pairs of projections on a hilbert space: properties...

Linköping Studies in Science and Technology. Thesis.No. 1525

Pairs of projections on a Hilbertspace: properties and generalized

invertibility

Sonja Radosavljevic

Department of MathematicsLinköping University, SE–581 83 Linköping, Sweden

Linköping 2012

Linköping Studies in Science and Technology. Thesis.No. 1525

Pairs of projections on a Hilbert space: properties and generalized invertibility

Sonja Radosavljevic

[email protected]

Mathematics and Applied MathematicsDepartment of Mathematics

Linköping UniversitySE–581 83 Linköping

Sweden

ISBN 978-91-7519-932-0 ISSN 0280-7971

Copyright c© 2012 Sonja Radosavljevic

Printed by LiU-Tryck, Linköping, Sweden 2012

To Aleksa, my own Oneiros,for passing through the gates of horn and ivory with me

Abstract

This thesis is concerned with the problem of characterizing sums, differences, and prod-ucts of two projections on a separable Hilbert space. Other objective is characterizing theMoore-Penrose and the Drazin inverse for pairs of operators. We use reasoning similar toone presented in the famous P. Halmos’ two projection theorem: using matrix represen-tation of two orthogonal projection depending on the relations between their ranges andnull-spaces gives us simpler form of their matrices and allows us to involve matrix theoryin solving problems. We extend research to idempotents, generalized and hypergeneral-ized projections and their combinations.

v

Acknowledgments

I would like to thank my supervisors, professors Vladimir Kozlov, Bengt-Ove Turessonand Uno Wennergren, for encouragement and guidance they have been giving since thefirst moment that I spent at LiU. Their support was invaluable for me in many ways.

I am also grateful to professor Dragan S. Djordjevic, who introduced me to the gener-alized inverses and under whose influence papers included in thesis are written.

Big thanks goes to Martin Singull for the help with LaTex files and to Fredrik Bernts-son for the inspiring talks about linear algebra.

I owe gratitude to all my colleagues for creating friendly athmosphere and makingMAI great working place.

To three people at home (to say nothing of the dog), I owe gratitude for understaningand support.

Linköping, March 12, 2012Sonja Radosavljevic

vii

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

A On pairs of generalized and hypergeneralized projections on a Hilbert space 71 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Characterization of generalized and hypergeneralized projections . . . . . 113 Properties of products, differences, and sums of generalized projections . 154 Properties of products, differences, and sums of hypergeneralized projec-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

B On the Moore-Penrose and the Drazin inverse of two projections on a Hilbertspace 251 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 The Moore-Penrose and the Drazin inverse of two orthogonal projections 334 The MP and the Drazin inverse of the generalized and hypergeneralized

projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

ix

x Contents

Introduction

In this thesis, we want to examine some properties of sums, differences, and products oftwo operators on Hilbert space. We start with the orthogonal projections, i.e., operatorssuch that P = P 2 = P ∗, but we also extend research to idempotent operator P (whichsatisfy P = P 2), generalized projections (for which P ∗ = P 2) and hypergeneralizedprojections (satisfying P † = P 2, where P † is the Moore-Penrose inverse of operatorP ). The method comes from P. Halmos’ two projection theorem, (see [16]), stating thatfor two orthogonal projections P and Q on a finite dimensional Hilbert space H thereexists suitable matrix representation in accordance to the relations between their rangesand null-spaces. We generalize this approach to the wider classes of operators and also toinfinite dimensional Hilbert spaces.

As we know, every Hilbert space can be represented as a direct sum of two orthogonalsubspaces, i.e.,

H = L⊕ L⊥,where L is arbitrary closed subspace of H . If P is an orthogonal projection andR(P ) ={Px : x ∈ H} its range and N (P ) = {x ∈ H : Px = 0} its null-space, then fromR(P )⊥ = N (P ∗) andR(P ∗)⊥ = N (P ) we get

H = R(P )⊕N (P ).

The matrix representation of P then becomes

P =

[P1 P2

P3 P4

]:

[R(P )N (P )

]→[R(P )N (P )

], (1)

and if we use the fact that P1 : R(P ) → R(P ), P2 : N (P ) → R(P ), P3 : R(P ) →N (P ) and P4 : N (P )→ N (P ), we conclude that P1 = IR(P ) and P2 = P3 = P4 = 0.If we denoteR(P ) = L, then

P =

[IL 00 0

]:

[LL⊥

]→[

LL⊥

]. (2)

1

2 Introduction

We can see now that representation for Q under the same decomposition of the space is

Q =

[A BB∗ D

]:

[LL⊥

]→[

LL⊥

], (3)

where A and D are self-adjoint operators.What Halmos theorem says, and what we presented here in the simplified form, is

that if there are two orthogonal projections on Hilbert space, one of them can be used togenerate “coordinates”. Expressing everything in the terms of the orthogonal projectionP gives us wanted coordinates and we are usually left with straightforward computations.(Similar relations can be seen between analytical and Euclidean geometry: coordinatesystem can make things easier.) We are now able to further discuss properties of theoperators A, B and D as well as properties of the sums, differences, and products of theprojections P and Q by studying their matrices. An advantage of using this method liesin the fact that the projection P is in its simplest form in (2). Any other representation ofP would have a more complicated form and would lead to a more complicated form ofother operators in which P appears.

The second and perhaps more direct influence to our work comes form D. Djordjevicand J. Koliha, (see [11], [12]), who gave matrix representation of a closed range operatorA ∈ L(H) on infinite dimensional Hilbert space H depending on the different decompo-sition of the space. For the proof of the following lemma see [12].

Lemma 0.1Let A ∈ L(H) be a closed range operator. Then the operator A has the following threematrix representations with respect to the orthogonal sums of subspaces H = R(A) ⊕N (A∗) = R(A∗)⊕N (A):

(a) We have

A =

[A1 00 0

]:

[R(A∗)N (A)

]→[R(A)N (A∗)

],

where A1 is invertible.

(b) We have

A =

[A1 A2

0 0

]:

[R(A)N (A∗)

]→[R(A)N (A∗)

],

where B = A1A∗1 +A2A

∗2 mapsR(A) onto itself and B > 0.

(c) Alternatively,

A =

[A1 0A2 0

]:

[R(A∗)N (A)

]→[R(A∗)N (A)

],

where B = A∗1A1 +A∗2A2 mapsR(A∗) onto itself and B > 0.

The operators Ai are different in all three cases.

Here we again use the fact thatR(A)⊥ = N (A∗) andR(A∗)⊥ = N (A). Hence,

H = R(A)⊕N (A∗) = R(A∗)⊕N (A).

3

Like in the case with orthogonal projections, here is obtained simpler form of operator Awith two generalizations: operator is in the infinite dimensional settings and it is closedrange operator, not necessarily orthogonal projection. Three different forms are the resultof different decompositions of the space.

Based on the previous lemma and results that deal with two operators and the suit-able decomposition of the underlying Hilbert space, we now shift to specific operatorclasses. Since the literature on two projection theory is quite vast (see [7] for introduc-tion to the topic), our idea is to extend research to pairs of idempotents, generalized and,hypergeneralized projections and their combinations. J. Gross and G. Trenkler in [15]and J. K. Baksalary, O. M. Baksalary, X. Liu and G. Trenkler in [2], [3], [4] examinedsome properties of such operators on Cn×n. We are concerned with the same classesof operators. However, the method from [12] and [11] gives us the opportunity to usematrix representation of these operators on an infinite dimensional Hilbert space and toinvolve matrix theory in solving problems coming from operator theory. Note that EPoperators are those satisfying R(A) = R(A∗). Denote by OP(H), GP(H), HGP(H),and EP(H) classes of orthogonal, generalized and hypergeneralized projections and EPoperators, respectively. Then,

OP(H) ⊂ GP(H) ⊂ HGP(H) ⊂ EP(H)

and generalization is justified.Generalized and hypergeneralized projections do not have all the properties of or-

thogonal projections, so it is important to know what properties they have and under whatconditions their product, sum and difference keep them. Among many properties that anoperator can have, we were especially interested in generalized invertibility.

Let A ∈ L(H) and b ∈ H . Consider the equation

Ax = b.

If A is invertible, then A−1b is the unique solution to the equation. If A is not invertibleand b ∈ R(A), then there exists solution (possible more than one). If b /∈ R(A), thenthere are no solutions. However, it is possible to obtain generalized solutions by using theMoore-Penrose inverse of A (see [1], [8], [13] for more details and proofs), denoted byA† and defined as the unique solution to the equations

AA†A = A, A†AA† = A†, (AA†)∗ = AA†, (A†A)∗ = A†A.

It turns out that the least square solution of the linear equation Ax = b is x0 = A†b,i.e.,

‖Ax0 − b‖ = minx‖Ax− b‖.

Moreover, if M is the set of all least square solutions of the linear equation Ax = b, thenx0 is the minimal norm least square solution, i.e.,

‖x0‖ = minx∈M‖x‖.

Depending on applications, sometimes we can give up the equation solving propertiesof the Moore-penrose inverse in exchange for commutativity or some spectral property

4 Introduction

that the ordinary inverse possess. Thus we come to Drazin inverse: for A ∈ L(H), AD isthe Drazin inverse if

ADAAD = AD, ADA = AAD, An+1AD = An,

for some non-negative integer n. The smallest such n is called the Drazin index of A.Several authors have results on Drazin inverse of the sum and difference of idempo-

tents (see [9], [10]). Using an algebraic approach, we are able to either simplify existingproofs or to give some new characterizations of the Moore-Penrose and the Drazin in-verse of sums, differences and products of projections, generalized and hypergeneralizedprojections. Similarly to Lemma (0.1), we find canonical forms of the Moore-Penroseinverse of a closed range operator A depending on decomposition of the space. So, ifA ∈ L(H) is a closed range operator and H = R(A∗)⊕N (A) = R(A)⊕N (A∗), thenwe have three different representations of operator and its inverse:

(a)

A =

[A1 00 0

]:

[R(A∗)N (A)

]→[R(A)N (A∗)

], A† =

[A−11 0

0 0

].

(b)

A =

[A1 A2

0 0

]:

[R(A)N (A∗)

]→[R(A)N (A∗)

], A† =

[A∗1B

−1 0A∗2B

−1 0

],

where B = A1A∗1 +A2A

∗2 mapsR(A) onto itself and B > 0.

(c)

A =

[A3 0A4 0

]:

[R(A∗)N (A)

]→[R(A∗)N (A)

], A† =

[C−1A∗3 C−1A∗4

0 0

],

where C = A∗3A3 +A∗4A4 mapsR(A∗) onto itself and C > 0.

We can use these representations for solving various problems. The algebraic charac-ter of the method provides transparency of the proofs. As an illustration of the method,we state the following theorem.

Theorem 0.1Let K ∈ L(H). Then K is idempotent if and only if it is expressible in the form K =(PQ)† for some orthogonal projections P,Q ∈ L(H). Moreover, K = QKP .

Proof: If K is idempotent,R(K) = L and H = L⊕ L⊥, then it has the form:

K =

[K1 0K2 0

],

where K21 = K1 and K2 is arbitrary. It is easy to see that

P =

[K†1K1 0

0 0

], Q =

[K1K

†1 0

0 0

]

5

are wanted orthogonal projections.Conversely, suppose that we have the representations (2) and (3) for the orthogonal

projections P and Q. From formula (PQ)† = (PQ)∗(PQ(PQ)∗)†, we obtain

(PQ)† =

[AA† 0B∗A† 0

]and a direct verification shows that (PQ)† is an idempotent.

Summary of papers

Two papers are included in thesis. Below is a short summary for each of the papers.

Paper A: On pairs of generalized and hypergeneralizedprojections on a Hilbert space

In this paper, we characterize generalized and hypergeneralized projections (bounded lin-ear operators which satisfy conditions A2 = A∗ and A2 = A†). We give their matrixrepresentations and examine under what conditions the product, difference and sum ofthese operators are operators of the same class.

Paper B: On the Moore-Penrose and the Drazin inverse of twoprojections on a Hilbert space

For two given orthogonal, generalized or hypergeneralized projections P and Q on aHilbert space H , we give their matrix representation. We also give canonical forms ofthe Moore-Penrose and the Drazin inverses of their product, difference, and sum. Inaddition, we provide results showing when these operators are EP operators and somesimple relationships between the mentioned operators are established.

Bibliography

[1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Compu-tations, Springer; 2nd edition (2003

[2] J.K. Baksalary and X. Liu, An alternative characterization of generalized pro-jectors, Linear Algebra and its Applications 388 (2004) 61-65

[3] J. K. Baksalary, O. M. Baksalary, X. Liu and G. Trenkler, Further results ongeneralized and hypergeneralized projectors, Linear Algebra and its Applica-tions 429 (2008) 1038-1050

[4] J. K. Baksalary, O. M. Baksalary and X. Liu, Further properties on general-ized and hypergeneralized projectors, Linear Algebra and its Applications 389(2004) 295303

6 Introduction

[5] O. M. Baskalary and G. Trenkler, Column space equalities for orthogonal pro-jectors, Applied Mathematics and Computations, 212 (2009) 519-529

[6] O. M. Baskalary and G. Trenkler, Revisitaton of the product of two orthogonalprojectors, Linear Algebra Appl. 430 (2009), 2813-2833.

[7] A. Bottcher, I. M. Spikovsky, A gentle guide to the basics of two projectionstheory, Linear Algebra Appl. (2009)

[8] S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transforma-tions, SIAM 2009.

[9] C. Deng, The Drazin inverses of sum and difference of idempotents, LinearAlgebra Appl. 430 (2009) 1282-1291

[10] C. Deng and Y. Wei, Characterizations and representations of the Drazin in-verse involving idempotents, Linear Algebra Appl. 431 (2009) 1526-1538

[11] D. S. Djordjevic and J. Koliha, Characterizing hermitian, normal and EP oper-ators, Filomat 21:1 (2007) 39-54

[12] D. S. Djordjevic, Characterizations of normal, hyponormal and EP operators,J. Math. Anal. Appl. 329 (2) (2007), 1181-1190

[13] D. S. Djordjevic and V. Rakocevic, Lectures on Generalized Inverses, Facultyof Sciences and Mathematics, University of Niš, 2008.

[14] H. Du and Y. Li, The spectral characterization of generalized projections, Lin-ear Algebra and its Applications 400 (2005) 313-318

[15] J. Gross and G. Trenkler, Generalized and hypergeneralized projectors, LinearAlgebra and its Applications 264 (1997) 463-474

[16] P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969) 381-389

[17] J. J. Koliha and V. Rakocevic, Range projections and the Moore-Penrose inversein rings with involution

[18] T. Kato, Perturbation theory for linear operators, Springer, Berlin, 1996.

[19] V. Rakocevic, Funkcionalna analiza, Naucna knjiga, Boegrad, 1994

Paper A

On pairs of generalized andhypergeneralized projections on a

Hilbert space

Authors: Sonja Radosavljevic and Dragan S. Djordjevic

Preliminary version published as a technical report LiTH-MAT-R–2012/01–SE, Depart-ment of Mathematics, Linköping University.

7

On pairs of generalized and hypergeneralizedprojections on a Hilbert space

Sonja Radosavljevic∗ Dragan S. Djordjevic†

∗Department of Mathematics,Linköping University,

SE–581 83 Linköping, Sweden.E-mail: [email protected]

†Faculty of Sciences and Mathematics,University of Nis, P.O.Box 224, 18000 Nis, Serbia

E-mail: [email protected]

Abstract

In this paper, we characterize generalized and hypergeneralized projections(bounded linear operators which satisfy conditions A2 = A∗ and A2 = A†).We give their matrix representations and examine under what conditions theproduct, difference and sum of these operators are operators of the sameclass.

Keywords: Generalized projections, hypergeneralized projections.

9

10 Paper A On pairs of generalized and hypergeneralized projections on a Hilbert space

1 Introduction

Let H be a separable Hilbert space and L(H) be a space of all bounded linear operatorsonH . The symbolsR(A),N (A) andA∗ denote range, null space and adjoint operator ofoperator A ∈ L(H). Operator A ∈ L(H) is a projection (idempotent) if A2 = A, whileit is an orthogonal projection ifA∗ = A = A2. Operator is hermitian (self adjoint) ifA =A∗, normal if AA∗ = A∗A and unitary if AA∗ = A∗A = I . All these operators havebeen extensively studied and there are plenty of characterizations both of these operatorsand their linear combinations ([5]).

The Moore-Penrose inverse of A ∈ L(H), denoted by A†, is the unique solution tothe equations


Notice thatA† exists if and only ifR(A) is closed. ThenAA† is the orthogonal projectionontoR(A) parallel toN (A∗), and A†A is the orthogonal projection ontoR(A∗) paralleltoN (A). Consequently, I −AA† is the orthogonal projection ontoN (A∗) and I −A†Ais the orthogonal projection onto N (A).

For A ∈ L(H), an element B ∈ L(H) is the Drazin inverse of A, if the followinghold:

BAB = B, BA = AB, An+1B = An,

for some non-negative integer n. The smallest such n is called the Drazin index of A. ByAD we denote Drazin inverse of A and by ind(A) we denote Drazin index of A.

If such n does not exist, ind(A) =∞ and operatorA is generalized Drazin invertible.Its invers is denoted by Ad.

Operator A is invertible if and only if ind(A) = 0.If ind(A) ≤ 1, operator A is group invertible and AD is its group inverse, usually

denoted by A#.Notice that if the Drazin inverse exists, it is unique. Drazin inverse exists ifR(An) is

closed for some non-negative integer n.Operator A ∈ L(H) is a partial isometry if AA∗A = A or, equivalently, if A† = A∗.

Operator A ∈ L(H) is EP if AA† = A†A, or, in the other words, if A† = AD = A#. Setof all EP operators on H will be denoted by EP(H). Self-adjoint and normal operatorswith closed range are important subset of set of all EP operators. However, converse isnot true even in a finite dimensional case.

In this paper we consider pairs of generalized and hypergeneralized projections on aHilbert space, whose concept for matrices A ∈ Cm×n was introduced by J. Gross and G.Trenkler in [6]. These operators extend the idea of orthogonal projections by deleting theidempotency requirement. Namely, we have the following definition:

Definition A.1. Operator A ∈ L(H) is

(a) a generalized projection if A2 = A∗;

(b) a hypergeneralized projection if A2 = A†.

The set of all generalized projecton on H is denoted by GP(H) and the set of all hyper-generalized projecton is denoted byHGP(H).

2 Characterization of generalized and hypergeneralized projections 11

2 Characterization of generalized andhypergeneralized projections

We begin this section by giving characterizations of generalized and hypergeneralizedprojections. Similarly to Theorem 2 in [4] and Theorem 1 in [6], we have the followingresult:

Theorem A.1Let A ∈ L(H). Then the following conditions are equivalent:

(a) A is a generalized projection.

(b) A is a normal operator and A4 = A.

(c) A is a partial isometry and A4 = A.

Proof: (a⇒ b) Since

AA∗ = AA2 = A3 = A2A = A∗A,

A4 = (A2)2 = (A∗)2 = (A2)∗ = (A∗)∗ = A,

the implication is obvious.(b ⇒ a) If AA∗ = A∗A, recall that then exists a unique spectral measure E on the

Borel subsets of σ(A) such that E(∆) is an orthogonal projection for every subset ∆ ⊂σ(A), E(∅) = 0, E(H) = I and if ∆i ∩∆j = ∅ for i 6= j, then E(∪∆i) = E(

∑∆i).

Moreover, A has the following spectral representation

A =

∫σ(A)

λdEλ,

where Eλ = E(λ) is the spectral projection associated with the point λ ∈ σ(A).FromA4 = A, we concludeA3|R(A) = IR(A) and λ3 = 1, or, equivalentely σ(A) ⊆

{0, 1, e2πi/3, e−2πi/3}. Now,

A = 0E(0)⊕ 1E(1)⊕ e2πi/3E(e2πi/3)⊕ e−2πi/3E(e−2πi/3),

where E(λ) is the spectral projection of A associated with the point λ ∈ σ(A) such thatE(λ) 6= 0 if λ ∈ σ(A),E(λ) = 0 if λ ∈ {0, 1, e2πi/3, e−2πi/3}\σ(A) andE(0)⊕E(1)⊕E(e2πi/3) ⊕ E(e−2πi/3) = I . From the fact that σ(A2) = σ(A∗) and from uniquenessof spectral representation, we get A2 = A∗.

(a ⇒ c) If A∗ = A2, then A4 = AA2A = AA∗A = A. Multiplying from the left(from the right) by A∗, we get A∗AA∗A = A∗A (AA∗AA∗ = AA∗), which proves thatA∗A (AA∗) is the orhtogonal projecton ontoR(A∗A) = R(A∗) = N (A)⊥ (R(AA∗) =R(A) = N (A∗)⊥), i.e., A (A∗) is a partial isometry.

(c ⇒ a) If A is a partial isometry, we know that AA∗ is orthogonal projection ontoR(AA∗) = R(A). Thus, AA∗A = PR(A)A = A and A4 = AA2A = A impliesA2 = A∗.


We give matrix representatons of generalized projections based on the previous char-acterizations.

Theorem A.2Let A ∈ L(H) be a generalized projection. Then A is a closed range operator and A3 isan orthogonal projection onR(A). Moreover, H has decomposition

H = R(A)⊕N (A)

and A has the following matrix representaton

A =

[A1 00 0

]:

[R(A)N (A)

]→[R(A)N (A)

],

where the restriction A1 = A|R(A) is unitary onR(A).

Proof: IfA is a generalized projection, thenAA∗A = A4 = A andA is a partial isometryimplying that

A3 = AA∗ = PR(A),

A3 = A∗A = PN (A)⊥ .

Thus, A3 is an orthogonal projection onto R(A) = N (A)⊥ = R(A∗). Consequently,R(A) is a closed subset in H as a range of an orthogonal projection on a Hilbert space.From Lemma (1.2) in [5] we get the following decomposition of the space

H = R(A∗)⊕N (A) = R(A)⊕N (A).

Now, A has the following matrix representation in accordance with this decomposition:

A =

[A1 00 0

]:

[R(A)N (A)

]→[R(A)N (A)

],

where A21 = A∗1, A4

1 = A1 and A1A∗1 = A∗1A1 = A3

1 = IR(A).

Similar to Theorem 2 in [6], we have:

Theorem A.3Let A ∈ L(H). Then the following conditions are equivalent:

(a) A is a hypergeneralized projecton.

(b) A3 is an orthogonal projection ontoR(A).

(c) A is an EP operator and A4 = A

Proof: (a⇒ b) If A2 = A†, then from A3 = AA† = PR(A) follows conclusion.(b ⇒ a) If A3 = PR(A), a direct verificaton of the Moore-Penrose equations shows

that A2 = A†.(a⇒ c) Since

AA† = AA2 = A3 = A2A = A†A,

2 Characterization of generalized and hypergeneralized projections 13

we conclude that A is EP operator, A† = A#, (A†)n = (An)† and

A4 = (A2)2 = (A†)2 = (A2)† = (A†)† = A.

(c ⇒ a) If A is an EP operator, then A† = A# and ind(A) = 1 or, equivalently,A2A† = A. Since A4 = A2A2 = A, from uniqueness of A† follows A2 = A†.

Theorem A.4Let A ∈ L(H) be a hypergeneralized projection. Then A is a closed range operator andH has decomposition

H = R(A)⊕N (A).

Also, A has the following matrix representaton with the respect to decomposition of thespace

A =

[A1 00 0

]:

[R(A)N (A)

]→[R(A)N (A)

],

where restriction A1 = A|R(A) satisfies A31 = IR(A).

Proof: If A is a hypergeneralized projecton, then A is EP operator, and using Lemma(1.2) in [5], we get the following decomposition of the space H = R(A)⊕N (A) and Ahas the required representation.

Notice that since R(A) is closed for both generalized and hypergeneralized projec-tions, these operators have the Moore-Penrose and Drazin inverses. Besides, they are EPoperators, which implies that A† = AD = A# = A2. For generalized projections we canbe more precise:

A† = AD = A# = A2 = A∗.

We can also writeGP(H) ⊆ HGP(H) ⊆ EP(H).

Parts (b) and (c) of the following two theorems are known for matrices A ∈ Cm×n,(see [2], [3]). Unlike their proof, which is based on representation of matrices, our proofrelies only on properties of generalized and hypergeneralized projections and basic prop-erties of the Moore-Penrose and the group inverse.

Theorem A.5Let A ∈ L(H). Then the following holds:

(a) A ∈ GP(H) if and only if A∗ ∈ GP(H).

(b) A ∈ GP(H) if and only if A† ∈ GP(H).

(c) If ind(A) ≤ 1, then A ∈ GP(H) if and only if A# ∈ GP(H).

Proof: (a) If A ∈ GP(H), then

(A∗)∗ = A = A4 = (A2)2 = (A∗)2,

meaning thatA∗ ∈ GP(H). Conversely, ifA∗ ∈ GP(H), then (A∗)4 = A∗ and (A∗)2 =(A∗)∗ = A, implying

A2 = (A∗)4 = A∗


and A ∈ GP(H).(b) If A ∈ GP(H), then A† = A∗ = A2 and

(A†)2 = (A2)2 = A = (A∗)∗ = (A†)∗,

implying A† ∈ GP .If A† ∈ GP(H), then (A†)2 = (A†)∗ = (A†)† = A and (A†)4 = A†. Thus,

A2 = (A†)4 = A†,

A∗ = ((A†)∗)∗ = A†

and A ∈ GP(H).(c) If A ∈ GP(H), then A† = A# and part (b) of this theorem implies that A# ∈

GP(H).To prove converse, it is enough to see that A# ∈ GP(H) implies (A#)2 = (A#)∗ =

(A#)† = (A#)# = A and (A#)4 = A# and

A2 = (A#)4 = A# = ((A#)∗)∗ = A∗.

Remark A.1. Let us mention an alternative proof for the previous theorem. If A† ∈GP(H), then A is normal andR(A) is closed and A, A† have representations

A =

[A1 A2

0 0

]:

[R(A)N (A∗)

]→[R(A)N (A∗)

], A† =

[A∗1B 0A∗2B 0

],

where B = (A1A∗1 +A2A

∗2)−1. From (A†)2 = (A†)∗, we get[A∗1BA

∗1B 0

A∗2BA∗1B 0

]=

[BA1 BA2

0 0

],

which implies A2 = 0, A∗2 = 0 and B = (A1A∗1)−1 and

A =

[A1 00 0

], A† =

[A−11 0

0 0

].

Since (A−11 )2 = (A−11 )∗, the same equality is also satisfied for A1 and A ∈ GP(H).Similarly, to prove that A# ∈ GP(H) implies A ∈ GP(H), assume that H =

R(A)⊕N (A∗). Then A# = A†

A =

[A1 A2

0 0

], A# =

[A#

1 (A#1 )2A2

0 0

].

Since (A#)2 = (A#)∗, we get A2 = 0 and (A#1 )2 = (A#

1 )∗. Fron the fact that A1

is surjective mapping on R(A) and R(A1) ∩ N (A1) = {0}, we have A#1 = A−11 .

Consequently, (A−11 )2 = (A−11 )∗ and A21 = A∗1, which proves that A ∈ GP(H).

3 Properties of products, differences, and sums of generalized projections 15

Theorem A.6Let A ∈ L(H). Then the following holds:

(a) A ∈ HGP(H) if and only if A∗ ∈ HGP(H).

(b) A ∈ HGP(H) if and only if A† ∈ HGP(H).

(c) If ind(A) ≤ 1, then A ∈ HGP(H) if and only if A# ∈ HGP(H).

Proof: Proofs of (a) and (b) are similar to proofs of Theorem A.5 (a) and (b).(c) We should only prove that A# ∈ HGP(H) implies A ∈ HGP(H), since the

”⇒ ” is analogous to the same part of Theorem A.5.Let H = R(A)⊕N (A∗) and ind(A) ≤ 1. Then

A =

[A1 A2

0 0

], A# =

[A−11 (A−11 )2A2

0 0

], (A#)† =

[(A−11 )∗B 0(A−12 )∗B 0

],

where B = (A−11 (A−11 )∗ + A−12 (A−12 )∗)−1. From (A#)† = (A#)2, we get A2 = 0and A1 = A−21 . Multiplying with A2

1, the last equation becomes A31 = IR(A) and A ∈

HGP(H).

3 Properties of products, differences, and sums ofgeneralized projections

In this section we study properties of products, differences, and sums of two generalizedprojections or of one orthogonal and one generalized projection. We begin with two veryuseful theorems which gives matrix representations of such pairs of operators. Also, weobtain basic properties of generalized projections which easily follow from their canonicalrepresentations.

Theorem A.7Let A,B ∈ GP(H) and H = R(A)⊕N (A). Then A and B has the following represen-tation with respect to decomposition of the space:

A =

[A1 00 0

]:

[R(A)N (A)

]→[R(A)N (A)

],

B =

[B1 B2

B3 B4

]:

[R(A)N (A)

]→[R(A)N (A)

],

where

B∗1 = B21 +B2B3,

B∗2 = B3B1 +B4B3,

B∗3 = B1B2 +B2B4,

B∗4 = B3B2 +B24 .

Furthermore, B2 = 0 if and only if B3 = 0.


Proof: Let H = R(A) ⊕ N (A). Then representation of A follows from Theorem A.2and let B has representation

B =

[B1 B2

B3 B4

].

Then, from

B2 =

[B2

1 +B2B3 B1B2 +B2B4

B3B1 +B4B3 B3B2 +B24

]=

[B∗1 B∗3B∗2 B∗4

]= B∗,

the conclusion follows directly.If B2 = 0, then B∗3 = B1B2 +B2B4 = 0 and B3 = 0. Analogously, B3 = 0 implies

B2 = 0.

Corollary A.1Under the assumptions of the previous theorem, operator B ∈ GP(H) has one of thefollowing matrix representations:

B =

[B1 B2

B3 B4

]or B =

[B1 00 B4

].

Theorem A.8LetP ∈ B(H) be an orthogonal projection,R(P ) = L andH = L⊕L⊥. IfA ∈ GP(H),then P and A has the following matrix representation with the respect to decompositionof the space

P =

[IL 00 0

]:

[LL⊥

]→[

LL⊥

],

A =

[A1 A2

A3 A4

]:

[LL⊥

]→[

LL⊥

],

where

A∗1 = A21 +A2A3,

A∗2 = A3A1 +A4A3,

A∗3 = A1A2 +A2A4,

A∗4 = A3A2 +A24.

Moreover,

A1 = PAP |L,A2 = PA(I − P )|L⊥ ,A3 = (I − P )AP |L,A4 = (I − P )A(I − P )|L⊥ .

Then A2 = 0 if and only if A3 = 0, or equivalentely, PA(I − P )|L⊥ = 0 if and only if(I − P )AP |L = 0.

Opreators P andA commute if and only if eitherA2 = 0 orA3 = 0, or equivalentely,if and only if either PA(I − P )|L⊥ = 0 or (I − P )AP |L = 0.


Proof: Matrix representation of A ∈ GP(H) and properties of Ai, i = 1, 4 can beobtained like in the proof of Theorem A.7.

Using matrix multiplication, it is easy to see that

PAP =

[A1 00 0

]and PAP |L = A1. The rest of the equalities are obtained in an analogous way.

If PA = AP , again using matrix multiplication, we get A2 = A3 = 0 which isequivalent to PA(I − P )|L⊥ = 0 or (I − P )AP |L = 0.

Corollary A.2Under the assumptions of the previous theorem, operator A ∈ GP(H) has the followingmatrix representations:

A =

[A1 A2

A3 A4

],

when P and A do not commute, or

A =

[A1 00 A4

],

when P and A commute.

As we know, if A is an orthogonal projection, I −A is also an orthogonal projection.It is of interest to examine whether generalized projections keep the same property.

Example A.1

If H = C2 and A =

[e

2πi3 0

0 0

], then A2 = A∗, but I − A =

[1− e 2πi

3 00 1

]and,

clearly, I −A 6= (I −A)4 implying that I −A is not a generalized projection.

Thus, we have the following theorem.

Theorem A.9(Theorem 6 in [2]) Let A ∈ L(H) be a generalized projection. Then I − A is a normaloperator. Moreover, I − A is a generalized projection if and only if A is an orthogonalprojection.

If I−A is a generalized projection, thenA is a normal operator andA is a generalizedprojection if and only if I −A is an orthogonal projecton.

Proof: If A is a generalized projection, then A is a normal operator and A4 = A, whichimplies

A = 0E(0)⊕ 1E(1)⊕ e2πi/3E(e2πi/3)⊕ e−2πi/3E(e−2πi/3),

where E(λ) is the orthogonal projection such that E(λ) 6= 0 if λ ∈ σ(A), E(λ) = 0if λ ∈ {0, 1, e2πi/3, e−2πi/3}\σ(A) and E(0) ⊕ E(1) ⊕ E(e2πi/3) ⊕ E(e−2πi/3) = I .Thus,

I−A = (1−0)E(0)⊕(1−1)E(1)⊕(1−e2πi/3)E(e2πi/3)⊕(1−e−2πi/3)E(e−2πi/3),


and

(I −A)2 = E(0)⊕ (1− e2πi/3)2E(e2πi/3)⊕ (1− e−2πi/3)2E(e−2πi/3),

(I −A)∗ = E(0)⊕ (1− e2πi/3)∗E(e2πi/3)⊕ (1− e−2πi/3)∗E(e−2πi/3).

Hence,(I −A)2 = (I −A)∗

if and only if(1− e2πi/3)2E(e2πi/3) = (1− e2πi/3)∗E(e2πi/3)

and(1− e−2πi/3)2E(e−2πi/3) = (1− e−2πi/3)∗E(e−2πi/3).

This is true if and only if E(e2πi/3) = 0 and E(e−2πi/3) = 0, which is equivalent toσ(A) = {0, 1} and A is an orthogonal projection.

Remark A.2. We can give another proof for this theorem. Let H = R(A) ⊕N (A) andaccording to Theorem A.7 generalized projection A has representation

A =

[A1 00 0

]:

[R(A)N (A)

]→[R(A)N (A)

].

Then

I −A =

[IR(A) −A1 0

0 IN (A)

]and it is obvious that normality of A implies normality of I −A. Also,

(I −A)2 =

[(IR(A) −A1)2 0

0 IN (A)

]=

[(IR(A) −A1)∗ 0

0 IN (A)

]= (I −A)∗

holds if and only if (IR(A) −A1)2 = (IR(A) −A1)∗. Since A2 = A∗, we get

IR(A) − 2A1 +A21 = IR(A) − 2A1 +A∗1 = IR(A) −A∗1,

which is satisfied if and only if A1 = A∗1. Hence, A = A∗ = A2.

Theorem A.10If P is an orthogonal projection and A is a generalized projection, then AP is a general-ized projection if and only if either PA(I − P ) = 0 or (I − P )AP = 0.

Proof: LetR(P ) = L and H = L⊕ L⊥. Then

P =

[IL 00 0

], A =

[A1 A2

A3 A4

].

From

AP =

[A1 0A3 0

], (AP )2 =

[A2

1 0A3A1 0

], (AP )∗ =

[A∗1 A∗30 0

]we conclude that (AP )2 = (AP )∗ if and only if A3 = 0, which is equivalent to (I −P )AP = 0.

Theorem A.7 provide us with the condition that A3 = 0 if and only if A2 = 0, or, inthe other words (I − P )AP = 0 if and only if PA(I − P ) = 0.


Theorem A.11Let P be an orthogonal projection and let A be a generalized projection. Then P − A isa generalized projection if and only if A is an orthogonal projection commuting with P .

Furthermore, if P is an orthogonal projection and A is a generalized projection suchthat PAP is orthogonal projection and either PA(I − P ) = 0 or (I − P )AP = 0, thenP −A is a generalized projection.

Proof: Obviously (P −A)2 = P −PA−AP +A2 = P ∗−A∗ = (P −A)∗ if and onlyif PA = AP = A∗. Since A2 = A∗, we conclude that A is an orthogonal projection.

To prove the second part of the theorem, let R(P ) = L and H = L ⊕ L⊥. IfPA(I − P ) = 0 or (I − P )AP = 0, then

P =

[IL 00 0

], A =

[A1 00 A4

], P −A =

[Il −A1 0

0 −A4

].

From the orthogonality of PAP comes A1 = A21 = A∗1 and

(P −A)2 =

[(Il −A1)2 0

0 A24

]=

[(Il −A1)∗ 0

0 A∗4

]= (P −A)∗.

Theorem A.12Let P be an orthogonal projection and letA be a generalized projection. Then P +A is ageneralized projection if PAP = 0. Moreover, if P +A is a generalized projection, thenPAP = 0, PA(I − P ) = 0 and (I − P )AP = 0.

Proof: From (P+A)2 = P+PA+AP+A2 = P+A∗ we conclude thatAP = PA = 0.This is equivalent to

PA =

[A1 A2

0 0

]=

[A1 0A3 0

]= 0,

which holds if and only if A1 = A2 = A3 = 0. Thus, PAP = 0, PA(I − P ) = 0, and(I − P )AP = 0.

Conversely, if PAP = 0, then A1 = 0 and by Theorem A.7 A2 = 0, and A3 = 0.Clearly,

P +A =

[IL 00 A4

]is a generalized projection.

Theorem A.13If P is an orthogonal projection and A is a generalized projection, then AP − PA is ageneralized projection if and only if PA(I − P ) = 0 or (I − P )AP = 0.

Proof: The matrix representations of the operators A, P , and AP imply that

PA−AP =

[0 A2

−A3 0

],


and it is clear that

(PA−AP )2 =

[−A2A3 0

0 −A3A2

]=

[0 −A∗3A∗2 0

]= (PA−AP )∗

if and only ifA2 = A3 = 0 which is equivalent to PA(I−P ) = 0 or (I−P )AP = 0.

The next theorem is not new. Actually, it is proved for matrices A ∈ Cm×n by J.Gross and G. Trenkler in [6] and it appears again in [2].

Theorem A.14Let A,B ∈ GP(H). Then AB ∈ GP(H) if AB = BA.

Proof: If

AB =

[A1B1 A1B2

0 0

]=

[B1A1 0B3A1 0

]= BA,

it is clear that A1B1 = B1A1, B2 = 0 and B3 = 0. Form Theorem A.7 we conclude thatB∗1 = B2

1 , B∗4 = B24 , and

(AB)2 =

[(A1B1)2 0

0 0

]=

[(A1B1)∗ 0

0 0

]= (AB)∗.

Theorem A.15Let A,B ∈ GP(H). Then A+B ∈ GP(H) if and only if AB = BA = 0.

Proof: If A,B ∈ GP(H) have the representations given in Theorem A.7, then

A+B =

[A1 +B1 B2

B3 B4

]and if

(A+B)2 =

[(A1 +B1)2 +B2B3 (A1 +B1)B2B4

B3(A1 +B1) +B4B3 B3B2 +B24

]=

[(A1 +B1)∗ B∗3

B∗2 B∗4

]= (A+B)∗,

it is clear that

(A1 +B1)2 = A21 +A1B1 +B1A1 +B2

1 +B2B3 = (A1 +B1)∗.

Since B∗1 = B21 + B2B3, we get A1B1 + B1A1 = 0. Thus, B1 = 0 which implies

B2 = B3 = 0, B24 = B∗4 . In this case we obtain AB = BA = 0.

Conversely, ifAB = BA = 0, thenA1B1 = B1A1 = 0, implyingB1 = B2 = B3 =0, B2

4 = B∗4 , and obviously, (A+B)2 = (A+B)∗.

The next theorem gives an answer when the difference of two generalized projectionsis a generalized projection itself. It can be proved using partial ordering on GP(H), likein [6] or [2]. We prefer using only the basic properties of the generalized projections andtheir matrix representation.

4 Properties of products, differences, and sums of hypergeneralized projections 21

Theorem A.16Let A,B ∈ GP(H). Then A−B ∈ GP(H) if and only if AB = BA = B∗.

Proof: If A,B have the representations given in Theorem A.7, then

A−B =

[A1 −B1 −B2

−B3 −B4

].

From

(A−B)2 =

[(A1 −B1)2 +B2B3 −(A1 −B1)B2 +B2B4

−B3(A1 −B1) +B4B3 B3B2 +B24

]=

[(A1 −B1)∗ −B∗3−B∗2 −B∗4

]= (A−B)∗,

we get B2 = 0, B3 = 0, B24 = −B∗4 and from Theorem A.7 comes B2

4 = B∗4 . Now,B4 = 0 and

(A1 −B1)2 = A21 −A1B1 −B1A1 +B2

1 = A∗1 −B∗1

follows. This is true if and only if A1B1 = B1A1 = B∗1 , and in that case AB = BA =B∗.

Theorem A.17LetA andB be two commuting generalized projections. ThenA(I−B) ∈ GP(H) if andonly if ABA = (AB)∗.

Proof: Since AB = BA, we know that AB is a generalized projection. Now,

(A(I −B))2 = (A−AB)2 = A2 − 2ABA+ (AB)2

= A∗ − 2ABA+ (AB)∗ = (A(I −B))∗

if and only if ABA = (AB)∗.

Theorem A.18If A ∈ GP(H) and α ∈ {1, e2πi/3, e−2πi/3}, then αA ∈ GP(H).

Proof: Since (αA)3 = α3A3 = A3, then (αA)3|R(A) = IR(A) and αA is a normaloperator, which completes the proof.

4 Properties of products, differences, and sums ofhypergeneralized projections

For hypergeneralized projections we have results similar to those for generalized projec-tions. In some theorems we are not able to establish equivalency like the one we estab-lish for the generalized projections because we need additional conditions to ensure that(A+B)† = A† +B† and (A−B)† = A† −B†.

We start with properties of pair of one orthogonal and one hypergeneralized projec-tion.


Theorem A.19Let P ∈ B(H) be an orthogonal projection and let A ∈ HGP(H). Then AP is a hyper-generalized projection if and only if (I−P )AP = 0. Similarly, PA is a hypergeneralizedprojection if and only if PA(I − P ) = 0.

Proof: Let H = L⊕ L⊥, whereR(P ) = L. Then

A =

[A1 A2

A3 A4

], AP =

[A1 0A3 0

].

If

(AP )2 =

[(A1)2 0A3A1 0

]=

[D†A∗1 D†A∗3

0 0

]= (PA)†,

then A3 = 0, which is equivalent to (I − P )AP = 0.Conversely, if (I − P )AP = 0 i.e. A3 = 0, then A has matrix form

A =

[A1 A2

0 A4

], A2 =

[A2

1 A1A2 +A2A4

0 A24

],

and it is easy to see that

A† =

[A†1 (2I −A1)†(I −A1)†A2A

†4

0 A†4

].

Since A is a hypergeneralized projection, it is clear that A21 = A†1 and consequently

(AP )2 =

[A2

1 00 0

]=

[A†1 00 0

]= (AP )†.

The following example shows that Theorem A.9 does not hold for hypergeneralizedprojections.

Example A.2

Let H = C2 and A =

[1 1

0 e2πi3

]. Then A2 =

[1 1 + e

2πi3

0 e−2πi

3

], A3 = IR(A),

A4 = A and A is a hypergeneralized projection. However, I − A =

[0 −1

0 1− e 2πi3

]and it is not normal.

Theorem A.20Let A,B ∈ HGP(H). If AB = BA, then AB ∈ HGP(H).

References 23

Proof: Let H = R(A)⊕N (A) and A,B ∈ HGP(H) have representations

A =

[A1 00 0

], B =

[B1 B2

B3 B4

].

Then

AB =

[A1B1 A1B2

0 0

], (AB)2 =

[A1B1A1B1 A1B1A1B2

0 0

].

A straightforward calculation using formula A† = A∗(AA∗)† shows that

(AB)† =

[(A1B1)∗D−1 0(A1B2)∗D−1 0

],

where D = A1B1(A1B1)∗ +A1B2(A1B2)∗ > 0 is invertible. Assume that hypergener-alized projections A,B commute, i.e., that

AB =

[A1B1 A1B2

0 0

]=

[B1A1 0B3A1 0

]= BA.

This implies B2 = 0, B3 = 0, A1B1 = B1A1 and it is easy to see that (AB)2 =(AB)†.

Theorem A.21Let A,B ∈ HGP(H). If AB = BA = 0, then A+B ∈ HGP(H).

Proof: Let H = R(A)⊕N (A). Then

A =

[A1 00 0

], B =

[B1 B2

B3 B4

].

From these matrix representations it is easy to see that AB = BA = 0 implies B1 =B2 = B3 = 0 and B2

4 = B†4. Now,

(A+B)2 = A2 +B2 = A† +B† = (A+B)†.

Theorem A.22Let A,B ∈ HGP(H). If AB = BA = B2, then A−B ∈ HGP(H).

Proof: Let H = R(A)⊕N (A). Then

A =

[A1 00 0

], B =

[B1 B2

B3 B4

].

From the condition AB = BA = B2, we get A1B1 = B1A1 = B21 , B2 = B3 = 0,

B24 = B†4, which implies (A−B)† = A† −B†. Hence

(A−B)2 = A2 −AB −BA+B2 = A2 −B2 = A† −B† = (A−B)†.


References

[1] J.K. Baksalary and X. Liu, An alternative characterization of generalized pro-jectors, Linear Algebra and its Applications 388 (2004) 61-65

[2] J. K. Baksalary, O. M. Baksalary, X. Liu and G. Trenkler, Further results ongeneralized and hypergeneralized projectors, Linear Algebra and its Applica-tions 429 (2008) 1038-1050

[3] J. K. Baksalary, O. M. Baksalary and X. Liu, Further properties on general-ized and hypergeneralized projectors, Linear Algebra and its Applications 389(2004) 295303

[4] H. Du and Y. Li, The spectral characterization of generalized projections, Lin-ear Algebra and its Applications 400 (2005) 313-318

[5] D. S. Djordjevic and J. Koliha, Characterizing hermitian, normal and EP oper-ators, Filomat 21:1 (2007) 39-54

[6] J. Gross and G. Trenkler, Generalized and hypergeneralized projectors, LinearAlgebra and its Applications 264 (1997) 463-474

Paper B

On the Moore-Penrose and the Drazininverse of two projections on a Hilbert

space

Authors: Sonja Radosavljevic and Dragan S. Djordjevic

Preliminary version published as a technical report LiTH-MAT-R–2012/03–SE, Depart-ment of Mathematics, Linköping University.

25

On the Moore-Penrose and the Drazin inverse oftwo projections on a Hilbert space

Sonja Radosavljevic∗ and Dragan S. Djordjevic†

∗Department of Mathematics,Linköping University,

SE–581 83 Linköping, Sweden.E-mail: [email protected]

†Faculty of Sciences and Mathematics,University of Nis, P.O.Box 224, 18000 Nis, Serbia

E-mail: [email protected]

Abstract

For two given orthogonal, generalized or hypergeneralized projections P andQ on a Hilbert space H , we give their matrix representation. We also givecanonical forms of the Moore-Penrose and the Drazin inverses of their prod-ucts, differences, and sums. In addition, it is showed when these operatorsare EP operators and some simple relations between the mentioned operatorsare established.

Keywords: Orthogonal projection, generalized projection, hypergeneralizedprojection, Moore-Penrose inverse, Drazin inverse

27

28Paper B

On the Moore-Penrose and the Drazin inverse of two projections on a Hilbert space

1 Introduction

Motivation for writing this paper comes from publicatons of C. Deng and Y. Wei ([5],[6]) and O. M. Baksalary and G. Trenkler, ([1], [2]). Namely, Deng and Wei studiedDrazin invertibility for products, differences, and sums of idempotents and Baksalaryand Trenkler used matrix representation of the Moore-Penrose inverse of products, differ-ences, and sums of orthogonal projections. Our main goal is to give canonical form of theMoore-Penrose and the Drazin inverse for products, differences, and sums of two orthog-onal, generalized or hypergeneralized projections on an arbitrary Hilbert space. Usingthe canonical forms, we examine when the Moore-Penrose and the Drazin inverese exist.Also, we describe the relation between inverses (if any), estimate the Drazin index andestablish necessary and sufficient conditions under which these operators are EP opera-tors. Although some of the results are the same or similar to the results in the mentionedpapers, there are differences since we use generalized and hypergeneralized pojectons,and not only orthogonal projections. We also examine different properties.

Throughout the paper, H stands for the Hilbert space and L(H) stands for set of allbounded linear operators on space H . The symbols R(A), N (A) and A∗ denote range,null space and adjoint operator of the operator A ∈ L(H).

Operator P ∈ L(H) is an idempotent if P = P 2 and it is an orthogonal projection ifP = P 2 = P ∗.

Generalized and hypergeneralized projections were inroduced in [7] by J. Gross andG. Trenkler.

Definition B.1. Operator G ∈ L(H) is

(a) a generalized projection if G2 = G∗,

(b) a hypergeneralized projection if G2 = G†.

The set of all generalized projecton on H is denoted by GP(H) and the set of all hyper-generalized projecton is denoted byHGP(H).

Here A† is the Moore-Penrose inverse of A ∈ L(H), i.e., the unique solution to theequations


Notice thatA† exists if and only ifR(A) is closed. ThenAA† is the orthogonal projectionontoR(A) parallel toN (A∗), and A†A is the orthogonal projection ontoR(A∗) paralleltoN (A). Consequently, I −AA† is the orthogonal projection ontoN (A∗) and I −A†Ais the orthogonal projection onto N (A). An essential property of any P ∈ L(H) is thatP is an orthogonal projection if and only if it is expressible as AA†, for some A ∈ L(H).

For A ∈ L(H), an element B ∈ L(H) is the Drazin inverse of A, if the followinghold:

BAB = B, BA = AB, An+1B = An,

for some non-negative integer n. The smallest such n is called the Drazin index of A. ByAD we denote the Drazin inverse of A and by ind(A) we denote Drazin index of A.

If such n does not exist, ind(A) =∞ and operatorA is generalized Drazin invertible.Its invers is denoted by Ad.

2 Auxiliary results 29

Operator A is invertible if and only if ind(A) = 0.If ind(A) ≤ 1,A is group invertible andAD is group inverse, usually denoted byA#.Notice that if the Drazin inverse exists, it is unique. Operator A ∈ L(H) is Drazin

invertible if and only if asc(A) < ∞ and dsc(A) < ∞, where asc(A) is the mini-mal integer such that N (An+1) = N (An) and dsc(A) is the minimal integer such thatR(An+1) = R(An). In this case, ind(A) = asc(A) = dsc(A) = n.

Recall that ifR(An) is closed for some integer n, then asc(A) = dsc(A) <∞.Operator A ∈ L(H) is EP operator if AA† = A†A, or, in the other words, if A† =

AD = A#. There are many characterization of EP operators. In this paper, we use resultsfrom D. Djordjevic and J. Koliha, (see [4]).

In what follows, A stands for I −A and PA stands for AA†.

2 Auxiliary results

Let P,Q ∈ L(H) be orthogonal projectons and R(P ) = L. Since H = R(P ) ⊕R(P )⊥ = L⊕ L⊥, we have the following representaton of the projections P, P ,Q,Q ∈L(H) with respect to the decomposition of space:

P =

[P1 00 0

]=

[IL 00 0

]:

[LL⊥

]→[

LL⊥

], (1)

P =

[0 00 IL⊥

]:

[LL⊥

]→[

LL⊥

], (2)

Q =

[A BB∗ D

]:

[LL⊥

]→[

LL⊥

], (3)

Q =

[IL −A −B−B∗ IL⊥ −D

]:

[LL⊥

]→[

LL⊥

], (4)

with A ∈ L(L) and D ∈ L(L⊥) being self adjoint and non-negative.The next two theorems are known for matrices on Cn, (see [2]).

Theorem B.1Let Q ∈ L(H) be represented as in (3). Then the following holds:

(a) A = A2 +BB∗, or, equivalently, AA = BB∗,

(b) B = AB +BD, or, equivalently, B∗ = B∗A+DB∗,

(c) D = D2 +B∗B, or, equivalently, DD = B∗B.

Proof: Since Q = Q2, we obtain[A BB∗ D

] [A BB∗ D

]=

[A2 +BB∗ AB +BDB∗A+DB∗ B∗B +D2

]=

[A BB∗ D

]implying that A = A2 +BB∗, B = AB +BD and D = D2 +B∗B.

30Paper B


Theorem B.2Let Q ∈ L(H) be represented as in (3). Then:

(a) R(B) ⊆ R(A),

(b) R(B) ⊆ R(A),

(c) R(B∗) ⊆ R(D),

(d) R(B∗) ⊆ R(D),

(e) A†B = BD†,

(f) A†B = BD†,

(g) A is a contraction,

(h) D is a contraction,

(i) A−BD†B∗ = IL −AA†.

Proof: (a) Since A = A2 +BB∗, we have

R(A) = R(A2 +BB∗) = R(AA∗ +BB∗).

To prove thatR(AA∗ +BB∗) = R(A) +R(B), observe the operator matrix

M =

[A B0 0

].

For any x ∈ R(MM∗), there exists y ∈ H such that x = MM∗y = M(M∗y) andx ∈ R(M). On the other hand, for x ∈ R(M), there is y ∈ H and x = My. Besides,MM†x = MM†My = My = x and MM† = MM∗(MM∗)† = PR(MM∗) implyingx ∈ R(MM∗). Hence,R(M) = R(MM∗) and

R(A) +R(B) = R(M) = R(MM∗) = R(AA∗ +BB∗),

and we haveR(A) = R(A) +R(B),

implyingR(B) ⊆ R(A).(b) Since A = I −A, from Theorem B.1 (a), we get A = A

2+BB∗. The rest of the

proof is analogously to the point (a) of this theorem.(c), (d) Similarly.(e) Since B = AB + BD, we have A†B = A†(AB + BD) = A†AB + A†BD,

and using the facts that A†A = PR(A∗) and R(B) ⊆ R(A∗), we get A†AB = B and

A†B = B + A†BD, or, equivalently B = A†BD. Postmultiplying this equation by D†

and using item (d) of this Theorem, in its equivalent form BDD†

= B, we obtain (e).(f) Analogously to the previous proof.

2 Auxiliary results 31

(g) Since A = A∗, from Theorem A.1 (a), we have that

IL −AA∗ = IL − (A−BB∗) = A+BB∗,

and the right hand side is nonnegative as a sum of two nonnegative operators implyingthat A is a contraction.

(h) This part of the proof is dual to the part (g).(i) From Theorem B.1 (a), item (f) of this theorem and the fact that self adjoint oper-

ator A commutes with its MP-inverse, it follows that

BD†B∗ = A†BB∗ = A

†AA = A

†(I −A)A = A

†A−A†AA = A

†A−A,

by taking into account that AA†

= A†A. Now we get

A−BD†B∗ = I −A†A,

establishing the condition.

Following the results of J. Gross and G. Trenkler for matrices, we formulate a fewtheorems for the generalized and hypergeneralized projections on arbitrary Hilbert space.We start with the result which is very similar to Theorem (1) in [7].

Theorem B.3Let G ∈ L(H) be a generalized projection. Then G is a closed range operator and G3 isan orthogonal projection onR(G). Moreover, H has the decomposition

H = R(G)⊕N (G)

and G has the following matrix representaton

G =

[G1 00 0

]:

[R(G)N (G)

]→[R(G)N (G)

],

where the restriction G1 = G|R(G) is unitary onR(G).

Proof: If G is a generalized projection, then G4 = (G2)2 = (G∗)2 = (G2)∗ = (G∗)∗ =G. From GG∗G = G4 = G, it follows that G is a partial isometry implying that

G3 = GG∗ = PR(G),

G3 = G∗G = PN (G)⊥ .

Thus, G3 is the orthogonal projection onto R(G) = N (G)⊥ = R(G∗). Consequently,R(G) is a closed subset in H as a range of an orthogonal projection on a Hilbert space.From Lemma (1.2) in [4] we get the following decomposition of the space

H = R(G∗)⊕N (G) = R(G)⊕N (G).

Now, G has the following matrix representation in accordance with this decomposition:

G =

[G1 00 0

]:

[R(G)N (G)

]→[R(G)N (G)

],

where G21 = G∗1, G4

1 = G1 and G1G∗1 = G∗1G1 = G3

1 = IR(G).

32Paper B


Theorem B.4Let G,H ∈ GP(H) and H = R(G) ⊕ N (G). Then G and H has the following repre-sentations with respect to the decomposition of the space:

G =

[G1 00 0

]:

[R(G)N (G)

]→[R(G)N (G)

],

H =

[H1 H2

H3 H4

]:

[R(G)N (G)

]→[R(G)N (G)

],

where

H∗1 = H21 +H2H3,

H∗2 = H3H1 +H4H3,

H∗3 = H1H2 +H2H4,

H∗4 = H3H2 +H24 .

Furthermore, H2 = 0 if and only if H3 = 0.

Proof: Let H = R(G)⊕N (G). Then representation of G follows from Theorem (1) in[7], and let H has the representaton

H =

[H1 H2

H3 H4

].

Then, from

H2 =

[H2

1 +H2H3 H1H2 +H2H4

H3H1 +H4H3 H3H2 +H24

]=

[H∗1 H∗3H∗2 H∗4

]= H∗,

conclusion follows directly.If H2 = 0, then H∗3 = H1H2 + H2H4 = 0 and H3 = 0. Analogously, H3 = 0

implies H2 = 0.

Theorem B.5Let G ∈ L(H) be a hypergeneralized projection. Then G is a closed range operator andH has the decomposition

H = R(G)⊕N (G).

Also, G has the following matrix representaton with respect to the decomposition of thespace

G =

[G1 00 0

]:

[R(G)N (G)

]→[R(G)N (G)

],

where the restriction G1 = G|R(G) satisfies G31 = IR(G).

Proof: IfG is a hypergeneralized projection, thenG andG† commute andG is EP. UsingLemma (1.2) in [4], we get decomposition of the space H = R(G) ⊕N (G), and G hasthe required representation.

3 The Moore-Penrose and the Drazin inverse of two orthogonal projections 33

3 The Moore-Penrose and the Drazin inverse of twoorthogonal projections

We start this secton with theorem which gives the matrix representation of the Moore-Penrose inverse of products, differences, and sums of orthogonal projections.

Theorem B.6Let orthogonal projections P,Q ∈ L(H) be represented as in (1) and (2). Then theMoore-Penrose inverse of PQ, P −Q and P +Q exists and the following holds:

(a) (PQ)† =

[AA† 0B∗A† 0

]:

[LL⊥

]→[

LL⊥

]and

R(PQ) = R(A)

(b) (P −Q)† =

[AA

† −BD†

−B∗A† −DD†

]:

[LL⊥

]→[

LL⊥

]and

R(P −Q) = R(A)⊕R(D)

(c) (P +Q)† =

[12 (I +AA

†) −BD†

−D†B∗ 2D† −DD†

]:

[LL⊥

]→[

LL⊥

]and

R(P +Q) = L⊕R(D).

Proof: (a) Using representatons (1) and (3) for orthogonal projections P,Q ∈ L(H), thewell known Harte-Mbekhta formula (PQ)† = (PQ)∗(PQ(PQ)∗)† and Theorem B.1(a),we obtain

(PQ)† =

[A 0B∗ 0

] [A2 +BB∗ 0

0 0

]†=

[AA† 0B∗A† 0

].

From PQ(PQ)† = PR(PQ), we obtain

(PQ)(PQ)† =

[A B0 0

] [AA† 0B∗A† 0

]=

[AA† 0

0 0

],

or, in the other words,R(PQ) = R(A).(b) Similarly to part (a), we can calculate the Moore-Penrose inverse of P − Q as

follows

(P −Q)† = (P −Q)∗((P −Q)(P −Q)∗)†

=

[A −B−B∗ −D

] [A

2+BB∗ −AB +BD

−B∗A+DB∗ B∗B +D2

]†

=

[A −B−B∗ −D

] [A†

00 D†

]

=

[AA

† −BD†

−B∗A† −DD†

].

34Paper B


For the range of P −Q we have

PR(P−Q) = (P −Q)(P −Q)†

=

[AAA

†+BD†B∗ −ABD† +BDD†

−B∗AA† +DD†B∗ B∗BD† +DDD†

]

=

[AA

†0

0 DD†

],

implyingR(P −Q) = R(A)⊕R(D).

(c) The Moore-Penrose inverse of P +Q has the following representation with the respectto decomposition of the space:

(P +Q)† =

[X1 X2

X3 X4

]:

[LL⊥

]→[

LL⊥

].

In order to calculate (P + Q)†, we use the Moore-Penrose equations. From the firstMoore-Penrose equation, (P +Q)(P +Q)†(P +Q) = P +Q, we have

((I +A)X1 +BX3)(I +A) + ((I +A)X2 +BX4)B∗ = I +A,

((I +A)X1 +BX3)B + ((I +A)X2 +BX4)D = B,

(B∗X1 +DX3)(I +A) + (B∗X2 +DX4)B∗ = B∗,

(B∗X1 +DX3)B + (B∗X2 +DX4)D = D.

The second Moore-Penrose equation, (P +Q)†(P +Q)(P +Q)† = (P +Q)†, implies

(X1(I +A) +X2B∗)X1 + (X1B +X2D)X3 = X1,

(X1(I +A) +X2B∗)X2 + (X1B +X2D)X4 = X2,

(X3(I +A) +X4B∗)X1 + (X3B +X4D)X3 = X3,

(X3(I +A) +X4B∗)X2 + (X3B +X4D)X4 = X4,

while the third and fourth Moore-Penrose equations, ((P+Q)(P+Q)†)∗ = (P+Q)(P+Q)† and ((P +Q)†(P +Q))∗ = (P +Q)†(P +Q), give X3 = X∗2 . Further calculationsshow that

(I +A)X1 +BX∗2 = IL,

(I +A)X2 +BX4 = 0,

B∗X1 +DX∗2 = 0,

B∗X2 +DX4 = DD†.

According to Theorem B.2 (b), (c), from B∗X1 +DX∗2 = 0 we get D†B∗X1 +X2 = 0,or equivalently, X∗2 = −D†B∗X1.


From (I + A)X1 + BX∗2 = IL and Theorem B.2 (i), we get (2I − AA†)X1 = IL

i.e. X1 = (2I − AA†)−1 = 12 (I + AA

†). Theorem B.1 (c) and B∗X2 +DX4 = DD†

imply −B∗BD† + DX4 = DD†. Finally, we have X2 = −BD†, X3 = −D†B∗,X4 = 2D† −DD† and

(P +Q)† =

[12 (I +AA

†) −BD†

−D†B∗ 2D† −DD†

].

Like in the proof of part (b) of this theorem,

PR(P+Q) = (P +Q)(P +Q)†

=

[12 (I +A)(I +AA

†)−BD†B∗ −(I +A)BD† + 2BD† −BDD†

12B∗(I +AA

†)−DD†B∗ −B∗BD† + 2DD† −DDD†

]

=

[IL 00 DD†

],

which assertsR(P +Q) = L⊕R(D).

To prove the existence of the Moore-Penrose inverse of PQ, P − Q and P + Q, it issufficient to prove that these operators have closed range. Since Q is the orthogonalprojection,R(Q) is closed subset of H . Also,

R(Q) = Q(H) =

[A BB∗ D

] [LL⊥

]=

[R(A) +R(B)R(B∗) +R(D)

]= R(A) +R(D),

because items (a), (c) of Theorem A.1 state that R(B) ⊆ R(A) and R(B∗) ⊆ R(D).This implies thatR(A) andR(D) are closed subsets of L and L⊥, respectively. IfR(A)is closed, then for every sequence (xn) ⊆ L, xn → x and Axn → y imply x ∈ L andAx = y. Now, (I −A)xn → x− y and x− y ∈ L, (I −A)x = x− y which proves thatR(I − A) is closed. Consequently, R(PQ), R(I − A) and R(I + A) are closed whichcompletes the proof.

Similarly to Theorem 3.1 in [6], we have the following result.

Theorem B.7Let orthogonal projections P,Q ∈ L(H) be represented as in (1) and (3). Then theDrazin inverses of PQ, P −Q and P +Q exist, P −Q and P +Q are EP operators andthe following holds:

(a) (PQ)D =

[AD (AD)2B0 0

]and ind(PQ) ≤ ind(A) + 1,

(b) (P −Q)D = (P −Q)† and ind(P −Q) ≤ 1,

(c) (P +Q)D = (P +Q)† and ind(P +Q) ≤ 1.

36Paper B


Proof: (a) Theorem B.6 proves that R(PQ) is the closed subset of H . Thus, the Drazininverse for this operators exists. According to representations (1) and (3) of projections Pand Q, their product PQ and the Drazin inverse (PQ)D can be written in the followingway:

PQ =

[A B0 0

], (PQ)D =

[X1 X2

X3 X4

]:

[LL⊥

]→[

LL⊥

].

Equations that describe Drazin inverse are

(PQ)DPQ(PQ)D =

[X1AX1 X1AX2

X3AX1 X3AX2

]=

[X1 X2

X3 X4

]= (PQ)D,

(PQ)DPQ =

[X1A X1BX3A X3B

]=

[AX1 AX2

0 0

]= PQ(PQ)D,

(PQ)n+1(PQ)D =

[An+1X1 An+1X2

0 0

]=

[An An−1B0 0

]= (PQ)n.

Thus, from the first equation we have

X1AX1 = X1, X1AX2 = X2, X3AX1 = X3, X3AX2 = X4,

from the second equation

X1A = AX1, AX2 = X1B, X3A = 0, X3B = 0,

and the third equation implies

An+1X1 = An, An+1X2 = An−1B.

It is easy to conclude that X1 = AD, X3 = 0, X4 = 0. Equations X1AX2 = X2 andAX2 = X1B give X2

1B = X2. Finally,

(PQ)D =

[AD (AD)2B0 0

].

To estimate the Drazin index of PQ, suppose that ind(A) = n. Then

(PQ)n+2(PQ)D =

[An+2 An+1B

0 0

] [AD (AD)2B0 0

]=

[An+1 An+1ADB

0 0

]=

[An+1 AnB

0 0

]= (PQ)n+1

implying that ind(PQ) ≤ ind(A) + 1.(b) Since (P −Q)(P −Q)∗ = (P −Q)∗(P −Q) andR(P −Q) = R(A)⊕R(D)

is closed , P − Q is the EP operator as a normal operator with the closed range and(P −Q)† = (P −Q)D. Besides,

(P −Q)2(P −Q)D = (P −Q)(P −Q)†(P −Q) = P −Q

and ind(P −Q) ≤ 1.(c) Similarly to (b), P +Q is EP operator and (P +Q)† = (P +Q)D, ind(P +Q) ≤

1.


Theorem B.8Let orthogonal projections P,Q ∈ L(H) be represented as in (1) and (3). Then thefollowing holds:

(a) If PQ = QP or PQP = PQ, then

(P +Q)D =

[IL − 1

2A 00 D

], (P −Q)D =

[A 00 −D

].

(b) If PQP = P , then

(P +Q)D =

[12IL 00 D

], (P −Q)D =

[0 00 −D

].

(c) If PQP = Q, then

(P +Q)D =

[IL − 1

2A 00 0

], (P −Q)D =

[A 00 0

]= P −Q.

(d) If PQP = 0, then

(P +Q)D =

[IL 00 D

]= P +Q, (P −Q)D =

[IL 00 −D

]= P −Q.

Proof: Let

(P +Q)D =

[X1 X2

X3 X4

]:

[LL⊥

]→[

LL⊥

].

(a) If

PQ =

[A B0 0

]=

[A 0B∗ 0

]= QP

or

PQP =

[A 00 0

]=

[A B0 0

]= PQ,

then B = B∗ = 0, IL + A is invertible and (IL + A)−1 = IL − 12A and according to

Theorem B.1 (c), D = D2. Thus, we can write

Q =

[A 00 D

], P +Q =

[IL +A 0

0 D

], (P +Q)n =

[(IL +A)n 0

0 D

].

Verifying the equation

(P +Q)2(P +Q)D =

[(IL +A)2X1 (IL +A)2X2

DX3 DX4

]=

[IL +A 0

0 D

]= P +Q

we getX2 = X3 = 0, DX4 = D.

38Paper B


The other two equations, (P +Q)D(P +Q)(P +Q)D = (P +Q)D and (P +Q)D(P +Q) = (P +Q)(P +Q)D, give

X4DX4 = X4, X4D = DX4

i.e. X4 = D. Thus,

(P +Q)D =

[IL − 1

2A 00 D

].

Formula

(P −Q)D =

[A 00 −D

]follows form Theorem B.7 (b) and the fact that A = A2 implies A

D= A

2= A.

(b) If PQP = P , then A = IL and Theorem B.1 implies B = B∗ = 0. Then,

Q =

[IL 00 D

]and from part (a) of this Theorem we conclude

(P +Q)D =

[12IL 00 D

], (P −Q)D =

[0 00 −D

].

(c) From PQP = Q we get B = B∗ = D = 0 and A = A2. Now, IL + A isinvertible and

(P +Q)D = (P +Q)−1 =

[(IL +A)−1 0

0 0

]=

[IL − 1

2A 00 0

]and

(P −Q)D =

[A 00 0

].

(d) If PQP = 0, then A = 0 and since R(B) ⊆ R(A), we conclude B = B∗ = 0.In this case,

Q =

[0 00 D

], P +Q =

[IL 00 D

]implying

(P +Q)D = P +Q =

[IL 00 D

], (P −Q)D = P −Q =

[IL 00 −D

].

Theorem B.9Let orthogonal projections P,Q ∈ L(H) be represented as in (1) and (3). Then

(PQ)D = (QP )†(PQ)†(QP )†.

Moreover, if PQ = QP , then PQ is the EP operator and

(PQ)D = (PQ)†, ind(PQ) ≤ 1.

4 The MP and the Drazin inverse of the generalized and hypergeneralized projections 39

Proof: Corollary 5.2 in [8] states that (PQ)† is idempotent for every orthogonal projec-tions P and Q. Thus we can write

(PQ)† =

[I 0K 0

]and

PQ = (PQ)†† =

[(I +K∗K)−1 (I +K∗K)−1K∗

0 0

].

Denote by A = (I +K∗K)−1 and B = (I +K∗K)−1K∗ = AK∗. Then

PQ =

[A B0 0

]and according to Theorem B.8 (a),

(PQ)D =

[AD (AD)2B0 0

]=

[I +K∗K (I +K∗K)2(I +K∗K)−1K∗

0 0

]=

[I +K∗K (I +K∗K)K∗

0 0

]=

[I +K∗K 0

0 0

] [I K∗

0 0

]=

[I K∗

0 0

] [I 0K 0

] [I K∗

0 0

]= (QP )†(PQ)†(QP )†,

where we used ((PQ)†)∗ = ((PQ)∗)† = (QP )†.If P and Q commute, then PQ is the normal operator with the closed range, which

means that it is an EP operator and (PQ)† = (PQ)D.

4 The Moore-Penrose and the Drazin inverse of thegeneralized and hypergeneralized projections

Some of the results obtained in the previous section we extend to generalized and hyper-generalized projections.

Theorem B.10Let G,H ∈ L(H) be two generalized or hypergeneralized projections. Then the Moore-Penrose inverse of GH exists and has the following matrix representation

(GH)† =

[(G1H1)∗D−1 0(G1H2)∗D−1 0

],

where D = G1H1(G1H1)∗ +G1H2(G1H2)∗ > 0 is invertible.

40Paper B


Proof: From Theorems B.3, B.4 and B.5, we see thatR(G) = R(G1) is closed and pairof generalized or hypergeneralized projections has the matrix form

G =

[G1 00 0

], H =

[H1 H2

H3 H4

].

Then,

GH =

[G1H1 G1H2

0 0

]and analogously to the proof of Theorem B.6 (a), we obtain the mentioned matrix form.SinceR(GH) = R(G1) is closed, the Moore-Penrose inverse (GH)† exists.

Theorem B.11Let G,H ∈ L(H) be two generalized or hypergeneralized projections. Then the Drazininverse of GH exists and has the following matrix representation

(GH)D =

[(G1H1)D ((G1H1)D)2G1H2

0 0

].

Proof: Similarly to the proof of Theorem B.7 (a) and using Theorem A.18.

Theorem B.12Let G,H ∈ L(H) be two generalized projections.

(a) If GH = HG, then GH is EP operator and

(GH)† = (GH)D = (GH)∗ = (GH)2 = (GH)−1,

(GH)† =

[(G1H1)−1 0

0 0

].

(b) If GH = HG = 0, then G+H is EP operator and

(G+H)† = (G+H)D = (GH)∗ = (G+H)2 = (G+H)−1,

(G+H)† =

[G−11 0

0 H−14

].

(c) If GH = HG = H∗, then G−H is EP operator and

(G−H)† = (G−H)D = (GH)∗ = (G−H)2 = (G−H)−1,

(G−H)† =

[(G1 −H1)−1 0

0 0

].

Proof: (a) If G,H ∈ L(H) are two commuting generalized projections, then from

(GH)∗ = H∗G∗ = H2G2 = (HG)2 = (GH)2

4 The MP and the Drazin inverse of the generalized and hypergeneralized projections 41

we conclude that GH is also generalized projection, and therefore EP operator. Checkingthe Moore-Penrose equations for (GH)2, we see that they hold. From the uniqueness ofthe Moore-Penrose inverse follows (GH)2 = (GH)† and

(GH)† = (GH)D = (GH)2

From GH(GH)† = PR(GH), using matrix form of GH , we get G1H1(G1H1)† = I , orequivalently, (G1H1)† = (G1H1)−1. Finally,

(GH)† = (GH)D = (GH)∗ = (GH)2 = (GH)−1.

(b) If GH = HG = 0, then (G + H)2 = G2 + H2 = G∗ + H∗ = (G + H)∗ andG+H is a generalized projection. The rest of the proof is similar to part (a).

(c) If GH = HG = H∗, then (G−H)2 = G2 −H2 = G∗ −H∗ = (G−H)∗ andthe rest of the proof is similar to part (a).

Matrix representations are easily obtained by using canonical forms ofG andH givenin Theorem B.4.

Theorem B.13Let G,H ∈ L(H) be two hypergeneralized projections.

(a) If GH = HG, then GH is EP operator and

(GH)† = (GH)D = (GH)2 = (GH)−1,

(GH)† =

[(G1H1)−1 0

0 0

].

(b) If GH = HG = 0, then G+H is EP operator and

(G+H)† = (G+H)D = (G+H)2 = (G+H)−1,

(G+H)† =

[G−11 0

0 H−14

].

(c) If GH = HG = H∗, then G−H is EP operator and

(G−H)† = (G−H)D = (G−H)2 = (G−H)−1,

(G−H)† =

[(G1 −H1)−1 0

0 0

].

Proof: (a) If GH = HG, then GH is an EP operator and (GH)4 = GH , so it is ahypergeneralized projection. Since (GH)2 = (GH)†, operator GH commutes with itsMoore-Penrose inverse and (GH)† = (GH)D. From

GH(GH)† =

[I 00 0

],

42Paper B


follows (GH)† = (GH)−1. Thus,

(GH)† = (GH)D = (GH)2 = (GH)−1.

(b) If GH = HG = 0, then (G+H)2 = (G+H)† and H1 = H2 = H3 = 0 implies

G+H =

[G1 00 H4

], (G+H)† =

[G†1 0

0 H†4

].

From (G+H)(G+H)† = PR(G+H) = PR(G) + PR(H) and

(G+H)(G+H)† =

[G1G

†1 0

0 H4H†4

]=

[I 00 I

]we conclude that G†1 = G−11 , H†4 = H−14 and (G+H)† = (G+H)−1.

(c) Similarly to (b).

References

[1] O. M. Baskalary and G. Trenkler, Column space equalities for orthogonal projectors,Applied Mathematics and Computations, 212 (2009) 519-529

[2] O. M. Baskalary and G. Trenkler, Revisitaton of the product of two orthogonal pro-jectors, Linear Algebra Appl. 430 (2009), 2813-2833.

[3] D. S. Djordjevic and V. Rakocevic, Lectures on generalized inverses, Faculty of Sci-ences and Mathematics, University of Niš, 2008.

[4] D. S. Djordjevic and J. Koliha, Characterizing hermitian, normal and EP operators,Filomat 21:1 (2007) 39-54

[5] C. Deng, The Drazin inverses of sum and difference of idempotents, Linear AlgebraAppl. 430 (2009) 1282-1291

[6] C. Deng and Y. Wei, Characterizations and representations of the Drazin inverseinvolving idempotents, Linear Algebra Appl. 431 (2009) 1526-1538

[7] J. Gross and G. Trenkler, Generalized and hypergeneralized projectors, Linear Alge-bra and its Applications 264 (1997) 463-474

[8] J. J. Koliha and V. Rakocevic, Range projections and the Moore-Penrose inverse inrings with involution

pairs of projections on a hilbert space: properties...

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