6_749_xie.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 4, 2010, 678-685

ISSN 1307-5543 – www.ejpam.com

Fredholmness of Combinations of Two Idempotents

Tao Xie∗, Kezheng Zuo

Math Department, Hubei Normal University, Hubei, Huangshi, 435002, China

Abstract. If P and Q are two idempotents on a Hilbert space, in this paper, we prove that Fredholmness

of aP + bQ− cPQ is independent of the choice of a, b, c with ab 6= 0.

2000 Mathematics Subject Classifications: 15A03, 15A24

Key Words and Phrases: Idempotent, Fredholmness, Combinations of idempotents

1. Introduction

Idempotents are important and have wide applications in the theory of linear algebra and

operator theorem. It is shown in [17] that every n× n matrix over a field of characteristic

zero is a linear combination of three idempotents and in [16] that every bounded linear

operator on a complex infinite Hilbert space is a sum of at most five idempotents. See also

[5],[18],[19].

Let X be a Banach space, and P,Q be two idempotent operators on X . Many researchers

(see [1]-[15] and the references within) have addressed stability properties of the linear

combination aP + bQ; it has been proved that some properties such as invertibility, nullity,

Fredholmness, closeness of the range and complementarity of the Kernel of linear combina-

tions of P and Q are independent of the choice of coefficients a and b, provided ab 6= 0 and

a+ b 6= 0.

A natural question is whether the results above can be extended to more general situations.

In this note we consider the Fredholmness of some special combinations aP + bQ− cQP and

aP + bQ− cPQ− dQP when P,Q are idempotents. We prove that Fredholmness and index of

any combinations aP + bQ− cQP are independent of the choice of a, b, c with ab 6= 0. As an

application, we obtain that the invertibility of combinations aP + bQ− cQP are equivalent to

the invertibility of P +Q for all a, b, c ∈ C with ab 6= 0, which generalizes the result of [4].

Moreover, counter examples are shown that the combination aP + bQ − cPQ − dQP fails to

retain any such properties.

∗Corresponding author.

Email addresses: xietao_1294�163.
om (T. Xie), xiangzuo28�yahoo.
n (K. Zuo)

http://www.ejpam.com 678 c© 2010 EJPAM All rights reserved.



T. Xie, K. Zuo / Eur. J. Pure Appl. Math, 3 (2010), 678-685 679

2. Preliminaries

LetH be a Hilbert space, and let all bounded linear operators onH be denoted byB(H ).
An operator P ∈ B(H ) is said to be idempotent if P2 = P. The set P of all idempotents in

B(H ) is invariant under similarity; that is, is P ∈ P and S ∈ B(H ) is an invertible operator,

then S−1PS is still an idempotent since (S−1PS)2 = S−1PSS−1PS = S−1P2S = S−1PS. An

idempotent P is called an orthogonal projection if P2 = P = P∗, where P∗ is the adjoint of P.

Moreover, for an idempotent P ∈ P , there exists an invertible operator U ∈ B(H ) such that

U−1PU is an orthogonal projection. In fact, if P ∈ P , then P can be written in the form of

P =

�
I P1

0 0

�

with respect to the space decompositionH =R(P)⊕R(P)⊥, whereR(M) denotes the range

of the operator M . In this case, we have

�
I P1

0 I

��
I P1

0 0

��
I −P1

0 I

�
=

�
I 0

0 0

�
,

where eP =
�

I −P1

0 I

�
is invertible and eP−1 =

�
I P1

0 I

�
. An operator A ∈ B(H ) is

said to be positive if (Ax , x) ≥ 0 for all x ∈ H . If A is positive, then A
1

2 denotes the positive

square root of A. An operator T is Fredholm if the nullities of T denoted by nul(T ) and T ∗

are finite and the range of T is closed. For a Fredholm operator T , its index, indT , is by

definition nul(T )-nul(T ∗). It is know that the Fredholmness of T is preserved under compact

perturbations and is equivalent to the existence of an operator T ′ with T T ′ − I and T ′T − I

being compact. For details of Fredholmness, see[3], Chapter XI.

For the proof of the main theorem we need the following two lemmas which are well

known, so the proofs are omitted.

Lemma 1 ([3]). Let A=

�
A11 A12

A21 A22

�
be a bounded linear operator on H ⊕K . Then A is a

positive operator if and only if A11 ≥ 0, A22 ≥ 0, A12 = A∗21 and there exists a contraction D from

K intoH such that

A=




A11 A
1

2

11DA
1

2

22

A
1

2

22D∗A
1

2

11 A22


 .

Lemma 2 ([3]). Let T =

�
A B

C D

�
be an operator on H ⊕K , where A is Fredholm with A′

act onH satisfying AA′ = I + K1 and A′A= I + K2 for some compact operators K1 and K2. Then

T is Fredholm if and only if D− CA′B is. In this case, indT = indA+ ind(D− CA′B).



T. Xie, K. Zuo / Eur. J. Pure Appl. Math, 3 (2010), 678-685 680

3. Main results

Theorem 1. Let P and Q inB(H ) be two idempotents, then the Fredholmness of aP+ bQ−cPQ

is independent of the choice of a, b, c with ab 6= 0 and ind(aP + bQ− cPQ) = ind(P +Q).

Proof. Let P and Q be two idempotents. By the discussion above, since aP + bQ− cPQ is

Fredholm if and only if aS−1PS + bS−1QS − c(S−1PS)(S−1PS) is Fredholm, to consider the

Fredholmness of aP + bQ− cPQ, without loss of generality, we can assume that one of P and

Q is an orthogonal projection. For example, assume that Q is an orthogonal projection. Of

course, Q is a positive operator. In this case, by Lemma 1, P and Q have the following operator

matrix forms:

P =

�
I P1

0 0

�
and Q =




Q1 Q
1

2

1 DQ
1

2

2

Q
1

2

2
D∗Q

1

2

1
Q2




with respect to the space decomposition H =R(P)⊕R(P)⊥, where Q1 and Q2 are positive

operators onR(P) andR(P)⊥, respectively, and D is a contraction operator from R(P)⊥ into

R(P). Furthermore, Q1 and Q2 have the following operator matrix forms:

Q1 =




0 0 0

0 I 0

0 0 Q11


 , Q2 =




Q22 0 0

0 I 0

0 0 0




respect to the space decomposition

R(P) =N (Q1)⊕N (I −Q1)⊕ (R(P)⊖ (N (Q1)⊕N (I −Q1)))

and the space decomposition

R(P)⊥ = (R(P)⊥⊖N (I −Q2))⊕N (I −Q2)⊕N (Q2),

respectively. Then denoteH0 =N (Q1),H1 =N (I−Q1),H2 =R(P)⊖(N (Q1)⊕N (I−Q1)),

H3 = R(P)
⊥ ⊖N (I −Q2) and H4 = N (I −Q2), H5 = N (Q2), therefore P and Q have the

following matrix representations:

Q =




0 0 0 0 0 0

0 I 0 0 0 0

0 0 Q11 Q
1

2

11D1Q
1

2

22 0 0

0 0 Q
1

2

22D∗1Q
1

2

11 Q22 0 0

0 0 0 0 I 0

0 0 0 0 0 0




and

P =




I 0 0 P11 P12 P13

0 I 0 P21 P22 P23

0 0 I P31 P32 P33

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0






T. Xie, K. Zuo / Eur. J. Pure Appl. Math, 3 (2010), 678-685 681

with respect to the space decomposition H = ⊕5
i=0
Hi for some contraction D1 from H3 to

H2. If we let

Q0 =




Q11 Q
1

2

11
D1Q

1

2

22

Q
1

2

22D∗1Q
1

2

11 Q22


 ,

then Q being an orthogonal projection implies that Q0 is also an orthogonal projection on

H2⊕H3. That is, Q0 = Q2
0. We obtain





Q11 = Q2
11 +Q

1

2

11D1Q22D∗1Q
1

2

11,

Q
1

2

11
D1Q

1

2

22
= Q

3

2

11
D1Q

1

2

22
+Q

1

2

11
D1Q

3

2

22
,

Q
1

2

22D∗1Q
1

2

11 = Q
3

2

22D∗1Q
1

2

11 +Q
1

2

22D∗1Q
3

2

11,

Q22 = Q2
22 +Q

1

2

22
D∗1Q11D1Q

1

2

22
.

It can be derived by using the injectivity of Q11, I −Q11, Q22 and I −Q22 that





D1D∗1 = I ,

D∗1D1 = I ,

Q22 = D∗1(I −Q11)D1.

(1)

Note that

aP + bQ− cPQ = (2)

=




U11 0 U13 U14 U15 U16

0 U22 U23 U24 U25 U26

0 0 V11 V12 U35 U36

0 0 V21 V22 0 0

0 0 0 0 U55 0

0 0 0 0 0 0




(3)

with respect to the space decomposition H = ⊕5
i=0Hi, where

U11 = aI , U13 = −cP11Q
1

2

22
D∗1Q

1

2

11
,

U14 = aP11 − cP11Q22, U15 = aP12 − cP12,

U16 = aP13, U22 = (a+ b− c)I ,

U23 = −cP21Q
1

2

22D∗1Q
1

2

11, U24 = aP21 − cP21Q22,

U25 = aP22 − cP22, U26 = aP23,

U35 = aP32 − cP32, U36 = aP33,

U55 = bI .



T. Xie, K. Zuo / Eur. J. Pure Appl. Math, 3 (2010), 678-685 682

and

V11 = aI + bQ11 − c(Q11 + P31Q
1

2

22D∗1Q
1

2

11)

= aI + bQ11 − c(Q11 + P31D∗1Q
1

2

11(I −Q11)
1

2 ),

V12 = aP31 + bQ
1

2

11D1Q
1

2

22 − c(Q
1

2

11D1Q
1

2

22 + P31Q22),

= aP31 + bQ
1

2

11(I −Q11)
1

2 D1 − c(Q
1

2

11(I −Q11)
1

2 D1

+ P31D∗1(I −Q11)
1

2 D1),

V21 = bQ
1

2

22D∗1Q
1

2

11 = bD∗1Q
1

2

11(I −Q11)
1

2 ,

V22 = bQ22 = bD∗1(I −Q11)D1.

We claim that aP+ bQ− cPQ is Fredholm if and only if I−Q11 is invertible and I− P31D∗1(I −

P11)
− 1

2 P
1

2

11 is Fredholm. Indeed, if aP + bQ− cPQ is Fredholm, then, letting A be an operator

onH such that

K = (aP + bQ− cPQ)A− I

is compact, we have, with

A=

�
A1 A2

A3 A4

�
and K =

�
K1 K2

K3 K4

�
onH =R(P)⊕R(P)⊥,

�
V11 V12

V21 V22

��
A1 A2

A3 A4

�
=

�
I + K1 K2

K3 I + K4

�
.

Carrying out the mulitiplication here yields

bQ
1

2

22
D∗1Q

1

2

11
A2 + bQ22A4 = I + K4

or

bQ
1

2

22(D
∗
1Q

1

2

11A2 +Q
1

2

22A4) = I + K4.

This shows that Q
1

2

22 is Fredholm and hence so is Q22. Therefore, Q22 is invertible and thus so

is I −Q11 by (1). The Fredholmness of aP + bQ− cPQ is equivalent to that of
�

V11 V12

V21 V22

�

by (3), which is in turn equivalent to that of

V11 − V12V ′22V21 = aI + bQ11 − (aP31 + bQ
1

2

11D1Q
1

2

22)(bQ22)
′(bQ

1

2

22D∗1Q
1

2

11)

by Lemma 2. But this letter operator is equal to

aI + bQ11 − (aP31 + bQ
1

2

11D1D∗1(I −Q11)
1

2 D1)D
∗
1(I −Q11)

− 1

2 D1D∗1Q
1

2

11,



T. Xie, K. Zuo / Eur. J. Pure Appl. Math, 3 (2010), 678-685 683

which can be further simplified to

a(I − P31D∗1(I −Q11)
− 1

2 Q
1

2

11)

by (1). This proves one direction. For the other, if I − Q11 is invertible and I − P31D∗1(I −

Q11)
− 1

2 Q
1

2

11
is Fredholm then we can reverse the above arguments to show that aP+ bQ− cPQ

is Fredholm. The equivalence of Fredholmness of aP + bQ − cPQ and P +Q follows easily.

Finally, we also have

ind(aP + bQ− cPQ) = ind(I − P31D∗1(I −Q11)
− 1

2 Q
1

2

11
) = ind(P +Q),

which complete the proof.

As an application, we immediately have the following corollary.

Corollary 1. Let P,Q be two idempotents inB(X ). Then

(i) the invertibility of aP + bQ− cQP is independent of the choice of a, b, c ∈ C and ab 6= 0.

(ii) the invertibility of aP+ bQ− cQP is equivalent to the invertibility of aP+ bQ for all choice

of a, b, c ∈ C and ab 6= 0.

Proof.

(i) Let a0P + b0Q− c0QP be invertible for some a0, b0, c0 ∈ C with a0 b0 6= 0. Then a0P +

b0Q−c0QP is Fredholm with the nullity and defect equal to zero. By the above Theorem

, aP + bQ− cQP is invertible for all a, b, c ∈ C with ab 6= 0.

(ii) Let c = 0, then the (ii) follows from (i).

Remark 1. Let c = 0, we obtain the Theorems of [4] and [7].

As to the invertibility of aP + bQ− cPQ, there is an natural question that does the com-

bination aP + bQ − cPQ − dQP retain the invertibility for any ab 6= 0 and a + b = c + d .

However, there is an counterexample to note that this is impossible. Let P =

�
1 0

0 0

�
,

Q =

�
2 1

−2 1

�
, then P,Q are idempotent and the determinant of aP + bQ− cPQ − dQP is

0 when a = 12, b = −5, c = 10, d = −3 with a+ b = c + d , and is −3 when a = 1, b = 1, c =

−1, d = −1 with a+ b = c+ d . So the invertibility of aP+ bQ− cPQ− dQP depending on the

choice of scalars a, b, c, d with a+ b = c+ d . Therefore the idea of generalize the invertibility

of aP + bQ − cPQ or aP + bQ − cQP to the invertibility of aP + bQ − cPQ − dQP or more

generally aP + bQ− cPQ− dQP − ePQP − f QPQ− · · · can not be achieved.



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