Relationships between DMP relation and some partial orders

Applied Mathematics and Computation 266 (2015) 41–53

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Relationships between DMP relation and some partial orders Chunyuan Deng∗, Anqi Yu School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China

a r t i c l e

i n f o

a b s t r a c t Let H be an infinite-dimensional complex Hilbert space and let B (H ) be the algebra of all bounded linear operators on H. The equivalent definition of the DMP relation on B (H ), using operator block matrices, is introduced. We present a new interpretation of this relation which allows to generalize many known results for matrices to general Hilbert spaces regarding the DMP relation and its relationship with the star order and minus partial order. Also some properties of DMP relation, defined with the help of idempotent operators, are investigated.

2000 MSC: 15A09 46L05 47A05 Keywords: DMP inverse Projection Star partial order Minus partial order

© 2015 Published by Elsevier Inc.

1. Introduction Let B (H ) denote the set of all bounded linear operators on a complex Hilbert space H. For A ∈ B (H ), we shall denote by A∗ , N (A ) and R(A ) the adjoint of A, the null space and the range of A, respectively. An operator P ∈ B(H ) is said to be idempotent if P 2 = P. An idempotent P is called an orthogonal projection if P 2 = P = P ∗ . The orthogonal projection onto closed subspace M ⊆ H is denoted by PM . The identity onto a closed subspace M is denoted by IM or I if there does not exist confusion. An operator T is called generalized inner invertible, if there is an operator S such that (I ) T ST = T, denoted by S = T − . The operator T − is not unique in general. In order to guarantee its uniqueness, further conditions have to be imposed. The most likely convenient additional conditions are

(II ) ST S = S,

(III ) (T S )∗ = T S,

(IV ) (ST )∗ = ST,

(V ) T S = ST.

Element S ∈ B (H ) satisfying (I, II, III, IV) is called the Moore–Penrose inverse (for short MP inverse), denoted by S = T † . It is well known that T † ∈ B (H ) if and only if R(T ) is closed, and the MP inverse of T is unique (see [3,6]). Similarly, (I, II, V) inverse is called the group inverse, denoted by S = T # . Moreover, (I, II, III, IV, V) inverse is called the EP element (i.e., T † = T # ). One also considers (Ik ) T k ST = T k with some k ∈ Z+ . Clearly, (I ) = (I1 ). And (Ik , II, V) inverse is called the Drazin inverse, denoted by S = T D (see [3,5–7]), where k is the Drazin index of T (ind(T ) = k). For some applications we refer the reader to Ref. [28]. Recently, Baksalary and Trenkler introduced in [1] a new pseudoinverse of a matrix named core inverse. Malik and Thome in [20] generalized this definition and defined a new generalized inverse of a square matrix of an arbitrary index. They used the Drazin inverse (D) and the MP inverse and therefore this new generalized inverse is called the DMP inverse. Definition 1.1 ([1,2,20]). Let closed range operator T ∈ B (H ) have index at most k. If an operator X ∈ B (H ) satisfies the following three operator equations

XT X = X,

XT = T T D

and

T kX = T kT †,

then X is called the DMP-inverse of T, denoted by X = T  . ∗

Corresponding author. Tel.: +86 15918691636. E-mail address: [email protected], [email protected] (C. Deng).

http://dx.doi.org/10.1016/j.amc.2015.05.023 0096-3003/© 2015 Published by Elsevier Inc.

(1)

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C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53

The solution of (1) is unique if system is consistent (see [20, Definition 2.3] for the matrix case). If a closed range operator T ∈ B (H ) has index k, then the DMP inverse of T is unique and

T  = T DT T † [20, Theorem 2.2]. There is another inverse associated with operator T, namely the dual DMP inverse

T  = T †T T D. Definition 1.2. (See, for example, [10], [19, Definition 2.1], [23]) Let T, B ∈ B (H ). 

• The DMP relation is defined as: T ≤ B if T  T = T  B and T T  = BT  . ∗ • The star partial order is defined as: T ≤ B if T ∗ T = T ∗ B and T T ∗ = BT ∗ . −

• The minus partial order is defined as: T ≤ B if T − T = T − B and T T − = BT − for some choice of a generalized inner inverse T − . #

It is clear, if T is group invertible and T − = T # , then the minus partial order reduces to the sharp order: T ≤ B if T T # = BT # and # T T = T # B. For each of relations that we work in this article (e.g. minus, star, DMP, etc.), it is easy to see that if the upper operator is zero, the lower is also zero. So we omit the trivial cases T = 0 and B = 0. Some results related to binary relations defined by the group and Drazin generalized inverses can be found in [13]. The purpose of this paper is to investigate some common characterizations and various individual properties of the DMP relation, the star order and the minus partial order of bounded operators on a Hilbert space. Section 2 is concerned with the problem of establishing matrix expressions and obtaining some relationships among these relations, with the problem of finding equivalent conditions for these relations. Sections 3 deals with properties of DMP inverse and relationships among these orders. Several generalizations of the results known in the literature and a number of new results are derived. We begin with the following lemma concerning the Drazin inverse of 2 × 2 block triangular matrices [22].



Lemma 1.1. If A ∈ B (X ) and B ∈ B (Y ) are Drazin invertible with ind (A ) = r and ind (B ) = s, C ∈ B (Y, X ), then M =



Drazin invertible and MD =

 X = ( AD ) 2

s−1 



AD 0

A 0



C B

is

X , where BD





(AD )nCBn (I − BBD ) + (I − AAD )

n=0

r−1 

 AnC (BD )n (BD )2 − ADCBD .

n=0

Let T ∈ B (H ) be Drazin invertible with ind (T ) = k. From T R(T k ) = R(T k+1 ) ⊆ R(T k ) we know R(T k ) is an invariant subspace of T. Hence T has the following operator matrix



T=

T11

T12

0

T22



(2)

k = 0 [9,11,26,27]. Throughout this with respect to the space decomposition H = R(T k ) ⊕ R(T k )⊥ , where T11 is invertible and T22 work we denote by

X0 =

k−1 

i−k−1 k−1−i T11 T12 T22 .

(3)

i=0

Then 2 T11 X0 = T12 + T11 X0 T22

(4)

−2 T12 if k = 1. This kind of representation of T has been found quite useful in studying the various generalized inand X0 = T11 verses. If T has the form (2) and R(T ) is closed, then it is straightforward to check that the generalized inner inverse T − can be represented in the form



T− =

−1 −1 T11 − T11 T12 X3

X2

X3

X4



,

(5)

where Xi , i = 2, 3, 4 are bounded linear operators on appropriate spaces satisfying

T22 X3 = 0,

T22 X4 T22 = T22 ,

T11 X2 T22 + T12 X4 T22 = 0.

By [8, Theorem 6], the operator T in (2) has bounded MP inverse if and only if R(T ) is closed if and only if R(T22 ) is closed. Then

 †

T =

∗ T11 † ∗ (I − T22 T22 )T12

† ∗ −T11 T12 T22 † † † ∗ T22 − (I − T22 T22 )T12 T12 T22

 ,

(6)

C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53

43

†

∗ + T (I − T T )T ∗ ]−1 and where = [T11 T11 12 22 22 12



TT† =

I

0

0

† T22 T22

By [9, Theorem 2.5],



TD =



−1 T11

X0

0

0

.

(7)

 .

(8)

T is DMP invertible if and only if T is both Drazin and MP invertible. If T has the representation (2), then



T  = T DT T † =

−1 T11

† X0 T22 T22

0

0

TT and T T are idempotent with



T T =

I

T11 X0

0

0



,



,



TT =

(9)

I

† T11 X0 T22 T22

0

0

 .

(10)

It is clear that the orthogonal projection PR(T k ) = I ⊕ 0 and

R ( T k ) = R ( T  ) = R ( T T  ) = R ( T  T ) = R ( T T D ) = R ( T D T ) = R ( T D ). T

TT−

= 0 if and only if T is a nilpotent operator. Since

R ( T T ) = R ( T ), −

(11)

is an idempotent with

R ( T ) = R ( T ) ⊆ R ( T ), 

k

we have T T − T  = T  for each inner inverse T − of T. Moreover,

T is EP ⇐⇒ R(T ) = R(T ∗ ) ⇐⇒ T12 = 0, T22 = 0 ⇐⇒ T † = T # = T  = T . 2. The factorizations of DMP relation and some partial orders In this section, we describe the matrix structures of operators which satisfy the DMP relation or the minus partial order or the star partial order. These matrix representations will be very important in the sequel. Theorem 2.1. Let T ∈ B (H ) be a closed range operator with index at most k. Then the following results are equivalent: 

(i) There exists B ∈ B (H ) such that T ≤ B. (ii) If T is written as (2), then B has the 2 × 2 operator matrix form



B=

T11

† 2 T11 X0 − T11 X0 T22 T22 B22

0

B22



(12)

for some B22 ∈ B (R(T k )⊥ ). (iii) There exists Y ∈ B (H ) such that

B = T + (I − T T  )Y (I − T  T ).



Proof. (i) ⇒ (ii) Let T have the form (2). Denote by B = (10),

 BT 

=

−1 B11 T11

† B11 X0 T22 T22

−1 B21 T11

† B21 X0 T22 T22



 =

I

† T11 X0 T22 T22

0

0

It follows that B11 = T11 and B21 = 0. Similarly, the condition



T B =

† −1 T11 B11 + X0 T22 T22 B21

† −1 T11 B12 + X0 T22 T22 B22

0

0 †

2 X − T X T T B . Hence, implies that B12 = T11 0 11 0 22 22 22



B=

T11

† 2 T11 X0 − T11 X0 T22 T22 B22

0

B22



.



 B12 . If T ≤ B, by Definition 1.2 and representations in (9) and B22

B11 B21





= T T .

 =

I

T11 X0

0

0

 = T T

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C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53

(ii) ⇒ (iii) Let



Y =



Y11

Y12

Y21

B22 − T22 + Y21 T11 X0

By (10),



(I − T T )Y (I − T T ) = 



∈ B ( H ).

0

† −T11 X0 T22 T22

0

I

Hence, by (4),



B=

T11

† 2 T11 X0 − T11 X0 T22 T22 B22

0

B22



0

Y

 =

B22

0

−T11 X0

0

I



 =

0

† −T11 X0 T22 T22 (B22 − T22 )

0

B22 − T22

† T12 + T11 X0 T22 − T11 X0 T22 T22 B22

T11

 † T12 − T11 X0 T22 T22 (B22 − T22 )

T11

=



 

0



=

T11

T12

0

T22

 +

.



B22







0

† −T11 X0 T22 T22 (B22 − T22 )

0

B22 − T22

= T + (I − T T )Y (I − T T ). 



(iii) ⇒ (i) From

T T T  = T DT T †T T  = T DT T  = T DT T DT T † = T DT T † = T  we know

T  (I − T T  ) = (I − T  T )T  = 0. Hence, if B = T + (I − T T  )Y (I − T  T ), then

BT  = [T + (I − T T  )Y (I − T  T )]T  = T T  and

T  B = T  [T + (I − T T  )Y (I − T  T )] = T  T.



From Theorem 2.1, if B22 = 0 in item (ii) or Y = −T in item (iii), one can get a particular result 

T ≤ T T  T. 

If T and B are square matrices of index 1 and of same order such that T ≤ B, then a matrix representation of B was given in 

in (12) exists. Suppose [1, Theorem 6]. If T, B ∈ B (H ) are DMP invertible and T ≤ B, then B has the matrix structure (12) and B 22 ind (B22 ) = s. Denote by † 2 Z0 = T11 X0 − T11 X0 T22 T22 B22 ,

Y0 =

s−1 

(13)

−i−2 −1 T11 Z0 Bi22 (I − B22 BD22 ) − T11 Z0 BD22 .

i=0

BB†

By (7),

†

= I ⊕ B22 B22 . By Lemma 1.1,



D

B = Hence,

T B =

−1 T11

Y0

0

BD22





,



D

 †

B = B BB =

I

T11Y0 B22 B†22 + T12 B 22

0

T22 B

−1 T11

Y0 B22 B†22

0

B 22



 ,

B T =

 .

(14)

I

−1 T11 T12 + Y0 B22 B†22 T22

0

B T 22 22

 .



If T ≤ B, then

BT  B = BT  T = T T  T = T T D T T † T = T T D T and

T  BT  = T  T T  = T  ,

i.e., B is a generalized inner inverse of T . We say T ≤ B if T T = T B and T T = BT . While proving Theorem 2.1 we show the following corollary. Corollary 2.1. Let T, B ∈ B (H ) and T be a closed range operator with index at most k. Let P be an orthogonal projection with R(P ) = R(T  ) and Q be an idempotent with N (Q ) = N (T  ) and (I − Q )P = (I − P )Q = 0. Then

C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53

45



(i) PT ≤ PT + (I − P )B(I − P ) for all B ∈ B (H ). (ii) BT  = T T  ⇐⇒ BP = T P. (iii) T  B = T  T ⇐⇒ QB = QT . 



(iv) T ≤ B ⇐⇒ T ∗ ≤ B∗ . Proof. (i) If T has closed range with index of T at most k and the orthogonal projection P satisfies range relation R(P ) = R(T  ), then T can be written as matrix form (2) and P has the corresponding matrix representation P = I ⊕ 0 with the index of



PT =

T11

T12

0

0





at most 1. If there exists B ∈ B (H ) such that PT ≤ B, by (12) in Theorem 2.1, we get B has the corresponding matrix representation



B=

T11

T12

0

B22



= PT + (I − P )B(I − P ).



Hence, PT ≤ PT + (I − P )B(I − P ) for all B ∈ B (H ). (ii) Applying the range relations in (11), we have

(B − T )T  = 0 ⇐⇒ (B − T )R(T  ) = 0 ⇐⇒ (B − T )R(P ) = 0 ⇐⇒ (B − T )P = 0. (iii) Let Q = T T  . By (9) and (10),



Q=

I

† T11 X0 T22 T22

0

0



is an idempotent with (I − Q )P = (I − P )Q = 0 and N (Q ) = N (T  ). If T  (B − T ) = 0, then

Q (B − T ) = T T  (B − T ) = 0. On the other hand, if T has the representation (2), then P = I ⊕ 0. Since

(I − Q )P = (I − P )Q = 0, Q has the form



Q=

I

Q1

0

0

 . †

Since N (Q ) = N (T  ), we derive that T  (I − Q ) = 0. It follows that Q1 = T11 X0 T22 T22 and Q = T T  by (9). Now,

Q (B − T ) = 0 ⇒ T T  (B − T ) = 0 ⇒ T D T T  (B − T ) = 0 ⇒ T  (B − T ) = 0. (iv) Note that (T  )∗ = (T D T T † )∗ = (T ∗ )† T ∗ (T ∗ )D = (T ∗ ) .



The main advantage of Corollary 2.1 over Theorem 2.1 is that equivalences (ii) and (iii) involve only one projection. Let us prove the following results, which are similar to [25, Theorem 2] and [10, Lemma 3]. We call that an operator T ∈ B (H ) is dense if the range of T is dense in H. Theorem 2.2. Let T ∈ B (H ). Then the following are equivalent: ∗

(i) There exists B ∈ B (H ) such that T ≤ B. (ii) T and B have the 2 × 2 operator matrix forms



T=



T11

0

0

0



,

B=

T11

0

0

B22

 (15)

for some choice of a bounded linear operator B22 ∈ B (N (T ), N (T ∗ )), where T11 ∈ B (R(T ∗ ), R(T )) is injective and dense. (iii) There exists Y ∈ B (H ) such that

B = T + (I − PR(T ) )Y (I − PR(T ∗ ) ).

(16)

Proof. Let T ∈ B (H ). Then T as an operator from R(T ∗ ) ⊕ N (T ) into R(T ) ⊕ N (T ∗ ) has the 2 × 2 matrix form T = T11 ⊕ 0, where T11 ∈ B (R(T ∗ ), R(T )) is injective and dense. Let PR(T ) and PR(T ∗ ) be two orthogonal projections on R(T ) and R(T ∗ ), respectively. Let B have the corresponding 2 × 2 matrix form



B=

B11

B12

B21

B22



.

46

C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53 ∗

(i) ⇒ (ii) If T ≤ B, then



∗

T T= and

∗ T11 T11

0

0

0

∗ T11 T11

0

0

0

 ∗

TT =



 =



∗ T11 B11

∗ T11 B12

0

0

 =

∗ B11 T11

0

∗ B21 T11

0

 = T ∗B

 = BT ∗ .

Since T11 is injective anddense, weget B11 = T11 , B12 = 0 and B21 = 0. Hence, B = T11 ⊕ B22 . Y11 Y12 ∈ B (H ). Note that PR(T ) as an operator on H = R(T ) ⊕ N (T ∗ ) can be written as PR(T ) (ii) ⇒ (iii) Put Y = Y21 B22 = IR(T ) ⊕ 0. And PR(T ∗ ) as an operator on H = R(T ∗ ) ⊕ N (T ) can be written as PR(T ∗ ) = IR(T ∗ ) ⊕ 0. Then



B = T11 ⊕ B22 =

T11

0

0

0





+



0

0

Y11

Y12

0

I

Y21

B22



0

0

0

I



= T + (I − PR(T ) )Y (I − PR(T ∗ ) ). ∗

(iii) ⇒ (i) Note that (I − PR(T ∗ ) )T ∗ = 0 and T ∗ (I − PR(T ) ) = 0. It is clear that (B − T )T ∗ = 0 and T ∗ (B − T ) = 0, i.e., T ≤ B.



∗

In Theorem 2.2, the operator T does not necessarily have closed range. Theorem 2.2 shows that, if T ≤ B, then R(T ) ⊆ R(B ) s

s

and R(T ∗ ) ⊆ R(B∗ ), denoted these two range relations by T ≤ B [4,14,15,21]. Therefore, when R(T ) is closed, T ≤ B with BT † B = T −

implies T ≤ B. Theorem 2.3. Let T ∈ B (H ) with T a closed range operator. Then the following results are equivalent: −

(i) There exists B ∈ B (H ) such that T ≤ B. (ii) T and B have the 2 × 2 operator matrix forms



T=



T11

0

0

0



,

B=

T11

0

0

B22

 ,

(17)

where T11 ∈ B (R(T0− T ), R(T T0− )) is invertible, B22 ∈ B (N (T0− T ), N (T T0− )) for some choice of a generalized inner inverse T0− −1 of T. Furthermore, T0− can be written as T0− = T11 ⊕ X22 , where X22 ∈ B (N (T T0− ), N (T0− T )) satisfying X22 B22 = 0 and B22 X22 = 0. (iii) There exists Y ∈ B (H ) such that

B = T + (I − T T0− )Y (I − T0− T )

(18)

for some choice of a generalized inner inverse T0− satisfying (I − T T0− )Y (I − T0− T )T0− = 0 and T0− (I − T T0− )Y (I − T0− T ) = 0. −

Proof. (i) ⇒ (ii) If T ≤ B, then T0− T = T0− B and T T0− = BT0− for some choice of a generalized inner inverse T0− of T. Since T T0− T = T, we know T T0− and T0− T are two idempotent operators. Hence, T T0− as an operator on H = R(T T0− ) ⊕ N (T T0− ) can be written as T T0− = I ⊕ 0. And T0− T as an operator on H = R(T0− T ) ⊕ N (T0− T ) can be written as T0− T = I ⊕ 0. From



T T0− T =

I

0

0

0





T =T

I

0

0

0



=T

we derive that T has the form T = T11 ⊕ 0, where T11 ∈ B (R(T0− T ), R(T T0− )). Since T T0− = I ⊕ 0 and T0− T = I ⊕ 0, we derive that T11 is invertible. By (5), the inner inverse T0− can be written as



T0− =

−1 T11

X12

X21

X22



for some suitable operators X12 , X21 and X22 . Moreover, T T0− = I ⊕ 0 and T0− T = I ⊕ 0 imply that X12 = 0 and X21 = 0. Hence, −1 T0− = T11 ⊕ X22 for some operator X22 ∈ B (N (T T0− ), N (T0− T )). Since T T0− B = T T0− T = BT0− T = T, we derive that



I

0

0

0



B=



T11

0

0

0





=B



I

0

0

0

.

It implies that B has the diagonal form B = T11 ⊕ B22 . Since T0− T = T0− B and T T0− = BT0− , we derive that B22 X22 = 0 and X22 B22 = 0.

C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53

(ii) ⇒ (iii) Put



Y11 Y21

Y =

Y12 B22

  :





R(T0− T ) N (T0− T )

→

47



R(T T0− ) . N (T T0− )

By item (ii),

B = T11 ⊕ B22 = T + (I − T T0− )Y (I − T0− T ), where T0− (I − T T0− )Y (I − T0− T ) = 0 ⊕ X22Y22 = 0 and (I − T T0− )Y (I − T0− T )T0− = 0 ⊕ Y22 X22 = 0. (iii) ⇒ (i) B has the forms as in (18), then for the inner inverse T0− , it is clear that (B − T )T0− = 0 and T0− (B − T ) = 0, i.e., −

T ≤ B.

 −

s

Remark 1. (i) Theorem 2.3 shows that, T ≤ B ⇐⇒ T ≤ B and T B− T = T . (ii) Note that R(T ) is closed if the inner inverse of T exists. In the proof of Theorem 2.3, for some choice of a generalized inner inverse T0− of T, T T0− and T0− T are two idempotent operators. If we denote the idempotent operators T T0− and T T0− with the same space decomposition as in Theorem 2.2, then



T T0− = and

−D

0

0

I

0

−S

0

 T0− T

=



 

I

:

R (T )



N (T ∗ )

 

R (T ∗ )

:



R (T )



N (T ∗ )





R (T ∗ )

→

N (T )





→

,

N (T )





where D ∈ B N (T ∗ ), R(T ) and S ∈ B R(T ∗ ), N (T ) . The condition T T0− T = T T0− B implies that





T11

0

0

0



=

B11 − DB21

B12 − DB22

0

0



.

We get B11 = T11 + DB21 and B12 = DB22 . Similarly, T T0− T = BT0− T implies that B21 = B22 S. Hence, if T has the representation as in −

(15), then T ≤ B implies that B can be represented as



B=

T11 + DB22 S

DB22

B22 S

B22

  :



R (T ∗ )



→

N (T )

R (T )



N (T ∗ )

B (N (T ), N (T ∗ )).

for some choice of B22 ∈ This representation is a generalization of the result in [12] which is developed by Hartwig and Styan in the finite dimensional case. Using this unified matrix structures, it is easy to see that, for closed range operator T, ∗

−

T ≤ B ⇒ T ≤ B.

(19)

The converse does not hold even assuming that T and B are selfadjoint. For example, let T, B as operators on H ⊕ H have the forms as



T=

I

0

0

0





and B =

2I

I

I

I

−



(T11 = B22 = D = S = I ).

∗

Then T, B are selfadjoint and T ≤ B. But T ≤ B does not hold since T∗ T = T∗ B. (iii) Note that



B=

T11 + DB22 S

DY22

B22 S

B22





=



I

D

T11

0

0

I

0

B22





I

0

S

I

.

−

If T ≤ B, then



c1 T + c2 B =





I

D

(c1 + c2 )T11

0

0

I

0

c2 B22

I

0

S

I



is invertible if and only if B22 is invertible if and only if B is invertible for every c1 , c2 ∈ C and c2 (c1 + c2 ) = 0; c1 T + c2 B has the inner inverse if and only if B22 has the inner inverse if and only if B has the inner inverse and





I

0

(c1 + c2 )# T11−1

0

−S

I

0

c2# B− 22





I

−D

0

I

is an inner inverse of c1 T + c2 B for every c1 , c2 ∈ C, where c# = c−1 if c = 0 and c# = 0 if c = 0. ´ V. Rakoˇcevic, ´ Lectures on Generalized Inverses, Faculty of Science and MatheSome related results appear in D.S. Djordjevic, matics, University of Niš, 2008 and in [17].

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C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53

3. Properties of DMP relation In this section we mainly consider properties of operators which have the DMP relation. First we show that the invertibility of a linear combination c1 T + c2 B in the case when T and B have one kind of relations of minus, star, DMP is independent of the choice of scalars c1 , c2 ∈ C\{0}, c2 (c1 + c2 ) = 0. Specially, in the case when the matrices T or B are EP-matrices which are ordered ´ c´ in [29]. Some related by star partial order or minus partial order, the similar results had been obtained by Tošic´ and Cvetkovic-Ili applications were obtained in [18]. Theorem 3.1. Let T, B ∈ B (H ) with T a closed range operator. Let c1 , c2 ∈ C be such that c2 (c1 + c2 ) = 0. Then 

(i) If ind (T ) = k and T ≤ B, then c1 T k + c2 Bk is invertible if and only if B is invertible. In this case,

(c1 T k + c2 Bk )−1 = (c1 + c2 )−1 (T k ) + c2−1 (Bk − T k ) −c2−1 (c1 + c2 )−1 (T k ) [c1 T k + c2 Bk ](Bk − T k ) . ∗

(ii) If T ≤ B, then c1 T + c2 B is invertible if and only if B is invertible. In this case,

(c1 T + c2 B )−1 = (c1 + c2 )−1 T † + c2−1 (I − T † T )B−1 (I − T T † ). −

(iii) If T ≤ B, then c1 T + c2 B is invertible if and only if B is invertible. In this case,

(c1 T + c2 B )−1 = (c1 + c2 )−1 T0− + c2−1 (I − T0− T )B−1 (I − T T0− ). for some inner inverse T0− such that BT0− = T T0− and T0− B = T0− T. 

Proof. (i) By (2) and (12), if ind (T ) = k and T ≤ B, then



T=

T11

T12

0

T22



,



B=

T11

B12

0

B22

k = 0 and T is invertible. Note that T22 11



Tk =

k T11

k−1 k−1−i i i=0 T11 T12 T22

0

0

and

 c1 T k + c2 Bk =



† 2 where B12 = T11 X0 − T11 X0 T22 T22 B22 ,

,

 ,

 Bk =

k T11

k−1 k−1−i i=0 T11 B12 Bi22

0

Bk22



(c1 + c2 )T11k

k−1 k−1−i k−1 k−1−i i c1 i=0 T11 T12 T22 + c2 i=0 T11 B12 Bi22

0

c2 Bk22

 .

It follows that c1 T k + c2 Bk is invertible if and only if B22 is invertible if and only if B is invertible. In this case, since T k (T k )† = PR(T k ) = I ⊕ 0 and (Bk − T k )† (Bk − T k ) = PR((Bk −T k )∗ ) = PN (T k )∗ ) = 0 ⊕ I, by Lemma 1.1,

(T k ) = (T k )D T k (T k )† = T11−k ⊕ 0, (Bk − T k ) = (Bk − T k )† (Bk − T k )(Bk − T k )D = 0 ⊕ B−k 22 . Now, denote by





−k k−1 k−1−i i k−1 k−1−i S = −c2−1 (c1 + c2 )−1 T11 c1 i=0 T11 T12 T22 + c2 i=0 T11 B12 Bi22 B−k 22 .

We have



(c1 T k + c2 Bk )−1 =

(c1 + c2 )−1 T11−k 0

S c2−1 B−k 22



= (c1 + c2 )−1 (T k ) − c2−1 (c1 + c2 )−1 (T k ) [c1 T k + c2 Bk ](Bk − T k ) + c2−1 (Bk − T k ) . ∗

(ii) If T ≤ B, by (15) in Theorem 2.2, it is obvious that c1 T + c2 B is invertible if and only if B is invertible. By (16), we know that (I − T T † )B = B(I − T † T ) and

(c1 T + c2 B )[(c1 + c2 )−1 T † + c2−1 (I − T † T )B−1 (I − T T † )] = c1 (c1 + c2 )−1 T T † + c2 (c1 + c2 )−1 BT † + B(I − T † T )B−1 (I − T T † ) = c1 (c1 + c2 )−1 T T † + c2 (c1 + c2 )−1 T T † + (I − T T † )BB−1 (I − T T † ) = I. Similarly we derive that

[(c1 + c2 )−1 T † + c2−1 (I − T † T )B−1 (I − T T † )](c1 T + c2 B ) = I.

C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53

49

The result holds. − (iii) If T ≤ B, by (17) in Theorem 2.3, it is obvious that c1 T + c2 B is invertible if and only if B is invertible. Since BT0− = T T0− and T0− B = T0− T,



(c1 T + c2 B ) (c1 + c2 )−1 T0− + c2−1 (I − T0− T )B−1 (I − T T0− )

= c1 (c1 + c2 )−1 T T0− + c2 (c1 + c2 )−1 BT0− + B(I − T0− T )B−1 (I − T T0− ) = c1 (c1 + c2 )−1 T T0− + c2 (c1 + c2 )−1 T T0− + (B − T T0− T )B−1 (I − T T0− ) = c1 (c1 + c2 )−1 T T0− + c2 (c1 + c2 )−1 T T0− + (B − T T0− B )B−1 (I − T T0− ) = T T0− + (I − T T0− )(I − T T0− ) = I. Similarly, we have



(c1 + c2 )−1 T0− + c2−1 (I − T0− T )B−1 (I − T T0− ) (c1 T + c2 B ) = I.

The result holds.  

In addition, we have the following equivalent conditions for T ≤ B. Theorem 3.2. Let T, B ∈ B (H ) and T be a closed range operator with index k. Then 

T ≤ B ⇐⇒ (T − B )T D = 0,

T D (I − T † B ) = 0.

Proof. Note that 

T ≤ B ⇐⇒ T T  = BT  ,

T T = T B

⇐⇒ T T D T T † = BT D T T † ,

T T D = T T DT †B

(by post-multiplying the first

equation by T and pre-multiplying the second equation by T D ) †

T D = T D T † B (by post-multiplying the first equation by T D )

⇐⇒ T T D T = BT D T, ⇐⇒ T T D = BT D ,

T D = T D T † B. 

The reasons for T T D = BT D and T D = T D T † B ⇒ T ≤ B are

T T D = BT D ⇒ T T D T T † = BT D T T † ⇒ T T  = BT  and

T D = T D T † B ⇒ T D T = T D T T † B ⇒ T D T T † T = T D T T † B ⇒ T  T = T  B. 

Hence T ≤ B.  

#

It is clear that, if T is EP, then T D = T † = T # and T ≤ B ⇐⇒ T ≤ B. In this case, T B = BT. In general, we have the following result. 

Theorem 3.3. Let T, B ∈ B (H ) and T be a closed range operator with index at most k. If T ≤ B, then

T B = BT ⇐⇒ (T D − T  )B = 0, Proof. By (8), (9) and (12) we have



( T − T )B = 

D

 =

0

−1 T11 0

X0 0



X0 (I −

0

PR(T k )⊥ (T B − BT ) = 0.

 −

−1 T11

† T22 T22

0

)B22

0

† X0 T22 T22



0

.

Similarly, by (2) and (12) we get



TB =

 BT =

 

2 T11

† 3 2 T11 X0 − T11 X0 T22 T22 B22 + T12 B22

0

T22 B22

 ,

2 T11

† 2 T11 T12 + T11 X0 T22 − T11 X0 T22 T22 B22 T22

0

B22 T22



T11

† 2 T11 X0 − T11 X0 T22 T22 B22

0

B22



50

C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53

and

 PR(T k )⊥ (T B − BT ) =



0

0

0

T22 B22 − B22 T22

.

Hence, T B = BT if and only if B22 T22 = T22 B22 and † † 3 2 2 T11 X0 − T11 X0 T22 T22 B22 + T12 B22 = T11 T12 + T11 X0 T22 − T11 X0 T22 T22 B22 T22 .

Since

3X T11 0

=

2X T T11 T12 + T11 0 22

(20)

and

2 = T11 X0 T22 B22 = (T11 X0 − T12 )B22

† T11 X0 T22 T22 B22 T22

†

by (4), the relation (20) reduces as X0 T22 T22 B22 = X0 B22 . So we get T B = BT if and only if (T D − T  )B = 0 and PR(T k )⊥ (T B − BT ) = 0.  It is enough to take two different invertible operators, say C and D, to see that even the subset of group invertible operators in B (H ) cannot be a lattice endowed with DMP-relation. Also, it is not difficult to see that, for example, the invertible operators are 

maximal elements with respect to the DMP-relation. Next, we will prove that, for group invertible operator, ≤ is indeed a partial order (see also [16, Theorem 3.8] and [24, Theorem 5.6]). 

Theorem 3.4. For group invertible operators, the DMP relation ≤, is a partial order. 

Proof. Let T, B, C ∈ B (H ) be group invertible operators. Then T22 = 0 in (2)–(12). Theorem 2.1 shows that, if T ≤ B, then there exists operator B22 ∈ B (N (T ∗ )) with index at most 1 such that



B=



T11 0

T12 . B22

(21) 







It evidently follows that T ≤ T when B22 = 0. Let now T ≤ B and B ≤ T . By ([9,14]) and Theorem 2.1, T ≤ B implies that



T =



−1 T11 0



0 , 0

B = B# BB† =

 

−1 T11

−1 −T11 T12 B22

0

B 22

.

(22)



B ≤ T implies that



0 0

( B − T ) B =

0 B22



−1 T11 0

−1 −T11 T12 B 22  B22



 =

0 0

0 B22 B 22



 =

0

0

0

B22 B†22

 = 0.

We conclude that B22 = 0 and T = B. 



Assume T ≤ B and B ≤ C with T, B, C having index at most 1. Then B22 has index at most 1 and B, as an operator on H = R(T ) ⊕ R(B22 ) ⊕ N (B∗22 ), can be written as



B=

T11 0



 where T12 = T12

T12 B22



⎛

T11 =⎝0 0



T11 C=⎝0 0 Then

 T12  B22 0

⎛

⎛

(C − T )T  





B22 B22 T with B22 is invertible. Since 11 0 0 0 ∈ B (N (B∗22 )) with index at most 1 such that

0 = ⎝0 0

and hence T ≤ C. 

⎞

 T12  ⎠ B22 . C33

−1 T11 T  (C − T ) = ⎝ 0 0

and

⎞

 T12 B22 ⎠, 0

 and B T12 22 =

that there exists operator C33

⎛



 T12 B22 0

0 0 0 0 B22 0

⎞⎛

0 0 0⎠⎝0 0 0

⎞⎛

⎞

0 B22 0

−1 0 T11  ⎠⎝ B22 0 C33 0

0 B22 ⎠ = 0 C33 0 0 0

⎞

0 0⎠ = 0 0

 T12  B22





is invertible, by Theorem 2.1, B ≤ C implies

C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53

51

For a proof of Theorem 3.4 on the complex matrix setting see [1]. 

Theorem 3.5. Let T ∈ B (H ) have index at most 1. Let B ∈ B (H ) be DMP invertible and T ≤ B. Then (i) Indices TB and BT at most 1 and the reverse order law (T B ) = B T  holds. 

(ii) B − T ≤ B ⇐⇒ (T B ) = T  B ⇐⇒ (BT ) = T  B ⇐⇒ B T  = T  B . 

Proof. If T has index at most 1 and T ≤ B, by the proof of Theorem 3.4, we have T22 = 0 in (2)–(12) and



T=

T11

T12

0

0





,

B=



T11

T12

0

B22



T =

,

where B22 is DMP invertible. Then it is clear that



TB =

2 T11

T11 T12 + T12 B22

0

0



−1 T11

0

0

0



and BT =



2 T11

T11 T12

0

0



B =

,

−1 T11

−1 −T11 T12 B 22

0

B 22



,

(23)



have the indices at most 1 since T11 is invertible. −2 −2 Applying formula in (9), we get (T B ) = (BT ) = T11 ⊕ 0 and (B − T ) = 0 ⊕ B . Note that (B − T ) T = 0, B T  = T11 ⊕ 0, 22



T  B =

−2 T11

−2 −T11 T12 B 22

0

0





T ( B − T ) =

,

0

T12 B 22

0

0



.

We have (T B ) = B T  always holds. (T B ) = T  B if and only if (BT ) = T  B holds if and only if T12 B = 0 if and only if B T  = T  B holds if and only if 22 

B − T ≤ B.  Let us prove the following results, which were invested in [24, Theorems 5.2 and 5.4] when the index is at most one. Theorem 3.6. Let T, B ∈ B (H ) and T be a closed range operator with index at most k. 

(i) If T ≤ B, then BT  B = T ⇐⇒ ind (T ) = 1. 

(ii) If T ≤ B and B is DMP invertible, then B BT  = T  . 

(iii) If ind (T ) = 1, then T ≤ B ⇐⇒ T T  B = BT  B = T . 

Proof. (i) if T ≤ B and BT  B = T, then

R(T ) = R(BT  B ) = R(T T  B ) ⊆ R(T T  ) = R(T k ) ⊆ R(T ). By (2), T22 = 0, i.e., ind (T ) = 1. On the other hand, if ind (T ) = 1, then T  = T # T T † and

BT  B = T T  B = T T  T = T T # T T † T = T. (ii) By (9), (12) (13) and (14),





B BT



=

−1 T11

Y0 B22 B†22

0

B 22



T11

Z0

0

B22



−1 T11

† X0 T22 T22

0

0

 = T .

−1 (iii) If ind (T ) = 1, then T22 = 0 in (2), T  = T11 ⊕ 0 and T T  = I ⊕ 0 by (9) and (10). Let B has the corresponding 2 × 2 matrix



structure B =



B11 B21

 

BT B =

T11

T12

B21

B22

 B12 = 0 and B =

B12 . Since T T  B = T, B11 = T11 and T12 = B12 . Since B22

T11 0





−1 T11

0

T11

T12

0

0

B21

B22







=

T11

T12

B21

−1 B21 T11 T12



 =

T11

T12

0

0

 ,

 T12 . By Theorem 2.1, we know T ≤ B. The necessity is obvious by Theorem 2.1.  B22

The following result gives a property about the inner inverse of T . Theorem 3.7. Let T ∈ B (H ) and T be a closed range operator with index at most k. Then the following results are equivalent: (i) There exists B ∈ B (H ) such that T  BT  = T  i.e., B is an inner inverse of T . (ii) B has the 2 × 2 operator matrix form



B=

† T11 − T11 X0 T22 T22 B21

B12

B21

B22



for some B12 ∈ B (R(T k )⊥ , R(T k )), B21 ∈ B (R(T k ), R(T k )⊥ ) and B22 ∈ B (R(T k )⊥ ).

(24)

52

C. Deng, A. Yu / Applied Mathematics and Computation 266 (2015) 41–53

(iii) There exists Y ∈ B (H ) such that

B = T + (I − T T  )Y + Y (I − T T  ).



Proof. (i) ⇒ (ii) Let T have the form (2). Denote by B =

 

T BT



=

−1 T11

† X0 T22 T22

0

0





B11

B12

B21

B22

† −1 T11 B11 + X0 T22 T22 B21

=

 =

0





B11 B21

B12 . By (9), if B22

−1 T11

† X0 T22 T22

0

0

† −1 T11 B12 + X0 T22 T22 B22





† −1 −1 T11 B11 + X0 T22 T22 B21 T11





0

−1 T11

† X0 T22 T22

0

0



† † −1 T11 B11 + X0 T22 T22 B21 X0 T22 T22

0





0



=T , −1 T11 B11

†

†

then + X0 T22 T22 B21 = I. It follows that B11 = T11 − T11 X0 T22 T22 B21 . Hence (ii) holds. (ii) ⇒ (iii) Let



Y =

0

† B12 + T11 X0 T22 T22 Y0 − T12

B21

Y0

1 2 [B22

where Y0 =

†



(I − T T )Y = and

0

† −T11 X0 T22 T22

0

I

 Y (I − T T  ) =

0

† B12 + T11 X0 T22 T22 Y0 − T12

B21

Y0

† B12 + T11 X0 T22 T22 Y0 − T12

B21

Y0

† T11 − T11 X0 T22 T22 B21

B12

B21

B22





0



+

,

+ B21 T11 X0 T22 T22 − T22 ]. By (10),



B=





 =

0

† B12 + T11 X0 T22 T22 Y0 − T12

0

† Y0 − B21 T11 X0 T22 T22





T11

T12

0

T22

0

† −T11 X0 T22 T22

0

I



 +



 =



 =

† −T11 X0 T22 T22 B12

† −T11 X0 T22 T22 Y0

B21

Y0

0

† B12 + T11 X0 T22 T22 Y0 − T12

0

† Y0 − B21 T11 X0 T22 T22

† −T11 X0 T22 T22 B12

† −T11 X0 T22 T22 Y0

B21

Y0







= T + (I − T T  )Y + Y (I − T T  ). (iii) ⇒ (i) Since TT and T T are idempotent with R(T  ) = R(T T  ) = R(T  T ) by (10) and (11), we get T T  T  = T  T T  = T  . If B = T + (I − T T  )Y + Y (I − T T  ), then

T  BT  = T  [T + (I − T T  )Y + Y (I − T T  )]T  = T  T T  = T  .



Acknowledgments This work was supported by the National Natural Science Foundation of China under grant 11171222. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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