On pairs of generalized and hypergeneralized projections on a Hilbert space

On pairs of generalized and hypergeneralized projections on a Hilbert space Sonja Radosavljević and Dragan S Djordjević February 16 01 Abstract In this paper we characterize generalized and hypergeneralized projections (bounded linear operators which satisfy conditions A A and A 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 1 Introducton 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 A A while it is an orthogonal projection if A A A 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 A n+1 B A n for some non-negative integer n The smallest such n is called the Drazin index of A By A D we denote Drazin inverse of A and by ind(a) we denote Drazin index of A 1

If such n does not exist ind(a) and operator A is generalized Drazin invertible Its invers is denoted by A d Operator A is invertible if and only if ind(a) 0 If ind(a) 1 operator A is group invertible and A D is its group inverse usually denoted by A # Notice that if the Drazin inverse exists it is unique Drazin inverse exists if R(A n ) 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 A D 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 C m n was 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 11 Operator A L(H) is (a) a generalized projection if A A ; (b) a hypergeneralized projection if A A The set of all generalized projecton on H is denoted by GP(H) and the set of all hypergeneralized projecton is denoted by HGP(H) Characterization of generalized and hypergeneralized projections We begin this section by giving characterizations of generalized and hypergeneralized projection Similar to Theorem in 4 and Theorem 1 in 6 we have the following result: Theorem 1 Let A L(H) Then the following conditions are equivalent: (a) A is a generalized projection (b) A is a normal operator and A 4 A (c) A is a partial isometry and A 4 A Proof (a b) Since AA AA A 3 A A A A A 4 (A ) (A ) (A ) (A ) A

the implication is obvious (b a) If AA A A recall that then there 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 j then E( i ) E( i ) Moreover A has the following spectral representation A λde λ σ(a) where E λ E(λ) is the spectral projection associated with the point λ σ(a) From A 4 A we conclude A 3 R(A) I R(A) and λ 3 1 or equivalentely σ(a) {0 1 e πi/3 e πi/3 } Now A 0E(0) 1E(1) e πi/3 E(e πi/3 ) e πi/3 E(e πi/3 ) where E(λ) is the spectral projection of A associated with the point λ σ(a) such that E(λ) 0 if λ σ(a) E(λ) 0 if λ {0 1 e πi/3 e πi/3 }\σ(a) and E(0) E(1) E(e πi/3 ) E(e πi/3 ) I From the fact that σ(a ) σ(a ) and from uniqueness of spectral representation we get A A (a c) If A A then A 4 AA A 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 ) ) ie 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 A 4 AA A A implies A A We can give matrix representatons of generalized projections based upon previous characterizations Theorem Let A L(H) be a generalized projection Then A is a closed range operator and A 3 is an orthogonal projection on R(A) Moreover H has decomposition H R(A) N (A) and A has the following matrix representaton R(A) R(A) A : N (A) N (A) where the restriction A 1 A R(A) is unitary on R(A) Proof If A is a generalized projection then AA A A 4 A and A is a partial isometry implying that A 3 AA P R(A) A 3 A A P N (A) Thus A 3 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 3

a Hilbert space From Lemma (1) 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: R(A) R(A) A : N (A) N (A) where A 1 A 1 A 4 1 A 1 and A 1 A 1 A 1A 1 A 3 1 I R(A) Similar to Theorem in 6 we have: Theorem 3 Let A L(H) Then the following conditions are equivalent: (a) A is a hypergeneralized projecton (b) A 3 is an orthogonal projection onto R(A) (c) A is an EP operator and A 4 A Proof (a b) If A A then from A 3 AA P R(A) follows conclusion (b a) If A 3 P R(A) a direct verificaton of the Moore-Penrose equations shows that A A (a c) Since AA AA A 3 A A A A we conclude that A is an EP operator A A # (A ) n (A n ) and A 4 (A ) (A ) (A ) (A ) A (c a) If A is an EP operator then A A # and ind(a) 1 or equivalently A A A Since A 4 A A A from uniqueness of A follows A A Theorem 4 Let A 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 R(A) R(A) A : N (A) N (A) where the restriction A 1 A R(A) satisfies A 3 1 I R(A) Proof If A is a hypergeneralized projecton then A is an EP operator and using Lemma (1) in 5 we get the following decomposition of the space H R(A) N (A) and A has the required representation 4

Notice that since R(A) is closed for both generalized and hypergeneralized projections these operators have the Moore-Penrose and the Drazin inverses Besides they are EP operators which implies that A A D A # A For generalized projections we can be more precise: We can also write A A D A # A A GP(H) HGP(H) EP(H) Parts (b) and (c) of the following two theorems are known for matrices A C m n see ( 3) Unlike their proof which is based on representation of matrices our proof relies only on properties of generalized and hypergeneralized projections and basic properties of the Moore-Penrose and group inverse Theorem 5 Let 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 A 4 (A ) (A ) meaning that A GP(H) Conversely if A GP(H) then (A ) 4 A and (A ) (A ) A implying A (A ) 4 A and A GP(H) (b) If A GP(H) then A A A and (A ) (A ) A (A ) (A ) implying A GP If A GP(H) then (A ) (A ) (A ) A and (A ) 4 A Thus A (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 # ) (A # ) (A # ) (A # ) # A and (A # ) 4 A # and A (A # ) 4 A # ((A # ) ) A 5

Remark 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 A : R(A) N (A ) R(A) N (A ) where B (A 1 A 1 + A A ) 1 From (A ) (A ) we get A 1 BA 1B 0 BA1 BA A BA 1B 0 which implies A 0 A 0 and B (A 1 A 1) 1 and A A A 1 1 0 A A 1 B 0 A B 0 Since (A 1 1 ) (A 1 1 ) the same equality is also satisfied for A 1 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 A A # A # 1 (A # 1 ) A Since (A # ) (A # ) we get A 0 and (A # 1 ) (A # 1 ) Fron the fact that A 1 is surjective on R(A) and R(A 1 ) N (A 1 ) {0} we have A # 1 A 1 1 Consequently (A 1 1 ) (A 1 1 ) and A 1 A 1 which proves that A GP(H) Theorem 6 Let 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 5 (a) and (b) (c) We should only prove that A # HGP(H) implies A HGP(H) since the is analogous to the sema part of Theorem 5 Let H R(A) N (A ) and ind(a) 1 Then A A1 A A # A 1 1 (A 1 1 ) A (A # ) (A 1 1 ) B 0 (A 1 ) B 0 where B (A 1 1 (A 1 1 ) + A 1 (A 1 ) ) 1 From (A # ) (A # ) we get A 0 and A 1 A 1 Multiplying with A 1 the last equation becomes A 3 1 I R(A) and A HGP(H) 6

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 31 Let A B GP(H) and H R(A) N (A) Then A and B has the following representation with respect to decomposition of the space: R(A) R(A) A : N (A) N (A) where B1 B B : B 3 B 4 R(A) N (A) B 1 B 1 + B B 3 B B 3 B 1 + B 4 B 3 B 3 B 1 B + B B 4 B 4 B 3 B + B 4 Furthermore B 0 if and only if B 3 0 R(A) N (A) Proof Let H R(A) N (A) Then representation of A follows from Theorem and let B has representaton B1 B B B 3 B 4 Then from B B1 + B B 3 B 1 B + B B 4 B 3 B 1 + B 4 B 3 B 3 B + B4 B 1 B3 B B4 B conclusion follows directly If B 0 then B 3 B 1 B + B B 4 0 and B 3 0 Analogously B 3 0 implies B 0 Corollary 31 Under the assumptions of the previous theorem B GP(H) has one of the following matrix representations: B1 B B B1 0 or B B 3 B 4 0 B 4 7

Theorem 3 Let P B(H) be an orthogonal projection R(P ) L and H L L If A GP(H) then P and A has the following matrix representation with respect to decomposition of the space IL 0 L L P : where Moreover A A1 A : A 3 A 4 L L L A 1 A 1 + A A 3 A A 3 A 1 + A 4 A 3 A 3 A 1 A + A A 4 A 4 A 3 A + A 4 A 1 P AP L A P A(I P ) L A 3 (I P )AP L L L L A 4 (I P )A(I P ) L Then A 0 if and only if A 3 0 or equivalentely P A(I P ) L 0 if and only if (I P )AP L 0 Opreators P and A commute if and only if either A 0 or A 3 0 or equivalentely if and only if either P A(I P ) L 0 or (I P )AP L 0 Proof Matrix representation of A GP(H) and properties of A i i 1 4 can be obtained like in the proof of Theorem 31 Using matrix multiplication it is easy to see that P AP and P AP L A 1 The rest of the equalities are obtained in analogous way If P A AP again using matrix multiplication we get A A 3 0 which is equivalent to P A(I P ) L 0 or (I P )AP L 0 Corollary 3 Under the assumptions of the previous theorem A GP(H) has matrix representations: A1 A A A 3 A 4 when P and A do not commute or A 0 A 4 when P and A commute 8

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 πi Example 31 If H C e 3 0 and A then A A but I A πi 1 e 3 0 and clearly I A (I A) 4 implying that I A is not a 0 1 generalized projection Thus we have the following theorem Theorem 33 (Theorem 6 in ) Let A 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 If I A is a generalized projection then A is a normal operator and A is a generalized projection if and only if I A is an orthogonal projecton Proof If A is a generalized projection then A is normal and A 4 A which implies A 0E(0) 1E(1) e πi/3 E(e πi/3 ) e πi/3 E(e πi/3 ) where E(λ) is the orthogonal projection such that E(λ) 0 if λ σ(a) E(λ) 0 if λ {0 1 e πi/3 e πi/3 }\σ(a) and E(0) E(1) E(e πi/3 ) E(e πi/3 ) I Thus I A (1 0)E(0) (1 1)E(1) (1 e πi/3 )E(e πi/3 ) (1 e πi/3 )E(e πi/3 ) and Hence (I A) E(0) (1 e πi/3 ) E(e πi/3 ) (1 e πi/3 ) E(e πi/3 ) (I A) E(0) (1 e πi/3 ) E(e πi/3 ) (1 e πi/3 ) E(e πi/3 ) if and only if and (I A) (I A) (1 e πi/3 ) E(e πi/3 ) (1 e πi/3 ) E(e πi/3 ) (1 e πi/3 ) E(e πi/3 ) (1 e πi/3 ) E(e πi/3 ) This is true if and only if E(e πi/3 ) 0 and E(e πi/3 ) 0 which is equivalent to σ(a) {0 1} and A is an orthogonal projection 9

Remark 31 We can give another proof for this theorem Let H R(A) N (A) and according to Theorem 31 generalized projection A has representation R(A) R(A) A : N (A) N (A) Then IR(A) A I A 1 0 0 I N (A) and it is obvious that normality of A implies normality of I A Also (I A) (IR(A) A 1 ) 0 (IR(A) A 1 ) 0 (I A) 0 I N (A) 0 I N (A) holds if and only if (I R(A) A 1 ) (I R(A) A 1 ) Since A A we get I R(A) A 1 + A 1 I R(A) A 1 + A 1 I R(A) A 1 which is satisfied if and only if A 1 A 1 Hence A A A Theorem 34 If P is an orthogonal projection and A is a generalized projection then AP is a generalized projection if and only if either P A(I P ) 0 or (I P )AP 0 Proof Let R(P ) L and H L L Then IL 0 A1 A P A A 3 A 4 From AP A 3 0 (AP ) A 1 0 A 3 A 1 0 (AP ) A 1 A 3 we conclude that (AP ) (AP ) if and only if A 3 0 which is equivalent to (I P )AP 0 Theorem 31 provide us with the condition that A 3 0 if and only if A 0 or in the other words (I P )AP 0 if and only if P A(I P ) 0 Theorem 35 Let P be an orthogonal projection and let A be a generalized projection Then P A is a 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 such that P 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) P P A AP + A P A (P A) if and only if P A AP A Since A A we conclude that A is an orthogonal projection 10

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 IL 0 Il A P A P A 1 0 0 A 4 0 A 4 From orthogonality of P AP comes A 1 A 1 A 1 and (P A) (Il A 1 ) 0 (Il A 0 A 1 ) 0 4 0 A 4 (P A) Theorem 36 Let P be an orthogonal projection and let A be a generalized projection Then P + A is a generalized projection if P 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) P + P A + AP + A P + A we conclude that AP P A 0 This is equivalent to A1 A P A 0 A 3 0 which holds if and only if A 1 A A 3 0 Thus P AP 0 P A(I P ) 0 and (I P )AP 0 Conversely if P AP 0 then A 1 0 and by Theorem 31 A 0 and A 3 0 Clearly IL 0 P + A 0 A 4 is a generalized projection Theorem 37 If P is an orthogonal projection and A is a generalized projection then AP P A is a generalized projection if and only if P A(I P ) 0 or (I P )AP 0 Proof The matrix representations of operators A P and AP imply that 0 A P A AP A 3 0 and it is clear that (P A AP ) A A 3 0 0 A 3 0 A 3 A A 0 (P A AP ) if and only if A A 3 0 which is equivalent to P A(I P ) 0 or (I P )AP 0 11

Next theorem is not new Actually it is proved for matrices A C m n by J Gross and G Trenkler in 6 and it appears again in Theorem 38 Let A B GP(H) Then AB GP(H) if AB BA Proof If AB A1 B 1 A 1 B B1 A 1 0 B 3 A 1 0 it is clear that A 1 B 1 B 1 A 1 B 0 and B 3 0 conclude that B1 B1 and B4 B4 and (AB) (A1 B 1 ) 0 (A1 B 1 ) 0 BA Form Theorem 31 we (AB) Theorem 39 Let A B GP(H) AB BA 0 Then A + B GP(H) if and only if Proof If A B GP(H) have the representations given in Theorem 31 then A1 + B A + B 1 B B 3 B 4 and if (A + B) (A1 + B 1 ) + B B 3 (A 1 + B 1 )B B 4 B 3 (A 1 + B 1 ) + B 4 B 3 B 3 B + B4 (A1 + B 1 ) B3 B B4 (A + B) it is clear that (A 1 + B 1 ) A 1 + A 1 B 1 + B 1 A 1 + B 1 + B B 3 (A 1 + B 1 ) and since B 1 B 1 + B B 3 we get A 1 B 1 + B 1 A 1 0 Thus B 1 0 which implies B B 3 0 B 4 B 4 In this case we obtain AB BA 0 Conversely if AB BA 0 then A 1 B 1 B 1 A 1 0 implying B 1 B B 3 0 B 4 B 4 and obviously (A + B) (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 We prefer using only basic propertie s of generalized projections and their matrix representation Theorem 310 Let A B GP(H) AB BA B Then A B GP(H) if and only if 1

Proof If A B have the representations given in Theorem 31 then A1 B A B 1 B B 3 B 4 From (A B) (A 1 B 1 ) + B B 3 (A 1 B 1 )B + B B 4 B 3 (A 1 B 1 ) + B 4 B 3 B 3 B + B4 (A1 B 1 ) B3 B B4 (A B) we get B 0 B 3 0 B 4 B 4 and from Theorem 31 comes B 4 B 4 Now B 4 0 and (A 1 B 1 ) A 1 A 1 B 1 B 1 A 1 + B 1 A 1 B 1 follows This is true if and only if A 1 B 1 B 1 A 1 B 1 and in that case AB BA B Theorem 311 Let A and B be two commuting generalized projections Then A(I B) GP(H) if and only if ABA (AB) Proof Since AB BA we know that AB is a generalized projection Now (A(I B)) (A AB) A ABA + (AB) if and only if ABA (AB) A ABA + (AB) (A(I B)) Theorem 31 If A GP(H) and α {1 e πi/3 e πi/3 } then αa GP(H) Proof Since (αa) 3 α 3 A 3 A 3 then (αa) 3 R(A) I R(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 projections In some theorems we are not able to establish equivalency like the one we establish 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 a pair of one orthogonal and one hypergeneralized projection Theorem 41 Let P B(H) be an orthogonal projection and let A HGP(H) Then AP is a hypergeneralized projection if and only if (I P )AP 0 Similarly P A is a hypergeneralized projection if and only if P A(I P ) 0 13

Proof Let H L L where R(P ) L Then A1 A A AP A 3 A 4 A 3 0 If (AP ) (A1 ) 0 A 3 A 1 0 D A 1 D A 3 (P A) then A 3 0 which is equivalent to (I P )AP 0 Conversely if (I P )AP 0 ie A 3 0 then A has matrix form A1 A A A A 1 A 1 A + A A 4 0 A 4 0 A 4 and it is easy to see that A A 1 (I A 1 ) (I A 1 ) A A 4 0 A 4 and conse- Since A is a hypergeneralized projection it is clear that A 1 A 1 quently (AP ) A 1 0 A 1 0 (AP ) The following example shows that Theorem 33 does not hold for hypergeneralized projections 1 1 Example 41 Let H C and A Then A 1 1 + e πi 3 0 e πi 3 0 e πi 3 A 3 I R(A) A 4 A and A is a hypergeneralized projection However I A 0 1 and it is not normal 0 1 e πi 3 Theorem 4 Let A B HGP(H) If AB BA then AB HGP(H) Proof Let H R(A) N (A) and A B HGP(H) have representations B1 B A B B 3 B 4 Then AB A1 B 1 A 1 B (AB) A1 B 1 A 1 B 1 A 1 B 1 A 1 B A straightforward calculation using formula A A (AA ) shows that (AB) (A1 B 1 ) D 1 0 (A 1 B ) D 1 0 14

where D A 1 B 1 (A 1 B 1 ) + A 1 B (A 1 B ) > 0 is invertible Assume that hypergeneralized projections A B commute ie that A1 B AB 1 A 1 B B1 A 1 0 BA B 3 A 1 0 This implies B 0 B 3 0 A 1 B 1 B 1 A 1 and it is easy to see that (AB) (AB) Theorem 43 Let A B HGP(H) If AB BA 0 then A+B HGP(H) Proof Let H R(A) N (A) Then B1 B A B B 3 B 4 From these matrix representations it is easy to see that AB BA 0 implies B 1 B B 3 0 and B 4 B 4 Now (A + B) A + B A + B (A + B) Theorem 44 Let A B HGP(H) HGP(H) If AB BA B then A B Proof Let H R(A) N (A) Then B1 B A B B 3 B 4 From condition AB BA B we get A 1 B 1 B 1 A 1 B 1 B B 3 0 B 4 B 4 which implies (A B) A B Hence (A B) A AB BA + B A B A B (A B) References 1 JK Baksalary and X Liu An alternative characterization of generalized projectors Linear Algebra and its Applications 388 (004) 61-65 J K Baksalary O M Baksalary X Liu and G Trenkler Further results on generalized and hypergeneralized projectors Linear Algebra and its Applications 49 (008) 1038-1050 3 J K Baksalary O M Baksalary and X Liu Further properties on generalized and hypergeneralized projectors Linear Algebra and its Applications 389 (004) 95303 15

4 H Du and Y Li The spectral characterization of generalized projections Linear Algebra and its Applications 400 (005) 313-318 5 D S Djordjević and J Koliha Characterizing hermitian normal and EP operators Filomat 1:1 (007) 39-54 6 J Gross and G Trenkler Generalized and hypergeneralized projectors Linear Algebra and its Applications 64 (1997) 463-474 Address: Sonja Radosavljevic MAI Linkoping University 58351 Sweden sonjaradosavljevic@liuse Dragan Djordjevic Faculty of Sciences and Mathematics University of Niš PO Box 4 18000 Niš Serbia dragan@pmfniacrs 16