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Linear Algebra and its Applications 47 0 76 87 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa On invertibility of combinations of k-potent operators Chunyuan Deng a,, Dragana S. Cvetković-Ilić b,,, Yimin Wei c,d, a School of Mathematics Science, South China Normal University, Guangzhou 506, PR China b Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš, Višegradska, 8000 Niš, Serbia c School of Mathematical Sciences, Fudan University, Shanghai 004, PR China d Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Ministry of Education, PR China ARTICLE INFO ABSTRACT Article history: Received 8 January 0 Accepted February 0 Availableonline0March0 SubmittedbyR.Brualdi AMS classification: 5A09 47A05 Some properties of combinations c P + c P c P s Pk s,where P and P are two different nonzero k-potents, c, c and c are three nonzero complex numbers and positive integers k ands k, are obtained. Furthermore, the invertibility, the group invertibility and the k-potency of the linear combinations of k-potents are investigated, under certain commutativity properties imposed on them. 0 Elsevier Inc. All rights reserved. Keywords: Inverse Group inverse Linear combination of k-potents. Introduction Let H and K be separable, infinite dimensional, complex Hilbert spaces. We denote the set of all bounded linear operators from H into K by BH, K and by BH when H K. ForA BH, K, let σa, RA and NA be the spectrum, the range and the null space of A, respectively.theidentity operator on the closed subspace U is denoted by I U or I if there does not exist confusion. Recall that an operator P BH is idempotent if P P, tripotentifp P, k-potent if P k P, where k Corresponding author. E-mail addresses: cydeng@scnu.edu.cn, cy-deng@6.net C. Deng, gagamaka@ptt.rs, dragana@pmf.ni.ac.rs D.S. Cvetković-Ilić, ymwei@fudan.edu.cn, yimin.wei@gmail.com Y. Wei. Supported by the National Natural Science Foundation of China under Grant 7 and the Doctoral Program of the Ministry of Education under Grant 0094407000. Supported by Grant No. 74007 of the Ministry of Science and Technological Development, Republic of Serbia. Supported by the National Natural Science Foundation of China under Grant 08705, Shanghai Municipal Education Committee and Doctoral Program of the Ministry of Education under Grant 009007000. 004-795/$ - see front matter 0 Elsevier Inc. All rights reserved. http://dx.doi.org/0.06/j.laa.0.0.0
C. Deng et al. / Linear Algebra and its Applications 47 0 76 87 77 is a positive integer. For T BH, the group inverse 0,,5,,6] of T is the unique if it exists element T # BH such that TT # T # T, T # TT # T #, T TT # T. If T is group invertible, then RT is closed and the spectral idempotent T π is given by T π I TT #. The operator matrix form of a group invertible operator T with respect to the space decomposition H RT NT is given by T T 0, where T is invertible in BRT and in that case T # T 0. We will use the usual notation P I P and P 0 I for any operator P BH.Fora positive integer k, the set of kthcomplexrootsofshallbedenotedby k {ω 0 k,ω k,...,ωk k }, where ω k expπi/k, i. We also use μ # μ if μ 0 0 ifμ 0, for arbitrary μ C. Linear combinations of k-potents have been considered in recent years 9,4,6,7,5 ]. For the cases: P and P are idempotents; P is idempotent and P is tripotent; P and P are commuting tripotent matrices; 4 P is idempotent and P is k-potent, the problem of characterizing when a linear combination c P + c P is an idempotent matrix or a tripotent matrix, or a group involutory matrix was studied in 7,,7], respectively. Benítez and Thome 8] extended generalized projectors to k-generalized projectors and listed all situations when a linear combination of commuting k-generalized projectors is a k-generalized projector. The purpose of this paper is to characterize the invertibility, the group invertibility and the k- potency of the combinations of k-potents and their products under certain commutativity properties imposed on some k-potents. Let P and P be nonzero k-potents, c, c C\{0}, c C and let s be a positive integer such that 0 < s < k. We will investigate properties of the combinations of the form c P + c P c P s Pk s. Firstly, we introduce some canonical decompositions. Lemma.. If P BH is a k-potent k, thenrp is closed and P k is idempotent with RP RP RP k, NP NP NP k. Using the space decomposition H RP NP, Pcanbewrittenas operator matrix form P A 0 00, where A BRP is invertible and A k I. Also, the following result holds: Lemma., Theorem.]. P BH is a k-potent if and only if i σp {0} k, ii There exists an invertible operator S such that SPS λ σp λeλ, where denotes the orthogonal direct sum and Eλ are orthogonal projections such that λ σp Eλ I, Eλ i Eλ j Eλ j Eλ i 0, λ i,λ j σp, λ i λ j.
78 C. Deng et al. / Linear Algebra and its Applications 47 0 76 87 m For an arbitrary finite commuting nonzero k-potent family {P i } m,wededucethatσ i P i i m c I n + c P i {c + c λ : λ {0} k }. Note that, for arbitrary λ k {0} k and σ i and c, c C\{0}, c + c λ 0 λ c c. Hence, c I n + c m i P i is invertible c c k. Any k-potent P is group invertible. If P P, thenp # P. IfP k P with k >, then P # P k. Moreover, we have the following observation. Lemma.. Let T, T, T BH be group invertible and commuting operators. Then e I T π I Tπ I Tπ, e I T π I Tπ Tπ, e I T π Tπ I Tπ, e 4 I T π Tπ Tπ, e 5 T π I Tπ I Tπ e 6 T π I Tπ Tπ, e 7 T π Tπ I Tπ, e 8 T π Tπ Tπ are idempotents with, H 8 Re i, i 8 i e i I, e i e j e j e i 0, i j, i, j, 8.. Characterizations of commuting k-potents In this section, we will characterize the invertibility, the group invertibility and idempotency of the linear combinations of group invertible and k-potent operators under certain commutativity properties imposed on them. We always assume F denotes a commuting family of nonzero k-potents and P, P and P are nonzero k-potents which are not scalar multiples of each other. Theorem.. Let T, T, T BH be group invertible and commuting operators and e i,i, 8 be given by. Let be a linear combination of the form at + bt + ct, a, b C\{0}, c C. Then is invertible if and only if b I + T # a T + c a T are invertible and e 8 0. e, I + b a T # T e, I + c a T # T e, I + c b T # T e 5 4 Proof. Let e i, i, 8bedefinedasin. Using the space decomposition H 8 Re i, we know e i is a block diagonal operator having the identity I in the ith block and zero otherwise, and T X X X X 4 0 0 0 0, i T Y Y 0 0 Y 5 Y 6 0 0, 5 T Z 0 Z 0 Z 5 0 Z 7 0,
C. Deng et al. / Linear Algebra and its Applications 47 0 76 87 79 where X i, Y j, Z l, i {,,, 4}, j {,, 5, 6} and l {,, 5, 7} are invertible. Hence, T # T # T # X X X X 4 0 0 0 0, Y Y 0 0 Y 5 Y 6 0 0, Z 0 Z 0 Z 5 0 Z 7 0. 6 Now, we obtain Note that at + bt + ct ax + by + cz e + ax + by e + ax + cz e +ax 4 e 4 + by 5 + cz 5 e 5 + by 6 e 6 + cz 7 e 7 + 0e 8. 7 I + T # b a T + c a T e I + b a X Y + c a X Z I I I I I I I, I + b a T # T e I I + b a X Y I I I I I I, I + c a T # T e I I I + c a X Z I I I I I, I + c b T # T e 5 I I I I I + c b Y 5 Z 5 I I I. 8 By 7 and 8, we get that is an invertible if and only if e 8 0 and 4 holds. Corollary.. Let k-potents P, P, P F and be a linear combination of the form ap + bp + cp, a, b C\{0}, c C with b a k, c a k and c b k. Then is invertible if and only if I + P # b a P + c a P e is invertible and e 8 0. Proof. We use the notions from the proof of Theorem..Since b a k and P # P e is k-potent, we have a / b k {0} and σp # P e k {0}. Hence, a b I P# P e is invertible, i.e., I + b a P# P e is invertible. Similarly, we have that I + c a P# P e, I + c b P# P e 5 are invertible. The result follows immediately by Theorem.. Let T 0inTheorem..Thene i 0, i {,, 5, 7} and e j, j {, 4, 6, 8} can be rewritten as e I Tπ I Tπ, e I Tπ Tπ, e Tπ I Tπ, e 4 Tπ Tπ, 9 respectively. The following results are derived by Theorem.. Corollary.. Let P, P BH be group invertible and commuting operators and let e i,i, 4 be defined as in 9 and be a linear combination of the form ap + bp, a, b C\{0}. is invertible if and only if I + b a P# P e is invertible and e 4 0. In addition, if P, P are k-potents and b a k,then is invertible e 4 0 NP NP {0}. 0 Proof. Let P 0inTheorem. and c 0 in Corollary..
80 C. Deng et al. / Linear Algebra and its Applications 47 0 76 87 Remark. In Corollary., by the proof of Theorem., P, P and reduce as P X X 4 0 0, P Y 0 Y 6 0, ax + by ax 4 by 6 0 with respect to the space decomposition H 4 i Re i, respectively. In the case when P, P are k-potents, e 4 0 Pk + I P k P is invertible P + I P k P is invertible. One special case for tripotent operators P, P, a and b in Corollary. has been proved in the paper of Liu et al. 8, Theorem.]. Corollary. 8, Theorem.]. Let P, P C n n be two commuting tripotents. Then P P is nonsingular if and only if I n P P and P + I n P P are nonsingular. The following results generalize, Theorem.] to k-potents, and we should remark that our proof is much simpler. Theorem.. Let k-potents P, P, P combination of the form ap + bp + cp, a, b, c C. F and e i,i, 8 be defined as in. Let be a linear Then is always group invertible. In particular, if k,then is m-potent if and only if a + b + c m a + b + c, when e 0, a + b m a + b, when e 0, a + c m a + c, when e 0, b + c m b + c, when e 5 0, a m a, when e 4 0, b m b, when e 6 0, c m c, when e 7 0 and # a + b + c # e + a + b # e + a + c # e + a # e 4 + b + c # e 5 + b # e 6 + c # e 7. Proof. We use the notions from the proof of Theorem.. Remark that in the case of k-potents, I P π i P k i, i,. By 7, we know that ax +by +cz e +ax +by e +ax +cz e +ax 4 e 4 +by 5 +cz 5 e 5 +by 6 e 6 +cz 7 e 7, where k-potents X i, Y j and Z l satisfy X k i I, Y k j I, Z k l I, i {,,, 4}, j {,, 5, 6} and l {,, 5, 7}. By Lemma.,ak-potent operator is diagonalizable. Since X, Y, Z resp. X and Y ; or X and Z ;ory 5 and Z 5 arek-potents and mutually commutative, they are simultaneously diagonalizable. Hence there is an invertible operator S such that S X S, S Y S and S Z S are all diagonal operators and their diagonal entries belong to k with proper multiplicities. Thus ax + by + cz is group invertible. Similarly, ax + by, ax + cz and gy 5 + cz 5 are group invertible, so is group invertible. If k, then X i, Y j and Z l in 7 are identity operators, so and hold. It is interesting to note that all sorts of situations in, Theorem.] are some special cases of in Theorem.. Furthermore, Theorem. also improves some recent results, Theorem,,
C. Deng et al. / Linear Algebra and its Applications 47 0 76 87 8 Theorem.4], with brief proof. The following theorem in the matrix case was for the first time given in, Theorem]: Corollary.4. Let two different nonzero idempotents P, P F, be their linear combination of the form ap + bp, a, b C\{0}. is idempotent if and only if any one of the following items holds: i a, b, P P 0; ii a, b, P P 0; iii a, b, P P 0. Proof. Take m and c 0inTheorem... Properties of combinations of the form c P + c P c P s Pk s Let P, P be k-potents. We first show that the kernel of c P +c P c +c P k P is independent of the choice of c, c C\{0}. Theorem.. Let P, P be k-potents and c, c C\{0}. Then the following statements hold: i NP P Nc P + c P c + c P k P. ii If P, P are commutative, then for every positive integer 0 < s < k, NP P Nc P + c P c + c P s Pk s NP s Pk s P. Proof. i By Lemma., P and P are given by P A 0, P X Y, 00 Z W where A BRP is invertible and A k I. Now, we have that P P A X Z Y W and c P + c P c + c P k P c A X c Y c Z c W, which immediately shows that NP P Nc P + c P c + c P k P. ii Since P and P are commutative, they are simultaneous diagonalizable, so it is sufficient to prove that for p, q, r : k {0} k {0} C, givenby pλ, μ λ μ, qλ, μ c λ + c μ c + c λ s μ k s, rλ, μ λ s μ k s μ, the following holds: pλ, μ 0 qλ, μ 0 and rλ, μ 0. 4
8 C. Deng et al. / Linear Algebra and its Applications 47 0 76 87 By distinguishing four cases: aλ μ 0, bλ 0, μ 0, cλ 0,μ 0, dλ 0,μ 0, it is evident that 4 holds, so we get that ii holds. Let P and P be nonzero k-potents. By, we get that P P 0impliesP get c P + c P c A 0 c Z c W and P + P P P k A 0 0 W c P + c P is invertible P + P P P k is invertible. 0 0,sowe Z W. Thus, for every c, c C\{0}, As for the case P P 0, we have the following theorem, which is a generalization of Theorem.5 in 8]. Theorem.. Let P, P be nonzero k-potents with P P 0 and k. Foreveryc, c C\{0}, c C and positive integer 0 < s < k : i If P k P P k P and c + c k c k,then c P + c P c P s Pk s is invertible P + P P P k is invertible. ii If P k P P k P and c + c k c k,thenatleastoneofthefollowing c P + c P c P s Pk s and is not invertible. k i Proof. i By, ifp k P P k P,wededucethat Y 0, X k A X 0, Since P is k-potent and P X 0 Z W k i0 X and W are k-potents and Z k and Z Z Note that k i0 k i0 k i0 W i ZX k i 0. X k 0 W i ZX k i W k i0 c k i i c P + c P + k c P s Pk s Pk, W i ZX k i.sox XA, k W i ZX k i W i ZX k i i0 W i ZX k i ZX k + W P + P P P k X + W k Z W k Z 5 k W i ZX k i i0 ZX k. 6 A 0. 7 0 W
C. Deng et al. / Linear Algebra and its Applications 47 0 76 87 8 Since RP RX NX and NP RW NW, P and P can be written as the 4 4 operator matrices: A A 4 00 A P A 00 0 0 0 0 0 0 0 0 where X k I and W k A X 0 A A, P X 0 0 0 0 0 0 0, 8 Z Z W 0 Z Z 4 0 0 I. FromX XA, we have that A X and A 4 0. Note that is invertible and A k I. SoA is invertible and A k I. By5 and 6, we get W k Z I 0 Z Z Z Z 00 Z Z 4 Z Z 4 ZX k It follows that Z 0, Z 0, and Z 4 0. Hence, by 8, c P + c P c P s Pk s Z Z I 0. Z Z 4 00 c + c X c I 0 0 0 s c A c A i A X k i c A 0 0 i0, 9 c Z 0 c W 0 0 0 0 0 where A, X and W are invertible k-potents. From c + c X c I + c + c X ] k 4c + +c k I c + c X ] k + c + c X ] k c c + c k c k I 0 and c + c k c k we get that c + c X c I is always invertible. Now 9 implies that c P + c P c P s Pk s is invertible if and only if NW {0}. By 7, NW {0} W is invertible P + P P P k is invertible. Hence c P + c P c P s Pk s is invertible if and only if P + P P P k is invertible. ii If c + c k c k,by0 it follows that at least one of the following operators c + c X c I and c + c X ] k + c + c X ] k c + c + c X ] k 4 c + +ck I is not invertible, which implies that one of the following operators c P + c P c P s Pk s and is not invertible. k i c k i i c P + c P + k c P s Pk s Remark. In Theorem., item i, the condition that c P + c P c P s Pk s is invertible can be replaced by the condition that c P +c P c I is invertible. Also, in the same theorem, item ii, the
84 C. Deng et al. / Linear Algebra and its Applications 47 0 76 87 condition that at least one of the operators c P +c P c P s Pk s and k i+ i ck i c P + c P c k P s Pk s is not invertible can be replaced by the condition that at least one of the operators i c P + c P is not invertible. c P + c P c I and k i0 ck i In Theorem.,itemi,ifP is invertible, then the second and the forth rows and columns in 8 and 9 will disappear, i.e., P X 0 and P X 0. In this case, it is clear that P + P 0 0 Z W P P k X 0 is invertible. Then c P + c P c P s Pk s c + c X c I 0 0 W c Z c W is always invertible. Note that and c + c k c k c + c X c I c + c X ] k +c + c X ] k c +c + c X ] k 4 c + +ck I c k i k i0 c + c X i c P + c P c P s Pk s Now, by computation we get that P c P + c P c P s Pk s ] I + P k P P P k + ] P k I P k. c c + c X c I W Z c + c X c I I 0 W Z I c k i c + c k c k 0 W c c + c X c I P k k i0 0 c + c P i If P is invertible in Theorem., itemi,p, P can be represented by P A 0, where A k 00 0 W c. X Y and Z W I and A BRP. The condition P k P P k P implies that Y 0 and X A. The invertibility of P implies that A and W are invertible k-potents. Now, it is clear that P + P P P k c P + c P c P s Pk s A 0 P is invertible. Then Z W c + c A c I 0 c Z c DA k s c W
is invertible, where P s As 0 D W s C. Deng et al. / Linear Algebra and its Applications 47 0 76 87 85 c + c A c I is invertible and, by a direct computation we get c P + c P c P s Pk s c P k I P k + c k i c +c k c k P k, for some operator D. Sincec + c k c k,wegetthat I c P k I P k c P c P s Pk s k i0 c + c P i ]. In particular, in the case when one of the operators P and P is invertible and k, if we apply Remark, we get 8, Theorem.5]. Corollary.. Let P, P be nonzero tripotents such that one of them is invertible, P P ] P P and c, c C\{0}, c C. Ifc + c c,thenc P + c P c P P or c P + c P + c P P is not invertible. If c + c c,thenc P + c P c P P is invertible. Furthermore, if P is invertible, then c P + c P c P P c P P P If P is invertible, then + c +c c c P + c P c P P I + P P P P I c P P P c P c P P c P + c +c P. c + c c ] c P + c + c P + c P I P ]. Let P, P satisfy the conditions in Corollary.. In8, Theorem.5], the authors got two representations of c P + c P c P P as follows: if P is nonsingular, then c + c c ]c P + c P c P P c + c P + c P + c +c c P P P + c c c P P P c + c c c If P is nonsingular, then c + c c ]c P + c P c P P P P P. c + c P c P P P + c c + 4 c c c P P P. We remark that the formulas and 4 are equivalent to the formulas in Corollary.. It is clear that the formulas in Corollary. are the particular case of the formulas and, which are valid for any k-potents k and any positive integer 0 < s < k. Moreover, in Corollary., ifc 0, then Corollary. reduces as the following one. Corollary. 9, Theorem.]. Let P, P be nonzero tripotents such that one of them is invertible, P P P P and c, c C\{0}.Thenc P + c P is invertible if and only if c + c 0.Inthiscase,if
86 C. Deng et al. / Linear Algebra and its Applications 47 0 76 87 P is nonsingular, then c P + c P c + c If P is nonsingular, then c P + c P c + c P + c c P I P. P + c c P I P. Proof. We only consider the case when P is invertible. The remaining case can be proved in the same way. The invertibility of P implies that P I, P P P and P P P. By Corollary., ] ] c P + c P I + P P P P c c +c + c P + P c I P ] ] I c +c P + P P P + c +c P c I P c +c P P + c +c c P I P ] c +c P + c c P I P ]. Let P and P be two k-potents k with P k P P k P, c, c C\{0}, c + c 0. Note that P # Pk and P # Pk.By9 we get c P + c P c + c X 0 0 0 c A c A 0 0 c Z 0 c W 0 0 0 0 0. Theorem.. Let P, P be nonzero k-potents, k such that P k P P k P.Letc, c C\{0}, c C be such that c + c c and let s be a positive integer such that 0 < s < k.then c P + c P c P s Pk s is invertible P + P P P k is invertible. Proof. If P k P P k P, by the proof of item i Theorem., weget s c A c c P + c P c P s Pk s c + c c X 0 0 0 A i A X k i c A 0 0 i0, 5 c Z 0 c W 0 0 0 0 0 where A, X and W are invertible k-potent operators. Since c +c c,wegetc P +c P c P s Pk s is invertible if and only if NW {0} if and only if W is invertible if and only if P + P P P k is invertible. Acknowledgements The authors would like to thank the anonymous reviewer for his/her very useful comments that helped to improve the presentation of this paper.
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