Linköping Studies in Science and Technology. Thesis.
No. 1525
Pairs of projections on a Hilbert
space: properties and generalized
invertibility
Sonja Radosavljevi ´c
Department of Mathematics
Linköping 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 Radosavljevi´c
mai-id@liu.se www.mai.liu.se
Mathematics and Applied Mathematics Department of Mathematics
Linköping University SE–581 83 Linköping
Sweden
ISBN 978-91-7519-932-0 ISSN 0280-7971
Copyright c 2012 Sonja Radosavljevi´c
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 the Moore-Penrose and the Drazin inverse for pairs of operators. We use reasoning similar to one presented in the famous P. Halmos’ two projection theorem: using matrix represen-tation of two orthogonal projection depending on the relations between their ranges and null-spaces gives us simpler form of their matrices and allows us to involve matrix theory in solving problems. We extend research to idempotents, generalized and hypergeneral-ized projections and their combinations.
Acknowledgments
I would like to thank my supervisors, professors Vladimir Kozlov, Bengt-Ove Turesson and Uno Wennergren, for encouragement and guidance they have been giving since the first moment that I spent at LiU. Their support was invaluable for me in many ways.
I am also grateful to professor Dragan S. Djordjevi´c, 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 making MAI great working place.
To three people at home (to say nothing of the dog), I owe gratitude for understaning and support.
Linköping, March 12, 2012 Sonja Radosavljevi´c
Contents
Introduction . . . 1 Bibliography . . . 5
A On pairs of generalized and hypergeneralized projections on a Hilbert space 7 1 Introduction . . . 10 2 Characterization of generalized and hypergeneralized projections . . . 11 3 Properties of products, differences, and sums of generalized projections . 15 4 Properties of products, differences, and sums of hypergeneralized
projec-tions . . . 21 References . . . 23 B On the Moore-Penrose and the Drazin inverse of two projections on a Hilbert
space 25
1 Introduction . . . 28 2 Auxiliary results . . . 29 3 The Moore-Penrose and the Drazin inverse of two orthogonal projections 33 4 The MP and the Drazin inverse of the generalized and hypergeneralized
projections . . . 39 References . . . 42
Introduction
In this thesis, we want to examine some properties of sums, differences, and products of two operators on Hilbert space. We start with the orthogonal projections, i.e., operators such that P = P2 = P∗, but we also extend research to idempotent operator P (which
satisfy P = P2), generalized projections (for which P∗ = P2) and hypergeneralized
projections (satisfying P† = P2, where P† is the Moore-Penrose inverse of operator
P ). The method comes from P. Halmos’ two projection theorem, (see [16]), stating that for two orthogonal projections P and Q on a finite dimensional Hilbert space H there exists suitable matrix representation in accordance to the relations between their ranges and null-spaces. We generalize this approach to the wider classes of operators and also to infinite dimensional Hilbert spaces.
As we know, every Hilbert space can be represented as a direct sum of two orthogonal subspaces, i.e.,
H = L ⊕ L⊥,
where L is arbitrary closed subspace of H. If P is an orthogonal projection and R(P ) = {P x : x ∈ H} its range and N (P ) = {x ∈ H : P x = 0} its null-space, then from R(P )⊥= N (P∗) and R(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 denote R(P ) = L, then P = IL 0 0 0 : L L⊥ → L L⊥ . (2) 1
2 Introduction
We can see now that representation for Q under the same decomposition of the space is
Q = A B B∗ D : L L⊥ → L L⊥ , (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 to generate “coordinates”. Expressing everything in the terms of the orthogonal projection P gives us wanted coordinates and we are usually left with straightforward computations. (Similar relations can be seen between analytical and Euclidean geometry: coordinate system can make things easier.) We are now able to further discuss properties of the operators A, B and D as well as properties of the sums, differences, and products of the projections P and Q by studying their matrices. An advantage of using this method lies in the fact that the projection P is in its simplest form in (2). Any other representation of P would have a more complicated form and would lead to a more complicated form of other operators in which P appears.
The second and perhaps more direct influence to our work comes form D. Djordjevi´c and J. Koliha, (see [11], [12]), who gave matrix representation of a closed range operator A ∈ 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 0.1
LetA ∈ L(H) be a closed range operator. Then the operator A has the following three matrix representations with respect to the orthogonal sums of subspacesH = R(A) ⊕ N (A∗) = R(A∗) ⊕ N (A): (a) We have A = A1 0 0 0 : R(A∗) N (A) → R(A) N (A∗) , whereA1is invertible. (b) We have A = A1 A2 0 0 : R(A) N (A∗) → R(A) N (A∗) ,
whereB = A1A∗1+ A2A∗2mapsR(A) onto itself and B > 0.
(c) Alternatively, A = A1 0 A2 0 : R(A∗) N (A) → R(A∗) N (A) ,
whereB = A∗1A1+ A∗2A2mapsR(A∗) onto itself and B > 0.
The operatorsAiare different in all three cases.
Here we again use the fact that R(A)⊥= N (A∗) and R(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 A with two generalizations: operator is in the infinite dimensional settings and it is closed range operator, not necessarily orthogonal projection. Three different forms are the result of 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 operator classes. 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] examined some properties of such operators on Cn×n. We are concerned with the same classes of operators. However, the method from [12] and [11] gives us the opportunity to use matrix representation of these operators on an infinite dimensional Hilbert space and to involve matrix theory in solving problems coming from operator theory. Note that EP operators 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 EP operators, 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 what conditions their product, sum and difference keep them. Among many properties that an operator 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 invertible and b ∈ R(A), then there exists solution (possible more than one). If b /∈ R(A), then there are no solutions. However, it is possible to obtain generalized solutions by using the Moore-Penrose inverse of A (see [1], [8], [13] for more details and proofs), denoted by A†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.,
kAx0− bk = min
x kAx − bk.
Moreover, if M is the set of all least square solutions of the linear equation Ax = b, then x0is the minimal norm least square solution, i.e.,
kx0k = min x∈Mkxk.
Depending on applications, sometimes we can give up the equation solving properties of 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), ADis the 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 existing proofs or to give some new characterizations of the Moore-Penrose and the Drazin in-verse of sums, differences and products of projections, generalized and hypergeneralized projections. Similarly to Lemma (0.1), we find canonical forms of the Moore-Penrose inverse of a closed range operator A depending on decomposition of the space. So, if A ∈ L(H) is a closed range operator and H = R(A∗) ⊕ N (A) = R(A) ⊕ N (A∗), then
we have three different representations of operator and its inverse: (a) A = A1 0 0 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 0 A∗ 2B−1 0 ,
where B = A1A∗1+ A2A∗2maps R(A) onto itself and B > 0.
(c) A = A3 0 A4 0 : R(A∗) N (A) → R(A∗) N (A) , A†= C−1A∗3 C−1A∗4 0 0 ,
where C = A∗3A3+ A∗4A4maps R(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 0.1
LetK ∈ L(H). Then K is idempotent if and only if it is expressible in the form K = (P Q)†for some orthogonal projectionsP, 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 0 K2 0 ,
where K12= K1and K2is arbitrary. It is easy to see that
P = K1†K1 0 0 0 , Q = K1K1† 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 (P Q)† = (P Q)∗(P Q(P Q)∗)†, we obtain
(P Q)† =
AA† 0
B∗A† 0
and a direct verification shows that (P Q)†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 hypergeneralized
projections 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 matrix
representations and examine under what conditions the product, difference and sum of these operators are operators of the same class.
Paper B: On the Moore-Penrose and the Drazin inverse of two
projections on a Hilbert space
For two given orthogonal, generalized or hypergeneralized projections P and Q on a Hilbert space H, we give their matrix representation. We also give canonical forms of the Moore-Penrose and the Drazin inverses of their product, difference, and sum. In addition, we provide results showing when these operators are EP operators and some simple 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 on generalized 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 hypergeneralgeneral-ized 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 orthogonal projectors, Linear Algebra Appl. 430 (2009), 2813-2833.
[7] A. Bottcher, I. M. Spikovsky, A gentle guide to the basics of two projections theory, 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, Linear Algebra 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. Djordjevi´c and J. Koliha, Characterizing hermitian, normal and EP
oper-ators, Filomat 21:1 (2007) 39-54
[12] D. S. Djordjevi´c, Characterizations of normal, hyponormal and EP operators, J. Math. Anal. Appl. 329 (2) (2007), 1181-1190
[13] D. S. Djordjevi´c and V. Rakoˇcevi´c, Lectures on Generalized Inverses, Faculty of 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, Linear Algebra 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. Rakocevi´c, Range projections and the Moore-Penrose inverse
in rings with involution
[18] T. Kato, Perturbation theory for linear operators, Springer, Berlin, 1996. [19] V. Rakoˇcevi´c, Funkcionalna analiza, Nauˇcna knjiga, Boegrad, 1994
Paper A
On pairs of generalized and
hypergeneralized projections on a
Hilbert space
Authors: Sonja Radosavljevi´c and Dragan S. Djordjevi´c
Preliminary version published as a technical report LiTH-MAT-R–2012/01–SE, Depart-ment of Mathematics, Linköping University.
On pairs of generalized and hypergeneralized
projections on a Hilbert space
Sonja Radosavljevi´c∗ Dragan S. Djordjevi´c†
∗Department of Mathematics,
Linköping University, SE–581 83 Linköping, Sweden. E-mail: sonja.radosavljevic@liu.se
†Faculty of Sciences and Mathematics,
University of Nis, P.O.Box 224, 18000 Nis, Serbia E-mail: dragan@pmf.ni.ac.rs
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 the product, difference and sum of these operators are operators of the same class.
Keywords: Generalized projections, hypergeneralized projections.
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 operators on H. The symbols R(A), N (A) and A∗denote range, null space and adjoint operator of operator A ∈ L(H). Operator A ∈ L(H) is a projection (idempotent) if A2= A, while it is an orthogonal projection if A∗= A = A2. Operator is hermitian (self adjoint) if A = A∗, normal if AA∗ = A∗A and unitary if AA∗ = A∗A = I. All these operators have been extensively studied and there are plenty of characterizations both of these operators and their linear combinations ([5]).
The Moore-Penrose inverse of A ∈ L(H), denoted by A†, is the unique solution to the equations
AA†A = A, A†AA† = A†, (AA†)∗= AA†, (A†A)∗= A†A.
Notice that A†exists if and only if R(A) is closed. Then AA†is the orthogonal projection onto R(A) parallel to N (A∗), and A†A is the orthogonal projection onto R(A∗) parallel to N (A). Consequently, I − AA†is the orthogonal projection onto N (A∗) and I − A†A is the orthogonal projection onto N (A).
For A ∈ L(H), an element B ∈ L(H) is the Drazin inverse of A, if the following hold:
BAB = B, BA = AB, An+1B = An,
for some non-negative integer n. The smallest such n is called the Drazin index of A. By ADwe denote Drazin inverse of A and by ind(A) we denote Drazin index of A.
If such n does not exist, ind(A) = ∞ and operator A 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 if R(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#. Set of all EP operators on H will be denoted by EP(H). Self-adjoint and normal operators with closed range are important subset of set of all EP operators. However, converse is not true even in a finite dimensional case.
In this paper we consider pairs of generalized and hypergeneralized projections on a Hilbert space, whose concept for matrices A ∈ Cm×nwas introduced by J. Gross and G.
Trenkler in [6]. These operators extend the idea of orthogonal projections by deleting the idempotency 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 by HGP(H).
2 Characterization of generalized and hypergeneralized projections 11
2
Characterization of generalized and
hypergeneralized projections
We begin this section by giving characterizations of generalized and hypergeneralized projections. Similarly to Theorem 2 in [4] and Theorem 1 in [6], we have the following result:
Theorem A.1
LetA ∈ 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(P ∆i).
Moreover, A has the following spectral representation
A = Z
σ(A)
λdEλ,
where Eλ= E(λ) is the spectral projection associated with the point λ ∈ σ(A).
From A4= A, we conclude A3|R(A) = I
R(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 that E(λ) 6= 0 if λ ∈ σ(A), E(λ) = 0 if λ ∈ {0, 1, e2πi/3, e−2πi/3}\σ(A) and E(0)⊕E(1)⊕ E(e2πi/3) ⊕ E(e−2πi/3) = I. From the fact that σ(A2) = σ(A∗) and from uniqueness of 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 that A∗A (AA∗) is the orhtogonal projecton onto R(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 onto R(AA∗) = R(A). Thus, AA∗A = P
R(A)A = A and A4 = AA2A = A implies
12 Paper A On pairs of generalized and hypergeneralized projections on a Hilbert space
We give matrix representatons of generalized projections based on the previous char-acterizations.
Theorem A.2
LetA ∈ L(H) be a generalized projection. Then A is a closed range operator and A3is
an orthogonal projection onR(A). Moreover, H has decomposition H = R(A) ⊕ N (A)
andA has the following matrix representaton
A = A1 0 0 0 : R(A) N (A) → R(A) N (A) ,
where the restrictionA1= A|R(A)is unitary onR(A).
Proof: If A is a generalized projection, then AA∗A = A4= A and A is a partial isometry
implying 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 0 0 0 : R(A) N (A) → R(A) N (A) , where A2 1= A∗1, A41= A1and A1A∗1= A∗1A1= A31= IR(A).
Similar to Theorem 2 in [6], we have: Theorem A.3
LetA ∈ L(H). Then the following conditions are equivalent: (a) A is a hypergeneralized projecton.
(b) A3is an orthogonal projection ontoR(A).
(c) A is an EP operator and A4= A
Proof: (a ⇒ b) If A2= A†, then from A3= AA†= P
R(A)follows conclusion.
(b ⇒ a) If A3 = P
R(A), a direct verificaton of the Moore-Penrose equations shows
that A2= A†. (a ⇒ c) Since
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.4
LetA ∈ L(H) be a hypergeneralized projection. Then A is a closed range operator and H has decomposition
H = R(A) ⊕ N (A).
Also,A has the following matrix representaton with the respect to decomposition of the space A = A1 0 0 0 : R(A) N (A) → R(A) N (A) ,
where restrictionA1= A|R(A)satisfiesA31= 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 A has 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 EP operators, which implies that A† = AD= A#= A2. For generalized projections we can be more precise:
A† = AD= A#= A2= A∗. We can also write
GP(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 proof relies only on properties of generalized and hypergeneralized projections and basic prop-erties of the Moore-Penrose and the group inverse.
Theorem A.5
LetA ∈ 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 that A∗∈ GP(H). Conversely, if A∗∈ GP(H), then (A∗)4= A∗and (A∗)2=
(A∗)∗= A, implying
14 Paper A On pairs of generalized and hypergeneralized projections on a Hilbert space
and A ∈ GP(H).
(b) If A ∈ GP(H), then A†= A∗= A2and
(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 and R(A) is closed and A, A†have representations
A = A1 A2 0 0 : R(A) N (A∗) → R(A) N (A∗) , A†= A∗1B 0 A∗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)−1and
A = A1 0 0 0 , A† = A−11 0 0 0 . Since (A−11 )2= (A−1
1 )∗, the same equality is also satisfied for A1and 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)2A 2 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 −1 1 .
3 Properties of products, differences, and sums of generalized projections 15
Theorem A.6
LetA ∈ 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 )2A 2 0 0 , (A#)†= (A−11 )∗B 0 (A−12 )∗B 0 , where B = (A−11 (A−11 )∗+ A−1 2 (A −1 2 )∗)−1. From (A #)† = (A#)2, we get A 2 = 0
and A1 = A−21 . Multiplying with A21, the last equation becomes A31 = IR(A)and A ∈
HGP(H).
3
Properties of products, differences, and sums of
generalized projections
In this section we study properties of products, differences, and sums of two generalized projections or of one orthogonal and one generalized projection. We begin with two very useful theorems which gives matrix representations of such pairs of operators. Also, we obtain basic properties of generalized projections which easily follow from their canonical representations.
Theorem A.7
LetA, 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 0 0 0 : R(A) N (A) → R(A) N (A) , B = B1 B2 B3 B4 : R(A) N (A) → R(A) N (A) , where B1∗ = B21+ B2B3, B2∗ = B3B1+ B4B3, B3∗ = B1B2+ B2B4, B4∗ = B3B2+ B42.
16 Paper A On pairs of generalized and hypergeneralized projections on a Hilbert space
Proof: Let H = R(A) ⊕ N (A). Then representation of A follows from Theorem A.2 and let B has representation
B = B1 B2 B3 B4 . Then, from B2= B2 1+ B2B3 B1B2+ B2B4 B3B1+ B4B3 B3B2+ B42 = B1∗ B3∗ B2∗ B4∗ = B∗,
the conclusion follows directly.
If B2= 0, then B3∗= B1B2+ B2B4= 0 and B3= 0. Analogously, B3= 0 implies
B2= 0.
Corollary A.1
Under the assumptions of the previous theorem, operatorB ∈ GP(H) has one of the following matrix representations:
B = B1 B2 B3 B4 or B = B1 0 0 B4 . Theorem A.8
LetP ∈ B(H) be an orthogonal projection, R(P ) = L and H = L⊕L⊥. IfA ∈ GP(H), thenP and A has the following matrix representation with the respect to decomposition of the space P = IL 0 0 0 : L L⊥ → L L⊥ , A = A1 A2 A3 A4 : L L⊥ → L L⊥ , where A∗1 = A21+ A2A3, A∗2 = A3A1+ A4A3, A∗3 = A1A2+ A2A4, A∗4 = A3A2+ A24. Moreover, A1 = P AP |L, A2 = P A(I − P )|L⊥, A3 = (I − P )AP |L, A4 = (I − P )A(I − P )|L⊥.
ThenA2= 0 if and only if A3 = 0, or equivalentely, P A(I − P )|L⊥ = 0 if and only if
(I − P )AP |L= 0.
OpreatorsP and A commute if and only if either A2= 0 or A3= 0, or equivalentely,
3 Properties of products, differences, and sums of generalized projections 17
Proof: Matrix representation of A ∈ GP(H) and properties of Ai, i = 1, 4 can be
obtained like in the proof of Theorem A.7.
Using matrix multiplication, it is easy to see that
P AP =
A1 0
0 0
and P AP |L= A1. The rest of the equalities are obtained in an analogous way.
If P A = AP , again using matrix multiplication, we get A2 = A3 = 0 which is
equivalent to P A(I − P )|L⊥ = 0 or (I − P )AP |L= 0.
Corollary A.2
Under the assumptions of the previous theorem, operatorA ∈ GP(H) has the following matrix representations: A = A1 A2 A3 A4 ,
whenP and A do not commute, or
A = A1 0 0 A4 ,
whenP 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 = C2and A = e2πi3 0 0 0 , then A2 = A∗, but I − A = 1 − e2πi3 0 0 1 and, clearly, I − A 6= (I − A)4implying that I − A is not a generalized projection.
Thus, we have the following theorem. Theorem A.9
(Theorem 6 in [2]) LetA ∈ L(H) be a generalized projection. Then I − A is a normal operator. Moreover,I − A is a generalized projection if and only if A is an orthogonal projection.
IfI −A is a generalized projection, then A is a normal operator and A is a generalized projection if and only ifI − A is an orthogonal projecton.
Proof: If A is a generalized projection, then A is a normal operator and A4= A, which
implies
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) = 0 if λ ∈ {0, 1, e2πi/3, e−2πi/3}\σ(A) and E(0) ⊕ E(1) ⊕ E(e2πi/3) ⊕ E(e−2πi/3) = I.
Thus,
18 Paper A On pairs of generalized and hypergeneralized projections on a Hilbert space
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 to
σ(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) and according to Theorem A.7 generalized projection A has representation
A = A1 0 0 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.10
IfP is an orthogonal projection and A is a generalized projection, then AP is a general-ized projection if and only if eitherP A(I − P ) = 0 or (I − P )AP = 0.
Proof: Let R(P ) = L and H = L ⊕ L⊥. Then
P = IL 0 0 0 , A = A1 A2 A3 A4 . From AP = A1 0 A3 0 , (AP )2= A2 1 0 A3A1 0 , (AP )∗= A∗1 A∗3 0 0
we conclude that (AP )2 = (AP )∗ if and only if A
3 = 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, in
3 Properties of products, differences, and sums of generalized projections 19
Theorem A.11
LetP be an orthogonal projection and let A be a generalized projection. Then P − A is a generalized projection if and only ifA is an orthogonal projection commuting with P .
Furthermore, ifP is an orthogonal projection and A is a generalized projection such thatP AP is orthogonal projection and either P A(I − P ) = 0 or (I − P )AP = 0, then P − A is a generalized projection.
Proof: Obviously (P − A)2= P − P A − AP + A2= P∗− A∗= (P − A)∗if and only
if P A = 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⊥. If P A(I − P ) = 0 or (I − P )AP = 0, then
P = IL 0 0 0 , A = A1 0 0 A4 , P − A = Il− A1 0 0 −A4 .
From the orthogonality of P AP comes A1= A21= A∗1and
(P − A)2= (Il− A1)2 0 0 A2 4 = (Il− A1)∗ 0 0 A∗4 = (P − A)∗. Theorem A.12
LetP be an orthogonal projection and let A be a generalized projection. Then P + A is a generalized projection ifP AP = 0. Moreover, if P + A is a generalized projection, then P AP = 0, P A(I − P ) = 0 and (I − P )AP = 0.
Proof: From (P +A)2= P +P A+AP +A2= P +A∗we conclude that AP = P A = 0. This is equivalent to P A = A1 A2 0 0 = A1 0 A3 0 = 0,
which holds if and only if A1 = A2 = A3 = 0. Thus, P AP = 0, P A(I − P ) = 0, and
(I − P )AP = 0.
Conversely, if P AP = 0, then A1 = 0 and by Theorem A.7 A2 = 0, and A3 = 0.
Clearly, P + A = IL 0 0 A4 is a generalized projection. Theorem A.13
IfP is an orthogonal projection and A is a generalized projection, then AP − P A is a generalized projection if and only ifP A(I − P ) = 0 or (I − P )AP = 0.
Proof: The matrix representations of the operators A, P , and AP imply that
P A − AP = 0 A2 −A3 0 ,
20 Paper A On pairs of generalized and hypergeneralized projections on a Hilbert space
and it is clear that
(P A − AP )2= −A2A3 0 0 −A3A2 = 0 −A∗ 3 A∗2 0 = (P A − AP )∗
if and only if A2= A3= 0 which is equivalent to P A(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.14
LetA, B ∈ GP(H). Then AB ∈ GP(H) if AB = BA. Proof: If AB = A1B1 A1B2 0 0 = B1A1 0 B3A1 0 = BA,
it is clear that A1B1= B1A1, B2= 0 and B3= 0. Form Theorem A.7 we conclude that
B1∗= B21, B4∗= B42, and (AB)2= (A1B1)2 0 0 0 = (A1B1)∗ 0 0 0 = (AB)∗. Theorem A.15
LetA, 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+ B42 = (A1+ B1)∗ B3∗ B2∗ B4∗ = (A + B)∗, it is clear that (A1+ B1)2= A21+ A1B1+ B1A1+ B12+ B2B3= (A1+ B1)∗.
Since B1∗ = B12+ B2B3, we get A1B1+ B1A1 = 0. Thus, B1 = 0 which implies
B2= B3= 0, B24= B4∗. In this case we obtain AB = BA = 0.
Conversely, if AB = BA = 0, then A1B1= B1A1= 0, implying B1= 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 projections is a generalized projection itself. It can be proved using partial ordering on GP(H), like in [6] or [2]. We prefer using only the basic properties of the generalized projections and their matrix representation.
4 Properties of products, differences, and sums of hypergeneralized projections 21
Theorem A.16
LetA, 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)∗ −B3∗ −B∗ 2 −B4∗ = (A − B)∗,
we get B2 = 0, B3 = 0, B42 = −B4∗ and from Theorem A.7 comes B42 = B4∗. Now,
B4= 0 and
(A1− B1)2= A21− A1B1− B1A1+ B21= A∗1− B∗1
follows. This is true if and only if A1B1= B1A1= B1∗, and in that case AB = BA =
B∗.
Theorem A.17
LetA and B be two commuting generalized projections. Then A(I − B) ∈ GP(H) if and only ifABA = (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.18
IfA ∈ 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 normal
operator, which completes the proof.
4
Properties of products, differences, and sums of
hypergeneralized 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.
22 Paper A On pairs of generalized and hypergeneralized projections on a Hilbert space
Theorem A.19
LetP ∈ 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, P A is a hypergeneralized projection if and only ifP A(I − P ) = 0.
Proof: Let H = L ⊕ L⊥, where R(P ) = L. Then
A = A1 A2 A3 A4 , AP = A1 0 A3 0 . If (AP )2= (A1)2 0 A3A1 0 = D†A∗1 D†A∗3 0 0 = (P A)†,
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= A21 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 A2 1= A † 1and consequently (AP )2= A2 1 0 0 0 = A†1 0 0 0 = (AP )†.
The following example shows that Theorem A.9 does not hold for hypergeneralized projections. Example A.2 Let H = C2 and A = 1 1 0 e2πi3 . Then A2 = " 1 1 + e2πi3 0 e−2πi3 # , A3 = I R(A),
A4 = A and A is a hypergeneralized projection. However, I − A =
0 −1 0 1 − e2πi3
and it is not normal.
Theorem A.20
References 23
Proof: Let H = R(A) ⊕ N (A) and A, B ∈ HGP(H) have representations
A = A1 0 0 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 0 B3A1 0 = BA.
This implies B2 = 0, B3 = 0, A1B1 = B1A1 and it is easy to see that (AB)2 =
(AB)†. Theorem A.21
LetA, B ∈ HGP(H). If AB = BA = 0, then A + B ∈ HGP(H). Proof: Let H = R(A) ⊕ N (A). Then
A = A1 0 0 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 B24= B † 4. Now, (A + B)2= A2+ B2= A†+ B†= (A + B)†. Theorem A.22 LetA, B ∈ HGP(H). If AB = BA = B2, thenA − B ∈ HGP(H). Proof: Let H = R(A) ⊕ N (A). Then
A = A1 0 0 0 , B = B1 B2 B3 B4 .
From the condition AB = BA = B2, we get A1B1 = B1A1 = B12, B2 = B3 = 0,
B2 4 = B
†
4, which implies (A − B)†= A†− B†. Hence
24 Paper A On pairs of generalized and hypergeneralized projections on a Hilbert space
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 on generalized 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 hypergeneralgeneral-ized 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. Djordjevi´c 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, Linear Algebra and its Applications 264 (1997) 463-474
Paper B
On the Moore-Penrose and the Drazin
inverse of two projections on a Hilbert
space
Authors: Sonja Radosavljevi´c and Dragan S. Djordjevi´c
Preliminary version published as a technical report LiTH-MAT-R–2012/03–SE, Depart-ment of Mathematics, Linköping University.
On the Moore-Penrose and the Drazin inverse of
two projections on a Hilbert space
Sonja Radosavljevi´c∗and Dragan S. Djordjevi´c†
∗Department of Mathematics,
Linköping University, SE–581 83 Linköping, Sweden. E-mail: sonja.radosavljevic@liu.se
†Faculty of Sciences and Mathematics,
University of Nis, P.O.Box 224, 18000 Nis, Serbia E-mail: dragan@pmf.ni.ac.rs
Abstract
For two given orthogonal, generalized or hypergeneralized projections P and Q on a Hilbert space H, we give their matrix representation. We also give canonical forms of the Moore-Penrose and the Drazin inverses of their prod-ucts, differences, and sums. In addition, it is showed when these operators are EP operators and some simple relations between the mentioned operators are established.
Keywords: Orthogonal projection, generalized projection, hypergeneralized projection, Moore-Penrose inverse, Drazin inverse
28
Paper 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 studied Drazin invertibility for products, differences, and sums of idempotents and Baksalary and 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 the Moore-Penrose and the Drazin inverse for products, differences, and sums of two orthog-onal, generalized or hypergeneralized projections on an arbitrary Hilbert space. Using the 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 and establish 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 mentioned papers, 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 all bounded 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 = P2and it is an orthogonal projection if P = P2= P∗.
Generalized and hypergeneralized projections were inroduced in [7] by J. Gross and G. 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 by HGP(H).
Here A†is the Moore-Penrose inverse of A ∈ L(H), i.e., the unique solution to the equations
AA†A = A, A†AA† = A†, (AA†)∗= AA†, (A†A)∗= A†A.
Notice that A†exists if and only if R(A) is closed. Then AA†is the orthogonal projection onto R(A) parallel to N (A∗), and A†A is the orthogonal projection onto R(A∗) parallel to N (A). Consequently, I − AA†is the orthogonal projection onto N (A∗) and I − A†A is the orthogonal projection onto N (A). An essential property of any P ∈ L(H) is that P 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 following hold:
BAB = B, BA = AB, An+1B = An,
for some non-negative integer n. The smallest such n is called the Drazin index of A. By ADwe 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 operator A 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 and ADis group inverse, usually denoted by A#. 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 that
R(An+1) = R(An). In this case, ind(A) = asc(A) = dsc(A) = n.
Recall that if R(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 results
from D. Djordjevi´c and J. Koliha, (see [4]).
In what follows, A stands for I − A and PAstands 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 0 0 0 = IL 0 0 0 : L L⊥ → L L⊥ , (1) P = 0 0 0 IL⊥ : L L⊥ → L L⊥ , (2) Q = A B B∗ D : L L⊥ → L L⊥ , (3) Q = IL− A −B −B∗ I L⊥− D : L L⊥ → L L⊥ , (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.1
LetQ ∈ 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 B B∗ D A B B∗ D = A2+ BB∗ AB + BD B∗A + DB∗ B∗B + D2 = A B B∗ D
30
Paper B On the Moore-Penrose and the Drazin inverse of two projections on a Hilbert space
Theorem B.2
LetQ ∈ 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− A A † .
Proof: (a) Since A = A2+ BB∗, we have
R(A) = R(A2+ BB∗) = R(AA∗+ BB∗).
To prove that R(AA∗+ BB∗) = R(A) + R(B), observe the operator matrix
M = A B 0 0 .
For any x ∈ R(M M∗), there exists y ∈ H such that x = M M∗y = M (M∗y) and x ∈ R(M ). On the other hand, for x ∈ R(M ), there is y ∈ H and x = M y. Besides, M M†x = M M†M y = M y = x and M M† = M M∗(M M∗)† = PR(M M∗)implying x ∈ R(M M∗). Hence, R(M ) = R(M M∗) and R(A) + R(B) = R(M ) = R(M M∗) = R(AA∗+ BB∗), and we have R(A) = R(A) + R(B), implying R(B) ⊆ R(A).
(b) Since A = I − A, from Theorem B.1 (a), we get A = A2+ 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 BD D†= B, we obtain (e).
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 implying that 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†A A = A†A − A, by taking into account that A A†= 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 few theorems 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.3
LetG ∈ L(H) be a generalized projection. Then G is a closed range operator and G3is
an orthogonal projection onR(G). Moreover, H has the decomposition H = R(G) ⊕ N (G)
andG has the following matrix representaton
G = G1 0 0 0 : R(G) N (G) → R(G) N (G) ,
where the restrictionG1= 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, G3is 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 0 0 0 : R(G) N (G) → R(G) N (G) , where G21= G∗1, G41= G1and G1G∗1= G∗1G1= G31= IR(G).
32
Paper B On the Moore-Penrose and the Drazin inverse of two projections on a Hilbert space
Theorem B.4
LetG, 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 0 0 0 : R(G) N (G) → R(G) N (G) , H = H1 H2 H3 H4 : R(G) N (G) → R(G) N (G) , where H1∗ = H12+ H2H3, H2∗ = H3H1+ H4H3, H3∗ = H1H2+ H2H4, H4∗ = H3H2+ H42.
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+ H42 = H1∗ H3∗ H2∗ H4∗ = H∗,
conclusion follows directly.
If H2 = 0, then H3∗ = H1H2+ H2H4 = 0 and H3 = 0. Analogously, H3 = 0
implies H2= 0.
Theorem B.5
LetG ∈ L(H) be a hypergeneralized projection. Then G is a closed range operator and H has the decomposition
H = R(G) ⊕ N (G).
Also,G has the following matrix representaton with respect to the decomposition of the space G = G1 0 0 0 : R(G) N (G) → R(G) N (G) ,
where the restrictionG1= G|R(G)satisfiesG31= IR(G).
Proof: If G is a hypergeneralized projection, then G and G†commute and G is EP. Using Lemma (1.2) in [4], we get decomposition of the space H = R(G) ⊕ N (G), and G has the required representation.
3 The Moore-Penrose and the Drazin inverse of two orthogonal projections 33
3
The Moore-Penrose and the Drazin inverse of two
orthogonal 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.6
Let orthogonal projections P, Q ∈ L(H) be represented as in (1) and (2). Then the Moore-Penrose inverse ofP Q, P − Q and P + Q exists and the following holds:
(a) (P Q)† = AA† 0 B∗A† 0 : L L⊥ → L L⊥ and R(P Q) = R(A) (b) (P − Q)†= " A A† −BD† −B∗A† −DD† # : L L⊥ → L L⊥ and R(P − Q) = R(A) ⊕ R(D) (c) (P + Q)†= " 1 2(I + A A † ) −BD† −D†B∗ 2D†− DD† # : L L⊥ → L L⊥ and R(P + Q) = L ⊕ R(D).
Proof: (a) Using representatons (1) and (3) for orthogonal projections P, Q ∈ L(H), the well known Harte-Mbekhta formula (P Q)†= (P Q)∗(P Q(P Q)∗)†and Theorem B.1(a), we obtain (P Q)†= A 0 B∗ 0 A2+ BB∗ 0 0 0 † = AA† 0 B∗A† 0 . From P Q(P Q)†= PR(P Q), we obtain (P Q)(P Q)†= A B 0 0 AA† 0 B∗A† 0 = AA† 0 0 0 ,
or, in the other words, R(P Q) = 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 " A2+ BB∗ −AB + BD −B∗A + DB∗ B∗B + D2 #† = A −B −B∗ −D " A† 0 0 D† # = " A A† −BD† −B∗A† −DD† # .
34
Paper B On the Moore-Penrose and the Drazin inverse of two projections on a Hilbert space
For the range of P − Q we have
PR(P −Q) = (P − Q)(P − Q)† = " A A A†+ BD†B∗ −ABD†+ BDD† −B∗A A†+ DD†B∗ B∗BD†+ DDD† # = " A A† 0 0 DD† # , implying R(P − Q) = R(A) ⊕ R(D).
(c) The Moore-Penrose inverse of P + Q has the following representation with the respect to decomposition of the space:
(P + Q)†= X1 X2 X3 X4 : L L⊥ → L L⊥ .
In order to calculate (P + Q)†, we use the Moore-Penrose equations. From the first Moore-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∗X
1+ 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= X2∗. Further calculations
show that
(I + A)X1+ BX2∗ = IL,
(I + A)X2+ BX4 = 0,
B∗X1+ DX2∗ = 0,
B∗X2+ DX4 = DD†.
According to Theorem B.2 (b), (c), from B∗X1+ DX2∗= 0 we get D†B∗X1+ X2= 0,
3 The Moore-Penrose and the Drazin inverse of two orthogonal projections 35
From (I + A)X1+ BX2∗ = ILand Theorem B.2 (i), we get (2I − A A †
)X1 = IL
i.e. X1= (2I − A A †
)−1 =12(I + A A†). 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)†= " 1 2(I + A A † ) −BD† −D†B∗ 2D†− DD† # .
Like in the proof of part (b) of this theorem,
PR(P +Q)= (P + Q)(P + Q)† = " 1 2(I + A)(I + A A † ) − BD†B∗ −(I + A)BD†+ 2BD†− BDD† 1 2B ∗(I + A A†) − DD†B∗ −B∗BD†+ 2DD†− DDD† # = IL 0 0 DD† , which asserts R(P + Q) = L ⊕ R(D).
To prove the existence of the Moore-Penrose inverse of P Q, P − Q and P + Q, it is sufficient to prove that these operators have closed range. Since Q is the orthogonal projection, R(Q) is closed subset of H. Also,
R(Q) = Q(H) = A B B∗ D L L⊥ = 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 that R(A) and R(D) are closed subsets of L and L⊥, respectively. If R(A) is closed, then for every sequence (xn) ⊆ L, xn → x and Axn → y imply x ∈ L and
Ax = y. Now, (I − A)xn→ x − y and x − y ∈ L, (I − A)x = x − y which proves that
R(I − A) is closed. Consequently, R(P Q), R(I − A) and R(I + A) are closed which completes the proof.
Similarly to Theorem 3.1 in [6], we have the following result. Theorem B.7
Let orthogonal projections P, Q ∈ L(H) be represented as in (1) and (3). Then the Drazin inverses ofP Q, P − Q and P + Q exist, P − Q and P + Q are EP operators and the following holds:
(a) (P Q)D=
AD (AD)2B
0 0
andind(P Q) ≤ ind(A) + 1,
(b) (P − Q)D= (P − Q)†andind(P − Q) ≤ 1,
36
Paper B On the Moore-Penrose and the Drazin inverse of two projections on a Hilbert space
Proof: (a) Theorem B.6 proves that R(P Q) is the closed subset of H. Thus, the Drazin inverse for this operators exists. According to representations (1) and (3) of projections P and Q, their product P Q and the Drazin inverse (P Q)Dcan be written in the following
way: P Q = A B 0 0 , (P Q)D= X1 X2 X3 X4 : L L⊥ → L L⊥ .
Equations that describe Drazin inverse are
(P Q)DP Q(P Q)D= X1AX1 X1AX2 X3AX1 X3AX2 = X1 X2 X3 X4 = (P Q)D, (P Q)DP Q = X1A X1B X3A X3B = AX1 AX2 0 0 = P Q(P Q)D, (P Q)n+1(P Q)D= An+1X 1 An+1X2 0 0 = An An−1B 0 0 = (P Q)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 = X2and
AX2= X1B give X12B = X2. Finally, (P Q)D= AD (AD)2B 0 0 .
To estimate the Drazin index of P Q, suppose that ind(A) = n. Then
(P Q)n+2(P Q)D = An+2 An+1B 0 0 AD (AD)2B 0 0 = An+1 An+1ADB 0 0 = An+1 AnB 0 0 = (P Q)n+1
implying that ind(P Q) ≤ ind(A) + 1.
(b) Since (P − Q)(P − Q)∗= (P − Q)∗(P − Q) and R(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) ≤
3 The Moore-Penrose and the Drazin inverse of two orthogonal projections 37
Theorem B.8
Let orthogonal projections P, Q ∈ L(H) be represented as in (1) and (3). Then the following holds: (a) If P Q = QP or P QP = P Q, then (P + Q)D= IL−12A 0 0 D , (P − Q)D= A 0 0 −D . (b) If P QP = P , then (P + Q)D= 1 2IL 0 0 D , (P − Q)D= 0 0 0 −D . (c) If P QP = Q, then (P + Q)D= IL−12A 0 0 0 , (P − Q)D= A 0 0 0 = P − Q. (d) If P QP = 0, then (P + Q)D= IL 0 0 D = P + Q, (P − Q)D= IL 0 0 −D = P − Q. Proof: Let (P + Q)D= X1 X2 X3 X4 : L L⊥ → L L⊥ . (a) If P Q = A B 0 0 = A 0 B∗ 0 = QP or P QP = A 0 0 0 = A B 0 0 = P Q,
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 0 0 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 get X2= X3= 0, DX4= D.
38
Paper B On the Moore-Penrose and the Drazin inverse of two projections on a Hilbert space
The other two equations, (P + Q)D(P + Q)(P + Q)D= (P + Q)Dand (P + Q)D(P +
Q) = (P + Q)(P + Q)D, give X4DX4= X4, X4D = DX4 i.e. X4= D. Thus, (P + Q)D= IL−12A 0 0 D . Formula (P − Q)D= A 0 0 −D
follows form Theorem B.7 (b) and the fact that A = A2implies AD= A2= A. (b) If P QP = P , then A = ILand Theorem B.1 implies B = B∗= 0. Then,
Q =
IL 0
0 D
and from part (a) of this Theorem we conclude
(P + Q)D= 1 2IL 0 0 D , (P − Q)D= 0 0 0 −D .
(c) From P QP = Q we get B = B∗ = D = 0 and A = A2. Now, I
L + A is invertible and (P + Q)D= (P + Q)−1 = (IL+ A)−1 0 0 0 = IL−12A 0 0 0 and (P − Q)D= A 0 0 0 .
(d) If P QP = 0, then A = 0 and since R(B) ⊆ R(A), we conclude B = B∗ = 0. In this case, Q = 0 0 0 D , P + Q = IL 0 0 D implying (P + Q)D= P + Q = IL 0 0 D , (P − Q)D= P − Q = IL 0 0 −D . Theorem B.9
Let orthogonal projectionsP, Q ∈ L(H) be represented as in (1) and (3). Then (P Q)D= (QP )†(P Q)†(QP )†.
Moreover, ifP Q = QP , then P Q is the EP operator and (P Q)D= (P Q)†, ind(P Q) ≤ 1.
4 The MP and the Drazin inverse of the generalized and hypergeneralized projections 39
Proof: Corollary 5.2 in [8] states that (P Q)†is idempotent for every orthogonal projec-tions P and Q. Thus we can write
(P Q)†= I 0 K 0 and P Q = (P Q)††= (I + K∗K)−1 (I + K∗K)−1K∗ 0 0 .
Denote by A = (I + K∗K)−1and B = (I + K∗K)−1K∗= AK∗. Then
P Q =
A B 0 0
and according to Theorem B.8 (a),
(P Q)D = AD (AD)2B 0 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 0 K 0 I K∗ 0 0 = (QP )†(P Q)†(QP )†, where we used ((P Q)†)∗= ((P Q)∗)†= (QP )†.
If P and Q commute, then P Q is the normal operator with the closed range, which means that it is an EP operator and (P Q)†= (P Q)D.
4
The Moore-Penrose and the Drazin inverse of the
generalized and hypergeneralized projections
Some of the results obtained in the previous section we extend to generalized and hyper-generalized projections.
Theorem B.10
LetG, H ∈ L(H) be two generalized or hypergeneralized projections. Then the Moore-Penrose inverse ofGH exists and has the following matrix representation
(GH)†= (G1H1)∗D−1 0 (G1H2)∗D−1 0 , whereD = G1H1(G1H1)∗+ G1H2(G1H2)∗> 0 is invertible.
40
Paper B On the Moore-Penrose and the Drazin inverse of two projections on a Hilbert space
Proof: From Theorems B.3, B.4 and B.5, we see that R(G) = R(G1) is closed and pair
of generalized or hypergeneralized projections has the matrix form
G = G1 0 0 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. Since R(GH) = R(G1) is closed, the Moore-Penrose inverse (GH)†exists.
Theorem B.11
LetG, H ∈ L(H) be two generalized or hypergeneralized projections. Then the Drazin inverse ofGH 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.12
LetG, 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 H4−1 .
(c) If GH = HG = H∗, thenG − 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
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. Checking the Moore-Penrose equations for (GH)2, we see that they hold. From the uniqueness of
the 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, or
equivalently, (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)∗and G + 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)∗and
the rest of the proof is similar to part (a).
Matrix representations are easily obtained by using canonical forms of G and H given in Theorem B.4.
Theorem B.13
LetG, 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 H4−1 .
(c) If GH = HG = H∗, thenG − 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 a
hypergeneralized projection. Since (GH)2 = (GH)†, operator GH commutes with its
Moore-Penrose inverse and (GH)† = (GH)D. From
GH(GH)†= I 0 0 0 ,
