have common ∗-upper bound, then PR(A) (A − B)B ∗ = 0,

B ∗ (A − B)PR(A∗ ) = 0.

Proof. As an example, we only prove PR(A) (A − B)B ∗ = 0. Suppose C is one common ∗-upper bound of A and B. By the proof of Theorem 2.3, the results hold when A, B, C belong to the cases (i)–(iii). Without loss of generality, let A, B, C have the matrix forms (2). We have ⎞⎛ ⎞⎛ ∗ ∗ A11 − B11 B11 0 B31 I 0 0 0 0 −B13 0 ⎜ ⎜0 I 0 0⎟⎜ 0 0⎟ 0 A22 0 0 ⎟⎜ ⎟⎜ 0 = ⎜ ∗ ∗ ⎝ ⎝ 0 0 0 0 ⎠ ⎝ −B31 ⎠ 0 −B33 0 B13 0 B33 0 0 0 0 0 0 0 0 0 0 0 ⎛ ⎞ ∗ ∗ ∗ ∗ (A11 − B11 )B11 − B13 B13 0 (A11 − B11 )B31 − B13 B33 0 ⎜ 0 0 0 0⎟ ⎟ = ⎜ ⎝ 0 0 0 0⎠ 0 0 0 0 ⎛

PR(A) (A − B)B ∗

by (3)

= 0.

2

Theorem 2.5. If A and B have common ∗-upper bound, then ∗

A≤B

⇐⇒

AA∗  BB ∗

and

A∗ A  B ∗ B.

⎞ 0 0⎟ ⎟ 0⎠ 0

C.Y. Deng / J. Math. Anal. Appl. 423 (2015) 32–40

38

Proof. Suppose C is one common ∗-upper bound of A and B. By (2), ∗

A≤B

⇐⇒ by (2)

⇐⇒

A∗ (B − A) = 0 and (B − A)A∗ = 0 ⎛ ∗ ⎞ A11 (B11 − A11 ) 0 A∗11 B13 0 ⎜ 0 −A∗22 A22 0 0⎟ ⎜ ⎟=0 ⎝ 0 0 0 0⎠ 0 0 0 0 ⎛ ⎞ ∗ (B11 − A11 )A11 0 0 0 ∗ ⎜ ⎟ A 0 −A 22 22 0 0 ⎟ ⎜ =0 ⎝ 0 0 0⎠ B31 A∗

and

11

 ⇐⇒ by (3)

⇐⇒



0

0

A∗11 A11 B31 A∗11

=

A∗11 A11 ∗ B11 B31

∗ = B11 B11 ∗ + B13 B33

=

0

A∗11 B11 , A11 A∗11 0, A∗11 B13 = 0, + =

0

= B11 A∗11 ; A22 = 0

∗ ∗ B31 B31 , A11 A∗11 = B11 B11 ∗ ∗ 0, B11 B13 + B31 B33 = 0,

∗ + B13 B13 ;

A22 = 0.

(5)

Note that AA∗ = A11 A∗11 ⊕ A22 A∗22 ⊕ 0 ⊕ 0, ⎛

∗ ∗ ∗ ∗ B11 B11 + B13 B13 0 B11 B31 + B13 B33 ⎜ 0 0 0 BB ∗ = ⎜ ⎝ B31 B ∗ + B33 B ∗ 0 B31 B ∗ + B33 B ∗ 11 13 31 33 0 0 0  ⎛ ⎞ ∗ ∗ B11 B11 + B13 B13 A11 A∗11 0 0 0 ⎜ 0 0 0 0⎟ ⎟   BB ∗ AA∗ = ⎜ ∗ ∗ ∗ ⎝ B31 B + B33 B A11 A 0 0 0⎠ 11 13 11 0 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

and   ∗ ∗ A11 A∗11 B11 B11 + B13 B13 ⎜ 0 AA∗ BB ∗ = ⎜ ⎝ 0 0 ⎛

  ∗ ∗ 0 A11 A∗11 B11 B31 + B13 B33 0 0 0 0 0 0

⎞ 0 0⎟ ⎟. 0⎠ 0

Sufficiency. If AA∗  BB ∗ , then AA∗ BB ∗ = (AA∗ )2 = BB ∗ AA∗ by Corollary 2.2, which implies ∗ ∗ A11 A∗11 = B11 B11 + B13 B13 ,

∗ ∗ B11 B31 + B13 B33 = 0,

A22 = 0.

(6)

Analogously, A∗ A  B ∗ B implies that ∗ ∗ A∗11 A11 = B11 B11 + B31 B31 ,

∗ ∗ B11 B13 + B31 B33 = 0.

∗

By equivalent relations in (5), we get A ≤ B. ∗

Necessity. If A ≤ B, by (3) and (5), we get AA∗ = A11 A∗11 ⊕ 0 ⊕ 0 ⊕ 0, A∗ A = A∗11 A11 ⊕ 0 ⊕ 0 ⊕ 0, ∗ ∗ BB ∗ = A11 A∗11 ⊕ 0 ⊕ C33 B33 ⊕ 0 and B ∗ B = A∗11 A11 ⊕ 0 ⊕ B33 C33 ⊕ 0.

We get AA∗ BB ∗ = (AA∗ )2 = BB ∗ AA∗ and A∗ AB ∗ B = (A∗ A)2 = B ∗ BA∗ A. By Corollary 2.2 we know AA∗  BB ∗ and A∗ A  B ∗ B. 2

C.Y. Deng / J. Math. Anal. Appl. 423 (2015) 32–40

39

In (6), the reason for getting first equation is that Aii , i = 1, 2, are injective dense defined. A22 = 0 means the second line and row in (2) will disappear. ∗

By (1), A ≤ B implies that A = PR(A) B = BPR(A∗ ) = PR(A) BPR(A∗ ) . The converse implication fails to ∗

hold in general. If there is another operator A0 also satisfying A0 ≤ B, then PR(A0 )∩R(A) A = PR(A0 )∩R(A) B = PR(A0 )∩R(A) A0 and APR(A∗ )∩R(A∗ ) = BPR(A∗ )∩R(A∗ ) = A0 PR(A∗ )∩R(A∗ ) . 0

0

0

Define the closed subspace set  P(A, B) =: M ⊆ H : M ⊆ R(A) ∩ R(B) : M is closed subspace,  M reduces AA∗ and BB ∗ , and PM A = PM B

(7)

with the usual order and common ∗-lower bound set ∗ ∗   Q(A, B) =: C ∈ B(H) : C ≤ A, C ≤ B

(8)

Φ : P(A, B) → Q(A, B) by Φ(M) = PM A.

(9)

with the ∗-order. Define

Then Φ : P(A, B) → Q(A, B) is well defined. In fact, let M ∈ P(A, B) and C = PM A. From C ∗ A = ∗

A∗ PM A = (PM A)∗ PM A = C ∗ C and AC ∗ = AA∗ PM = PM AA∗ PM = CC ∗ , we get C ≤ A. Similarly ∗

we have C ≤ B. Hence, Φ(M) = C ∈ Q(A, B) is well defined. If M, N ∈ P(A, B) such that M ⊆ N , then Φ(M)∗ [Φ(M) − Φ(N )] = A∗ PM [PM − PN ]A = 0 and [Φ(M) − Φ(N )]Φ(M)∗ = [PM − PN ]AA∗ PM = ∗

[PM − PN ]PM AA∗ = 0, i.e., Φ(M) ≤ Φ(N ). Φ is bijective from P(A, B) onto Q(A, B). On one hand, if M, N ∈ P(A, B) such that M = N , then ∗

∗

Φ(M) = PM A = Φ(N ) = PN A. On the contrary, if Φ(M) = Φ(N ), i.e., PM A ≤ PN A and PN A ≤ PM A, then R(PM A) ⊆ R(PN A) and R(PN A) ⊆ R(PM A) by Corollary 2.1. Note that R(PM A) = R(PM PR(A) ) = M,

R(PN A) = R(PN PR(A) ) = N .

We get M ⊆ N and N ⊆ M, i.e., M = N , which is contradictory. Hence, Φ is injective. On the other hand, ∀C ∈ Q(A, B), let M = R(C). Then M ⊆ R(A) ∩ R(B) by Corollary 2.1 and C = PM A. Now, PM AA∗ = CA∗ = AC ∗ = AA∗ PM . Similarly we have M reduces BB ∗ . Hence there exists M ∈ P(A, B) such that Φ(M) = C, i.e. Φ is surjective. In [12], the authors showed that the infimum of a bounded observable and a quantum event is a quantum event. To be more precise, they found that the structure of A B with respect to logic order can be characterized by using the invariant subspaces of A and B. In fact, ∗-order has similar nice properties too.

C.Y. Deng / J. Math. Anal. Appl. 423 (2015) 32–40

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Theorem 2.6. Let P(A, B), Q(A, B) and Φ be defined by (7), (8) and (9), respectively. Then ∗

A ∧ B = PM0 A = PM0 B, where M0 is the maximum element of the partially ordered set (P(A, B), ⊆).  Proof. Suppose that {Mi : i ∈ I} is a chain in P(A, B). Define M = {Mi , i ∈ I}. Since PMi AA∗ = AA∗ PMi , PMi BB ∗ = BB ∗ PMi , PMi A = PMi B and PMi → PM in the strong operator topology, we derive that M ∈ P(A, B) and hence, every chain {Mi : i ∈ I} has an upper bound in P(A, B). By Zorn’s lemma we know that P(A, B) has a maximum element M0 . In fact, U ⊆ M0 for every U ∈ P(A, B). If U  M0 , we get M0  M00 , where M00 = span(U, M0 ), the closure of the subspace {x1 + x2 : x1 ∈ U, x2 ∈ M0 }. Since PM0 AA∗ = AA∗ PM0 and PU AA∗ = AA∗ PU , we get PM00 AA∗ = AA∗ PM00 . In this vein, M00 ∈ P(A, B), which contradicts with the fact that M0 is a maximal element of P(A, B). ∗

We say A ∧ B = PM0 A = PM0 B. In fact, PM0 A ∈ Q(A, B) since Φ is bijection. For arbitrary C ∈ Q(A, B), there exists closed subspace M ∈ P(A, B) such that C = PM A = PM B. Since M ⊆ M0 , we get Φ(M)∗ Φ(M0 ) = A∗ PM PM0 A = A∗ PM A = Φ(M)∗ Φ(M) and Φ(M0 )Φ(M)∗ = PM0 AA∗ PM = PM0 PM AA∗ = PM AA∗ = Φ(M)Φ(M)∗ . ∗

∗

So Φ(M) ≤ Φ(M0 ), i.e., C = PM A ≤ PM0 A. 2 Theorem 2.6 shows that the minus ∗-order is equivalent with the set P(H) inheritance type properties. If A and B ∈ S(H), the condition that M reduces AA∗ and BB ∗ is equivalent to that M reduces A and B. Then Theorem 2.6 reduces to following corollary. Corollary 2.6. (See [12, Theorem 2.7].) Let A, B ∈ S(H). Then A B = PM A = PM B with respect to the logic order, where M is the maximum element in (7). Acknowledgment The author would like to express his hearty thanks to the referee for his/her valuable comments and suggestions which greatly improved the presentation of this paper. References [1] J. Antezana, C. Cano, I. Mosconi, D. Stojanoff, A note on the star order in Hilbert spaces, Linear Multilinear Algebra 58 (2010) 1037–1051. [2] J.K. Baksalary, O.M. Baksalary, X.J. Liu, Further properties of the star, left-star, right-star, and minus partial orderings, Linear Algebra Appl. 375 (2003) 83–94. [3] M.P. Drazin, Natural structures on semigroups with involution, Bull. Amer. Math. Soc. 84 (1978) 139–141. [4] S. Gudder, An order for quantum observables, Math. Slovaca 56 (2006) 573–589. [5] R.E. Hartwig, M.P. Drazin, Lattice properties of the ∗-order for complex matrices, J. Math. Anal. Appl. 86 (1982) 359–378. [6] H. Li, T. Huang, X. Liu, H. Li, On the inverses of general tridiagonal matrices, Linear Algebra Appl. 433 (2010) 965–983. [7] Y. Li, X. Sun, A note on the logic of bounded quantum observables, J. Math. Phys. 50 (2009) 122107. [8] W. Liu, J. Wu, A representation theorem of infimum of bounded quantum observables, J. Math. Phys. 49 (2008) 073521. [9] J.K. Merikoski, X. Liu, On the partial ordering of normal matrices, JIPAM. J. Inequal. Pure Appl. Math. 7 (1) (2006), Art. 17. [10] C.R. Putnam, Commutation Properties of Hilbert Space Operators, Springer-Verlag New York, New York, 1967. [11] P. Šemrl, Automorphisms of B(H) with respect to minus partial order, J. Math. Anal. Appl. 369 (2010) 205–213. [12] X.M. Xu, H.K. Du, X.C. Fang, On the infimum of bounded quantum observables, J. Math. Phys. 51 (2010) 093522. [13] X.M. Xu, H.K. Du, X.C. Fang, Y. Li, The supremum of linear operators for the ∗-order, Linear Algebra Appl. 433 (2010) 2198–2207.