Representation for the generalized Drazin inverse of block matrices in Banach algebras
|
|
- Poppy Lucas
- 5 years ago
- Views:
Transcription
1 Representation for the generalized Drazin inverse of block matrices in Banach algebras Dijana Mosić and Dragan S. Djordjević Abstract Several representations of the generalized Drazin inverse of a block matrix with a group invertible generalized Schur complement in Banach algebra are presented. Key words and phrases: generalized Drazin inverse, Schur complement, block matrix Mathematics subject classification: 46H05, 47A05, 15A09. 1 Introduction The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, iterative methods in numerical linear algebra, etc. Representations for the Drazin inverse of block matrices under certain conditions where given in the literature 1, 3, 4, 5, 6, 7, 9, 14, 17. In this paper, we present formulas for the generalized Drazin inverse of block matrix with generalized Schur complement being group invertible in Banach algebra. Moreover, necessary and sufficient conditions for the existence as well as the expressions for the group inverse of triangular matrices are obtained. Let A be a complex unital Banach algebra with unit 1. For a A, we use σa) and ρa), respectively, to denote the spectrum and the resolvent set of a. The sets of all nilpotent and quasinilpotent elements σa) {0}) of A will be denoted by A nil and A qnil, respectively. The generalized Drazin inverse of a A or Koliha Drazin inverse of a) is the element b A which satisfies bab b, ab ba, a a 2 b A qnil. The authors are supported by the Ministry of Education and Science, Republic of Serbia, grant no
2 If the generalized Drazin inverse of a exists, it is unique and denoted by a d, and a is generalized Drazin invertible see 11). Recall that p 1 aa d is the spectral idempotent of a corresponding to the set {0}, and it will be denoted by a π. The set of all generalized Drazin invertible elements of A is denoted by A d. The Drazin inverse is a special case of the generalized Drazin inverse for which a a 2 b A nil instead of a a 2 b A qnil. Obviously, if a is Drazin invertible, then it is generalized Drazin invertible. The group inverse is the Drazin inverse for which the condition a a 2 b A nil is replaced with a aba. We use a to denote the group inverse of a, and we use A to denote the set of all group invertible elements of A. Let p p 2 A be an idempotent. Then we can represent element a A as a11 a a 12, a 21 a 22 where a 11 pap, a 12 pa1 p), a 21 1 p)ap, a 22 1 p)a1 p). We use the following lemmas. Lemma 1.1. Let b A d and a A qnil. i) 2, Corollary 3.4 If ab 0, then ab A d and ab) d b d ) n1 a n. ii) If ba 0, then a b A d and a b) d a n b d ) n1. Specializing 2, Corollary 3.4 with multiplication reversed) to bounded linear operators Castro González et al. 2 recovered 8, Theorem 2.2 which is a spacial case of Lemma 1.1ii). By the following lemma, Castro González et al. 2 recovered 10, Theorem 2.1 for matrices and 8, Theorem 2.3 for bounded linear operators. Lemma , Example 4.5 Let a, b A d and let ab 0. Then a b) d b d ) n1 a n a π b π b n a d ) n1. The following result is well-known for complex matrices see 16) and it is proved for elements of Banach algebra in 12. a b Lemma , Lemma 2.2 Let x A relative to the idempotent p A, a pap) d and let w aa d a d bca d be such that aw pap) d c d. 2
3 If ca π 0, a π b 0 and the generalized Schur complement s d ca d b is equal to 0, then x d p 0 aw) d 2 a 0 p a d b ca d. 1) 0 Let a b x c d A 2) relative to the idempotent p A, a pap) d and let the generalized Schur complement s d ca d b 1 p)a1 p)). The generalized Schur complement s plays an important role in the representations for x d in many cases 9, 14, 15, 17. Hartwig et al. 9 presented representations for the Drazin inverse of a 2 2 block matix under conditions which involve W AA D A D BCA D and that the generalized Schur complement is equal to 0. Li 13 investigated a representation for the Drazin inverse of block matrices with a singular and group invertible generalized Schur complement, recovering the formula 1) for complex matrices 16. We investigate representations of the generalized Drazin inverse of a block matrix x in 2) with a group invertible generalized Schur complement s d ca d b under different conditions. Thus, we extend some results from 13, 16 to more general settings. Also, we obtain equivalent condition for the existence and representations for the group inverse of triangular matrices in Banach algebra. The paper s aim is to further weaken the conditions on the elements needed to produce explicit formulae for the generalized Drazin inverse of x compared to those known from the literature. Such formulae are very complicated, but the main goal is to establish that x has the generalized Drazin inverse, and the formulae are the means to produce that result. 2 Results In this section, we assume that i) the element x is defined as in 2) relative to the idempotent p A, ii) a pap) d, iii) s d ca d b and s 1 p)a1 p)), 3
4 iv) w aa d a d bs π ca d. First, we give a formula for the generalized Drazin inverse of block matrix x in 2) in terms of w aa d a d bs π ca d and the group invertible Schur complement s. Theorem 2.1. If aw pap) d, a π b 0, bs π ca π 0, wbss 0, ss ca d bss 0, 3) then x A d and ) x d 0 bs 1 0 ca d bs s cwaw) d 2 a s s cwaw) d 2 bs π ) s ) n1 ca n 1 a π v n a v n bs π n1 p bs 0 1 p) ds r, 4) where v n s ) n1 ca d aw) n 1 aw) π, n 1, 2,... ), and aw) r d 2 a aw) d 2 bs π ca d aw) d 2 a ca d aw) d 2 bs π. Proof. Using the equalities aa d a π p, ss s π 1 p and ss π 0, note that a x 2 a d bs π aa π bss caa d ca d bs π ca π dss : y z. By a d a π 0 and 3), we obtain bs yz π ca π awbss ca d bs π ca π cwbss 0. To check that y A d, set A y a 2 a d, B y bs π, C y caa d and D y ca d bs π. Since a 2 a d ) a d, A y pap) and S y D y C y A y B y 0. Also, A π y B y a π bs π 0, C y A π y 0 and W y A y A y A y B y C y A y w. Applying Lemma 1.3, observe that y A d and y d p 0 ca d 0 aw) d 2 a 0 4 p a d bs π r.
5 In order to prove that z A d, we write z aa π bss ca π s 0 ca d bss : z 1 z 2 z 3. m t Recall that, for u, 0 n λ ρ pap m) ρ 1 p)a1 p) n) λ ρu), i.e. σu) σ pap m) σ 1 p)a1 p) n). From aa π pap) qnil and s A, we conclude that z 1 A qnil, z 2 A and z 2 0 s. Using Lemma 1.1i), by z 1 z 2 0, we get z 1 z 2 A d and z 1 z 2 ) d z 2 )n1 z1 n. Further, z2 3 0, i.e. z 3 A nil and z 1 z 2 )z 3 0. By Lemma 1.1ii), z A d and z d z 1 z 2 ) d z 3 z 1 z 2 ) d 2. Therefore, by Lemma 1.2, we deduce that x A d and x d z d ) n1 y n y π z π z n y d ) n1 1 z 3 z 1 z 2 ) d )z 1 z 2 ) d n1 y n y π z π z n y d ) n1 : X 1 X 2, 5) where X 1 1 z 3 z 1 z 2 ) d )x 1 y π, x 1 z 1 z 2 ) d n1 y n, X 2 z π y d z π z n y d ) n1. n1 5
6 Since z 1 y 0, then z 1 z 2 ) d y 0 s x 1 z 1 z 2 ) d z 1 z 2 ) d z 1 z 2 ) d Now, we have X z 1 z 2 ) d n z 2 yn n1 n1 z 2 k1 y z 2 y and ) n z 2 )k1 z1 k z 2 yn ) z 2 )n z 2 )n 1 z 2 )k1 z1 k n1 z 2 )n1 z1 n z 2 )n1 y n. 0 b 0 ca d b 0 b 0 ca d b k0 n1 k1 z 2 yn ) ) z 2 )k1 z1 k z 2 )n1 z1 n z 2 )n1 y n ) z 2 z 2 )k1 z1 k k1 ) z 2 )n z1 n z 2 )n y n bs n1 0 ca d bs 0 bs 0 ca d bs y π ) z 2 yπ ) z 2 yπ n1 z 2 n1 ) z 2 )n1 z1 n y π z 2 )n1 y n y π n1 n1 y π ) z 2 )n1 z1 n z 2 )n1 y n y π.6) n1 Observe that aa d aw) aw aw)aa d, y π 1 yy d p aw) d a cwaw) d 2 a aw) d bs π 1 p) cwaw) d 2 bs π and y n y π aa d aw) n 1 aw) π a aw) n 1 aw) π bs π ca d aw) n 1 aw) π a ca d aw) n 1 aw) π bs π n 1, 2,... ). 6
7 Hence, these equalities and 6) give ) 0 bs X ca d bs s cwaw) d 2 a s s cwaw) d 2 bs π a n a π 0 0 s ) n1 ca n 1 a π 0 n1 ) s ) n1 ca d aw) n 1 aw) π a s ) n1 ca d aw) n 1 aw) π bs π n1 ) 0 bs 1 0 ca d bs s cwaw) d 2 a s s cwaw) d 2 bs π ) s ) n1 ca n 1 a π v n a v n bs π. 7) From n1 zz d y z 1 z 2 )z 1 z 2 ) d z 3 z 1 z 2 ) d y z 1 z 2 )z 2 y z 3z 2 y z 2 z 3 )z 2 y, we get zz d y d z 2 z 3 )z p bs 2 yd and z π y d 0 1 p) ds r. Thus, by p bs z 0 1 p) ds z 1 and a π aw) d a π aa d awaw) d 2 0, notice p bs that zz π y d z 0 1 p) ds r z 1 r 0 and X 2 z π y d So, 5), 7) and 8) imply 4). z n z π y d ) n1 z π r. 8) n1 Our conditions a π b 0, bs π ca π 0, wbss 0, ss ca d bss 0 in Theorem 2.1 can be formulated geometrically as ba aa, s π ca π A b, bss A w, ca d bss A s, where e {f A : ef 0}. If we assume that a is a group invertible and w 1 in Theorem 2.1, we obtain the following result as a consequence. 7
8 Corollary 2.1. Let a pap) and let w aa a bca 1. If a π b 0, bca π 0 and bss 0, then x A d and x d a a ) 2 b s ca s ) 2 ca π s π ca ) 2 s s ca ) 2 b s π ca ) 3. b In the following theorem, the other formula for the generalized Drazin inverse of block matrix is presented. Theorem 2.2. If aw pap) d, ca π 0, a π bs π c 0, ss cw 0, ss ca d bss 0, then x A d and x d 0 aw) d bs 0 s s π ca d aw) d bs 0 a n 1 a π bs ) n1 aa d aw) n 1 aw) π bs ) n1 ) 0 s π ca d aw) n 1 aw) π bs ) n1 n1 ) p 0 1 s c s ca d t b s c 1 p) s, 9) d where t aw) d 2 a s π ca d aw) d 2 a aw) d 2 b s π ca d aw) d 2 b. Proof. If we write a x 2 a d aa d b s π c s π ca d b aa π a π b ss c ss d : y z, then zy 0. Using Lemma 1.3, we conclude that y A d and y d t. To show that z A d, let aa π a z π b 0 s ss c ss ca d : z b 1 z 2 z 3. Then z 1 A qnil, z 2 A and z 2 z 1 0. By Lemma 1.1ii), z 1 z 2 A d and z 1 z 2 ) d z1 nz 2 )n1. From z3 2 0, z 3z 1 z 2 ) 0 and Lemma 1.1i), z A d and z d z 1 z 2 ) d z 1 z 2 ) d 2 z 3. 8
9 Applying Lemma 1.2, observe that x A d and x d y d ) n1 z n z π y π y n z d ) n1 : X 1 X 2. 10) The equalities yz 1 0 and yz 1 z 2 ) d yz 2 imply y d z π p 0 t s c 1 p) s and y d d z π z 0. Next, we obtain X 2 X 1 y d z π y π y n z d ) n1 Since y π y π z 1 z 2 ) d y π y d ) n1 z π z n tz π. 11) n1 y π y n z 1 z 2 ) d n1 1 z 1 z 2 ) d z 3 ) ) y π y n z 2 z 1 z 2 ) d n 1 z 1 z 2 ) d z 3 ) n1 ) z1 n z 2 )n1 y π y n z 2 z 2 )n z1 k z 2 )kn 1 z 2 z 3 y π n1 ) z1 n z 2 )n1 z 3 n1 z n 1 z 2 )n1 n1 k1 y π y n z 2 )n1 1 z 2 z 3) n1 y π z 2 z1 n z 2 )n1 y π y n z 2 )n1 1 z 2 z 3). p aw) d a s π ca d aw) d a n1 aw) d b 1 p) s π ca d aw) d and b y n y π aa d aw) n 1 aw) π a aa d aw) n 1 aw) π b s π ca d aw) n 1 aw) π a s π ca d aw) n 1 aw) π b n 1, 2,... ), 9
10 we get X 2 0 aw) d bs 0 s s π ca d aw) d bs 0 a n 1 a π bs ) n1 aa d aw) n 1 aw) π bs ) n1 0 s π ca d aw) n 1 aw) π bs ) n1 n1 ) 1 s c s ca d b ) 12) Thus, from 10), 11) and 12), we obtain 9). Notice that explicit formulae 4) and 9) for the generalized Drazin inverse of x are complicated, but the conditions on the elements needed to produce these formulae are weaken than those known from the literature and x has the generalized Drazin inverse under these conditions. In Theorem 2.2, supposing that a pap) and w 1, the next corollary follows. Corollary 2.2. Let a pap) and let w aa a bca 1. If ca π 0, a π bc 0 and ss c 0, then x A d and x d a a bs a π bs ) 2 a ) 2 bs π s π ca ) 2 s ca ) 2 bs s π ca ) 3 bs π. The following result is a consequence of Theorem 2.1 and Theorem 2.2. Corollary 2.3. If aw pap) d, s 0 and if one of the following conditions holds: i) a π b 0 and ca π 0, ii) a π b 0 and bca π 0, iii) ca π 0 and a π bc 0, then x A d and x d aw) d 2 a ca d aw) d 2 a aw) d 2 b ca d aw) d 2 b. Notice that, the preceding corollary recover Lemma 1.3 for elements of Banach algebra and the analogy result for matrices 16. We will use Theorem 2.1 to find the group inverse of a triangular block matrix. Precisely, for b 0 in Theorem 2.1, we obtain the next result. 10
11 a 0 Theorem 2.3. Let x A relative to the idempotent p A, c s a pap) d and s 1 p)a1 p)). Then i) x A d and x d a d 0 s π ca d ) 2 s ca d s n1 s ) n1 ca n 1 a π 0 ; 13) ii) x A if and only if a pap) and s π ca π 0. Furthermore, if a pap) and s π ca π 0, then x a 0 s π ca ) 2 s ca s ) 2 ca π s Proof. i) If b 0 in Theorem 2.1, then s d, w aa d, aw a 2 a d pap), aw) a d and a d aw) π a d a π 0 implying 13). ii) By the part i), note that a 2 a d 0 x 2 x d caa d s 2 s ) n1 ca n 1 a π. s n1 Consequently, x 2 x d x a 2 a d a and caa d ss ) n ca n 1 a π c. n1 Hence, x A is equivalent to a pap) and ss ca π ca π. In the same manner as in the proof of Theorem 2.3, if c 0 in Theorem 2.2, we verify the following theorem in which necessary and sufficient condition for the existence and representation of the group inverse are considered. a b Theorem 2.4. Let x A relative to the idempotent p A, 0 s a pap) d and s 1 p)a1 p)). Then. i) x A d and x d a d a d ) 2 bs π a d bs 0 s 0 a n 1 a π bs ) n1 n1 ; 11
12 ii) x A if and only if a pap) and a π bs π 0. Furthermore, if a pap) and a π bs π 0, then x a a ) 2 bs π a bs a π bs ) 2 0 s. Observe that the part i) of Theorem 2.3 and the same part of Theorem 2.4 are the special cases of 2, Theorem 2.3 for Banach algebra elements and 8, Theorem 2.2 for bounded linear operators. Finally, we give an example to illustrate our results. p b Example 2.1. In Banach algebra A, if x A or x p 0 A) relative to the idempotent p A, then a c 0 d a p, a π 0, s 0 s, s π 1 p and w p aw aw) d. Using Theorem 2.1 or p b p 0 Theorem 2.2, we get that x A d and x d or x d ). c 0 References 1 N. Castro González, E. Dopazo, J. Robles, Formulas for the Drazin inverse of special block matrices, Appl. Math. Comput ) N. Castro González, J.J. Koliha, New additive results for the g-drazin inverse, Proc. Roy. Soc. Edinburgh Sect. A ) D.S. Cvetkovic-Ilić, Y. Wei, Representations for the Drazin inverse of bounded operators on Banach space, Electron. J. Linear Algebra ) C. Deng, A note on the Drazin inverses with Banachiewicz-Schur forms, Appl. Math. Comput ) C. Deng, D.S. Cvetkovic-Ilić, Y.Wei, Some results on the generalized Drazin inverse of operator matrices, Linear and Multilinear Algebra ) C. Deng, Y. Wei, Representations for the Drazin inverse of 2 2 blockoperator matrix with singular Schur complement, Linear Algebra and its Applications )
13 7 D.S. Djordjević, P.S. Stanimirović, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J ) D.S. Djordjević, Y. Wei, Additive results for the generalized Drazin inverse, J. Austral. Math. Soc ) R.E. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of 2 2 block matrix, SIAM J. Matrix Anal. Appl ) R.E. Hartwig, G. Wang and Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl ) J.J. Koliha, A generalized Drazin inverse, Glasgow Math. J ), M.Z. Kolundžija, D. Mosić, D.S. Djordjević, Generalized Drazin inverse of certain block matrices in Banach algebras, preprint). 13 X. Li, A representation for the Drazin inverse of block matrices with a singular generalized Schur complement, Appl. Math. Comput ) 2011) X. Li, Y. Wei, A note on the representations for the Drazin inverse of 2 2 block matrix, Linear Algebra Appl ) M.F. Martínez-Serrano, N. Castro-González, On the Drazin inverse of block matrices and generalized Schur complement, Appl. Math. Comput ) J. Miao, Results of the Drazin inverse of block matrices, J. Shanghai Normal Univ ) Y. Wei, Expressions for the Drazin inverse of 2 2 block matrix, Linear and Multilinear Algebra ) Address: Faculty of Sciences and Mathematics, University of Niš, P.O. Box 224, Niš, Serbia D. Mosić: dijana@pmf.ni.ac.rs D. S. Djordjević: dragan@pmf.ni.ac.rs 13
Formulae for the generalized Drazin inverse of a block matrix in terms of Banachiewicz Schur forms
Formulae for the generalized Drazin inverse of a block matrix in terms of Banachiewicz Schur forms Dijana Mosić and Dragan S Djordjević Abstract We introduce new expressions for the generalized Drazin
More informationAdditive results for the generalized Drazin inverse in a Banach algebra
Additive results for the generalized Drazin inverse in a Banach algebra Dragana S. Cvetković-Ilić Dragan S. Djordjević and Yimin Wei* Abstract In this aer we investigate additive roerties of the generalized
More informationELA
REPRESENTATONS FOR THE DRAZN NVERSE OF BOUNDED OPERATORS ON BANACH SPACE DRAGANA S. CVETKOVĆ-LĆ AND MN WE Abstract. n this paper a representation is given for the Drazin inverse of a 2 2operator matrix
More informationSome results on the reverse order law in rings with involution
Some results on the reverse order law in rings with involution Dijana Mosić and Dragan S. Djordjević Abstract We investigate some necessary and sufficient conditions for the hybrid reverse order law (ab)
More informationADDITIVE RESULTS FOR THE GENERALIZED DRAZIN INVERSE
ADDITIVE RESULTS FOR THE GENERALIZED DRAZIN INVERSE Dragan S Djordjević and Yimin Wei Abstract Additive perturbation results for the generalized Drazin inverse of Banach space operators are presented Precisely
More informationWeighted generalized Drazin inverse in rings
Weighted generalized Drazin inverse in rings Dijana Mosić and Dragan S Djordjević Abstract In this paper we introduce and investigate the weighted generalized Drazin inverse for elements in rings We also
More informationFormulas for the Drazin Inverse of Matrices over Skew Fields
Filomat 30:12 2016 3377 3388 DOI 102298/FIL1612377S Published by Faculty of Sciences and Mathematics University of Niš Serbia Available at: http://wwwpmfniacrs/filomat Formulas for the Drazin Inverse of
More informationarxiv: v1 [math.ra] 15 Jul 2013
Additive Property of Drazin Invertibility of Elements Long Wang, Huihui Zhu, Xia Zhu, Jianlong Chen arxiv:1307.3816v1 [math.ra] 15 Jul 2013 Department of Mathematics, Southeast University, Nanjing 210096,
More informationThe Expression for the Group Inverse of the Anti triangular Block Matrix
Filomat 28:6 2014, 1103 1112 DOI 102298/FIL1406103D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat The Expression for the Group Inverse
More informationMoore-Penrose-invertible normal and Hermitian elements in rings
Moore-Penrose-invertible normal and Hermitian elements in rings Dijana Mosić and Dragan S. Djordjević Abstract In this paper we present several new characterizations of normal and Hermitian elements in
More informationPerturbation of the generalized Drazin inverse
Electronic Journal of Linear Algebra Volume 21 Volume 21 (2010) Article 9 2010 Perturbation of the generalized Drazin inverse Chunyuan Deng Yimin Wei Follow this and additional works at: http://repository.uwyo.edu/ela
More informationFactorization of weighted EP elements in C -algebras
Factorization of weighted EP elements in C -algebras Dijana Mosić, Dragan S. Djordjević Abstract We present characterizations of weighted EP elements in C -algebras using different kinds of factorizations.
More informationarxiv: v1 [math.ra] 27 Jul 2013
Additive and product properties of Drazin inverses of elements in a ring arxiv:1307.7229v1 [math.ra] 27 Jul 2013 Huihui Zhu, Jianlong Chen Abstract: We study the Drazin inverses of the sum and product
More informationEP elements in rings
EP elements in rings Dijana Mosić, Dragan S. Djordjević, J. J. Koliha Abstract In this paper we present a number of new characterizations of EP elements in rings with involution in purely algebraic terms,
More informationInner image-kernel (p, q)-inverses in rings
Inner image-kernel (p, q)-inverses in rings Dijana Mosić Dragan S. Djordjević Abstract We define study the inner image-kernel inverse as natural algebraic extension of the inner inverse with prescribed
More informationRepresentations for the Drazin Inverse of a Modified Matrix
Filomat 29:4 (2015), 853 863 DOI 10.2298/FIL1504853Z Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Representations for the Drazin
More informationDrazin Invertibility of Product and Difference of Idempotents in a Ring
Filomat 28:6 (2014), 1133 1137 DOI 10.2298/FIL1406133C Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Drazin Invertibility of
More informationRe-nnd solutions of the matrix equation AXB = C
Re-nnd solutions of the matrix equation AXB = C Dragana S. Cvetković-Ilić Abstract In this article we consider Re-nnd solutions of the equation AXB = C with respect to X, where A, B, C are given matrices.
More informationThe characterizations and representations for the generalized inverses with prescribed idempotents in Banach algebras
Filomat 27:5 (2013), 851 863 DOI 10.2298/FIL1305851C Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat The characterizations and
More informationPartial isometries and EP elements in rings with involution
Electronic Journal of Linear Algebra Volume 18 Volume 18 (2009) Article 55 2009 Partial isometries and EP elements in rings with involution Dijana Mosic dragan@pmf.ni.ac.yu Dragan S. Djordjevic Follow
More informationA Note on the Group Inverses of Block Matrices Over Rings
Filomat 31:1 017, 3685 369 https://doiorg/1098/fil171685z Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat A Note on the Group Inverses
More informationGeneralized B-Fredholm Banach algebra elements
Generalized B-Fredholm Banach algebra elements Miloš D. Cvetković, Enrico Boasso, Snežana Č. Živković-Zlatanović Abstract Given a (not necessarily continuous) homomorphism between Banach algebras T : A
More informationGroup inverse for the block matrix with two identical subblocks over skew fields
Electronic Journal of Linear Algebra Volume 21 Volume 21 2010 Article 7 2010 Group inverse for the block matrix with two identical subblocks over skew fields Jiemei Zhao Changjiang Bu Follow this and additional
More informationDragan S. Djordjević. 1. Introduction
UNIFIED APPROACH TO THE REVERSE ORDER RULE FOR GENERALIZED INVERSES Dragan S Djordjević Abstract In this paper we consider the reverse order rule of the form (AB) (2) KL = B(2) TS A(2) MN for outer generalized
More informationREPRESENTATIONS FOR THE GENERALIZED DRAZIN INVERSE IN A BANACH ALGEBRA (COMMUNICATED BY FUAD KITTANEH)
Bulletin of Mathematical Analysis an Applications ISSN: 1821-1291, UL: http://www.bmathaa.org Volume 5 Issue 1 (2013), ages 53-64 EESENTATIONS FO THE GENEALIZED DAZIN INVESE IN A BANACH ALGEBA (COMMUNICATED
More informationOn EP elements, normal elements and partial isometries in rings with involution
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012 Article 39 2012 On EP elements, normal elements and partial isometries in rings with involution Weixing Chen wxchen5888@163.com Follow this
More informationEP elements and Strongly Regular Rings
Filomat 32:1 (2018), 117 125 https://doi.org/10.2298/fil1801117y Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat EP elements and
More informationWeighted Drazin inverse of a modified matrix
Functional Analysis, Approximation Computation 6 (2) (2014), 23 28 Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac Weighted Drazin inverse
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 432 21 1691 172 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Drazin inverse of partitioned
More informationELA ON THE GROUP INVERSE OF LINEAR COMBINATIONS OF TWO GROUP INVERTIBLE MATRICES
ON THE GROUP INVERSE OF LINEAR COMBINATIONS OF TWO GROUP INVERTIBLE MATRICES XIAOJI LIU, LINGLING WU, AND JULIO BENíTEZ Abstract. In this paper, some formulas are found for the group inverse of ap +bq,
More informationarxiv: v1 [math.ra] 25 Jul 2013
Representations of the Drazin inverse involving idempotents in a ring Huihui Zhu, Jianlong Chen arxiv:1307.6623v1 [math.ra] 25 Jul 2013 Abstract: We present some formulae for the Drazin inverse of difference
More informationGENERALIZED HIRANO INVERSES IN BANACH ALGEBRAS arxiv: v1 [math.fa] 30 Dec 2018
GENERALIZED HIRANO INVERSES IN BANACH ALGEBRAS arxiv:1812.11618v1 [math.fa] 30 Dec 2018 HUANYIN CHEN AND MARJAN SHEIBANI Abstract. Let A be a Banach algebra. An element a A has generalized Hirano inverse
More informationOn 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
More informationOperators with Compatible Ranges
Filomat : (7), 579 585 https://doiorg/98/fil7579d Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat Operators with Compatible Ranges
More informationRepresentations of the (b, c)-inverses in Rings with Involution
Filomat 31:9 (217), 2867 2875 htts://doiorg/12298/fil179867k Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt://wwwmfniacrs/filomat Reresentations of the (b,
More informationarxiv: v1 [math.ra] 28 Jan 2016
The Moore-Penrose inverse in rings with involution arxiv:1601.07685v1 [math.ra] 28 Jan 2016 Sanzhang Xu and Jianlong Chen Department of Mathematics, Southeast University, Nanjing 210096, China Abstract:
More informationarxiv: v1 [math.ra] 7 Aug 2017
ON HIRANO INVERSES IN RINGS arxiv:1708.07043v1 [math.ra] 7 Aug 2017 HUANYIN CHEN AND MARJAN SHEIBANI Abstract. We introduce and study a new class of Drazin inverses. AnelementainaringhasHiranoinversebifa
More informationOn Sum and Restriction of Hypo-EP Operators
Functional Analysis, Approximation and Computation 9 (1) (2017), 37 41 Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac On Sum and
More informationThe Moore-Penrose inverse of 2 2 matrices over a certain -regular ring
The Moore-Penrose inverse of 2 2 matrices over a certain -regular ring Huihui Zhu a, Jianlong Chen a,, Xiaoxiang Zhang a, Pedro Patrício b a Department of Mathematics, Southeast University, Nanjing 210096,
More informationMoore Penrose inverses and commuting elements of C -algebras
Moore Penrose inverses and commuting elements of C -algebras Julio Benítez Abstract Let a be an element of a C -algebra A satisfying aa = a a, where a is the Moore Penrose inverse of a and let b A. We
More informationThe Moore-Penrose inverse of differences and products of projectors in a ring with involution
The Moore-Penrose inverse of differences and products of projectors in a ring with involution Huihui ZHU [1], Jianlong CHEN [1], Pedro PATRÍCIO [2] Abstract: In this paper, we study the Moore-Penrose inverses
More informationOn the Moore-Penrose and the Drazin inverse of two projections on Hilbert space
On the Moore-Penrose and the Drazin inverse of two projections on Hilbert space Sonja Radosavljević and Dragan SDjordjević March 13, 2012 Abstract For two given orthogonal, generalized or hypergeneralized
More informationDrazin invertibility of sum and product of closed linear operators
Malaya J. Mat. 2(4)(2014) 472 481 Drazin invertibility of sum and product of closed linear operators Bekkai MESSIRDI a,, Sanaa MESSIRDI b and Miloud MESSIRDI c a Département de Mathématiques, Université
More informationNonsingularity and group invertibility of linear combinations of two k-potent matrices
Nonsingularity and group invertibility of linear combinations of two k-potent matrices Julio Benítez a Xiaoji Liu b Tongping Zhu c a Departamento de Matemática Aplicada, Instituto de Matemática Multidisciplinar,
More informationEXPLICIT SOLUTION OF THE OPERATOR EQUATION A X + X A = B
EXPLICIT SOLUTION OF THE OPERATOR EQUATION A X + X A = B Dragan S. Djordjević November 15, 2005 Abstract In this paper we find the explicit solution of the equation A X + X A = B for linear bounded operators
More informationMultiplicative Perturbation Bounds of the Group Inverse and Oblique Projection
Filomat 30: 06, 37 375 DOI 0.98/FIL67M Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Multiplicative Perturbation Bounds of the Group
More informationTHE EXPLICIT EXPRESSION OF THE DRAZIN INVERSE OF SUMS OF TWO MATRICES AND ITS APPLICATION
italian journal of pure an applie mathematics n 33 04 (45 6) 45 THE EXPLICIT EXPRESSION OF THE DRAZIN INVERSE OF SUMS OF TWO MATRICES AND ITS APPLICATION Xiaoji Liu Liang Xu College of Science Guangxi
More informationSCHUR COMPLEMENTS IN C ALGEBRAS
mn header will be provided by the publisher SCHUR COMPLEMENTS IN C ALGEBRAS Dragana S. Cvetković-Ilić 1, Dragan S. Djordjević 1, and Vladimir Rakočević 1 1 Department of Mathematics, Faculty of Science
More informationResearch Article Partial Isometries and EP Elements in Banach Algebras
Abstract and Applied Analysis Volume 2011, Article ID 540212, 9 pages doi:10.1155/2011/540212 Research Article Partial Isometries and EP Elements in Banach Algebras Dijana Mosić and Dragan S. Djordjević
More informationThe DMP Inverse for Rectangular Matrices
Filomat 31:19 (2017, 6015 6019 https://doi.org/10.2298/fil1719015m Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://.pmf.ni.ac.rs/filomat The DMP Inverse for
More informationDragan S. Djordjević. 1. Introduction and preliminaries
PRODUCTS OF EP OPERATORS ON HILBERT SPACES Dragan S. Djordjević Abstract. A Hilbert space operator A is called the EP operator, if the range of A is equal with the range of its adjoint A. In this article
More informationAPPLICATIONS OF THE HYPER-POWER METHOD FOR COMPUTING MATRIX PRODUCTS
Univ. Beograd. Publ. Eletrotehn. Fa. Ser. Mat. 15 (2004), 13 25. Available electronically at http: //matematia.etf.bg.ac.yu APPLICATIONS OF THE HYPER-POWER METHOD FOR COMPUTING MATRIX PRODUCTS Predrag
More informationDragan S. Djordjević. 1. Motivation
ON DAVIS-AAN-WEINERGER EXTENSION TEOREM Dragan S. Djordjević Abstract. If R = where = we find a pseudo-inverse form of all solutions W = W such that A = R where A = W and R. In this paper we extend well-known
More informationGeneralized left and right Weyl spectra of upper triangular operator matrices
Electronic Journal of Linear Algebra Volume 32 Volume 32 (2017 Article 3 2017 Generalized left and right Weyl spectra of upper triangular operator matrices Guojun ai 3695946@163.com Dragana S. Cvetkovic-Ilic
More informationNote on paranormal operators and operator equations ABA = A 2 and BAB = B 2
Note on paranormal operators and operator equations ABA = A 2 and BAB = B 2 Il Ju An* Eungil Ko PDE and Functional Analysis Research Center (PARC) Seoul National University Seoul, Korea ICM Satellite Conference
More informationRonalda Benjamin. Definition A normed space X is complete if every Cauchy sequence in X converges in X.
Group inverses in a Banach algebra Ronalda Benjamin Talk given in mathematics postgraduate seminar at Stellenbosch University on 27th February 2012 Abstract Let A be a Banach algebra. An element a A is
More informationof a Two-Operator Product 1
Applied Mathematical Sciences, Vol. 7, 2013, no. 130, 6465-6474 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.39501 Reverse Order Law for {1, 3}-Inverse of a Two-Operator Product 1 XUE
More informationExpressions and Perturbations for the Moore Penrose Inverse of Bounded Linear Operators in Hilbert Spaces
Filomat 30:8 (016), 155 164 DOI 10.98/FIL1608155D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Expressions and Perturbations
More informationSome Expressions for the Group Inverse of the Block Partitioned Matrices with an Invertible Subblock
Ordu University Journal of Science and Technology/ Ordu Üniversitesi ilim ve Teknoloji Dergisi Ordu Univ. J. Sci. Tech., 2018; 8(1): 100-113 Ordu Üniv. il. Tek. Derg., 2018; 8(1): 100-113 e-issn: 2146-6459
More informationOn the spectrum of linear combinations of two projections in C -algebras
On the spectrum of linear combinations of two projections in C -algebras Julio Benítez and Vladimir Rakočević Abstract In this note we study the spectrum and give estimations for the spectral radius of
More informationPredrag S. Stanimirović and Dragan S. Djordjević
FULL-RANK AND DETERMINANTAL REPRESENTATION OF THE DRAZIN INVERSE Predrag S. Stanimirović and Dragan S. Djordjević Abstract. In this article we introduce a full-rank representation of the Drazin inverse
More informationDecomposition of a ring induced by minus partial order
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012) Article 72 2012 Decomposition of a ring induced by minus partial order Dragan S Rakic rakicdragan@gmailcom Follow this and additional works
More informationarxiv: v1 [math.ra] 21 Jul 2013
Projections and Idempotents in -reducing Rings Involving the Moore-Penrose Inverse arxiv:1307.5528v1 [math.ra] 21 Jul 2013 Xiaoxiang Zhang, Shuangshuang Zhang, Jianlong Chen, Long Wang Department of Mathematics,
More informationSome additive results on Drazin inverse
Linear Algebra and its Applications 322 (2001) 207 217 www.elsevier.com/locate/laa Some additive results on Drazin inverse Robert E. Hartwig a, Guorong Wang a,b,1, Yimin Wei c,,2 a Mathematics Department,
More informationThe (2,2,0) Group Inverse Problem
The (2,2,0) Group Inverse Problem P. Patrício a and R.E. Hartwig b a Departamento de Matemática e Aplicações, Universidade do Minho, 4710-057 Braga, Portugal. e-mail: pedro@math.uminho.pt b Mathematics
More informationThe Drazin inverses of products and differences of orthogonal projections
J Math Anal Appl 335 7 64 71 wwwelseviercom/locate/jmaa The Drazin inverses of products and differences of orthogonal projections Chun Yuan Deng School of Mathematics Science, South China Normal University,
More informationDRAZIN INVERTIBILITY OF PRODUCTS
http://www.mathematik.uni-karlsruhe.de/ semlv Seminar LV, No. 26, 5 pp., 1.6.2006 DRAZIN INVERTIBILITY OF PRODUCTS CHRISTOPH SCHMOEGER Abstract. If a b are elements of an algebra, then we show that ab
More informationSOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES. S. S. Dragomir
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 5: 011), 151 16 DOI: 10.98/FIL110151D SOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR
More informationEXPLICIT SOLUTION TO MODULAR OPERATOR EQUATION T XS SX T = A
EXPLICIT SOLUTION TO MODULAR OPERATOR EQUATION T XS SX T A M MOHAMMADZADEH KARIZAKI M HASSANI AND SS DRAGOMIR Abstract In this paper by using some block operator matrix techniques we find explicit solution
More informationarxiv: v1 [math.fa] 5 Nov 2012
ONCE MORE ON POSITIVE COMMUTATORS ROMAN DRNOVŠEK arxiv:1211.0812v1 [math.fa] 5 Nov 2012 Abstract. Let A and B be bounded operators on a Banach lattice E such that the commutator C = AB BA and the product
More informationEXPLICIT SOLUTION TO MODULAR OPERATOR EQUATION T XS SX T = A
Kragujevac Journal of Mathematics Volume 4(2) (216), Pages 28 289 EXPLICI SOLUION O MODULAR OPERAOR EQUAION XS SX A M MOHAMMADZADEH KARIZAKI 1, M HASSANI 2, AND S S DRAGOMIR 3 Abstract In this paper, by
More informationWorkshop on Generalized Inverse and its Applications. Invited Speakers Program Abstract
Workshop on Generalized Inverse and its Applications Invited Speakers Program Abstract Southeast University, Nanjing November 2-4, 2012 Invited Speakers: Changjiang Bu, College of Science, Harbin Engineering
More informationSome Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces
Some Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces S.S. Dragomir Abstract. Some new inequalities for commutators that complement and in some instances improve recent results
More informationSingular Value Inequalities for Real and Imaginary Parts of Matrices
Filomat 3:1 16, 63 69 DOI 1.98/FIL16163C Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Singular Value Inequalities for Real Imaginary
More informationResearch Article On the Idempotent Solutions of a Kind of Operator Equations
International Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 84665, pages doi:0.540/0/84665 Research Article On the Idempotent Solutions of a Kind of Operator Equations Chun Yuan
More informationNew Extensions of Cline s Formula for Generalized Inverses
Filomat 31:7 2017, 1973 1980 DOI 10.2298/FIL1707973Z Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat New Extensions of Cline s
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationON OPERATORS WITH AN ABSOLUTE VALUE CONDITION. In Ho Jeon and B. P. Duggal. 1. Introduction
J. Korean Math. Soc. 41 (2004), No. 4, pp. 617 627 ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION In Ho Jeon and B. P. Duggal Abstract. Let A denote the class of bounded linear Hilbert space operators with
More informationOn the invertibility of the operator A XB
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 29; 16:817 831 Published online 8 April 29 in Wiley InterScience www.interscience.wiley.com)..646 On the invertibility of the operator
More informationThe Residual Spectrum and the Continuous Spectrum of Upper Triangular Operator Matrices
Filomat 28:1 (2014, 65 71 DOI 10.2298/FIL1401065H Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat The Residual Spectrum and the
More informationarxiv: v1 [math.ra] 14 Apr 2018
Three it representations of the core-ep inverse Mengmeng Zhou a, Jianlong Chen b,, Tingting Li c, Dingguo Wang d arxiv:180.006v1 [math.ra] 1 Apr 018 a School of Mathematics, Southeast University, Nanjing,
More informationarxiv: v1 [math.ra] 8 Apr 2016
ON A DETERMINANTAL INEQUALITY ARISING FROM DIFFUSION TENSOR IMAGING MINGHUA LIN arxiv:1604.04141v1 [math.ra] 8 Apr 2016 Abstract. In comparing geodesics induced by different metrics, Audenaert formulated
More informationVariations on Weyl Type Theorems
Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 4, 189-198 HIKARI Ltd, www.m-hikari.com Variations on Weyl Type Theorems Anuradha Gupta Department of Mathematics Delhi College of Arts and Commerce University
More informationNIL, NILPOTENT AND PI-ALGEBRAS
FUNCTIONAL ANALYSIS AND OPERATOR THEORY BANACH CENTER PUBLICATIONS, VOLUME 30 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1994 NIL, NILPOTENT AND PI-ALGEBRAS VLADIMÍR MÜLLER Institute
More informationThe generalized Schur complement in group inverses and (k + 1)-potent matrices
The generalized Schur complement in group inverses and (k + 1)-potent matrices Julio Benítez Néstor Thome Abstract In this paper, two facts related to the generalized Schur complement are studied. The
More informationand Michael White 1. Introduction
SPECTRAL CHARACTERIZATION OF ALGEBRAIC ELEMENTS Thomas Ransford 1 and Michael White 1. Introduction Let A be a unital Banach algebra. An element a A is algebraic if there exists a nonconstant polynomial
More informationJacobson s lemma via Gröbner-Shirshov bases
Jacobson s lemma via Gröbner-Shirshov bases arxiv:1702.06271v1 [math.ra] 21 Feb 2017 Xiangui Zhao Department of Mathematics, Huizhou University Huizhou, Guangdong Province 516007, China zhaoxg@hzu.edu.cn
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 433 (2010) 476 482 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Nonsingularity of the
More informationJENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES FOR OPERATORS IN HILBERT SPACE
JENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES FOR OPERATORS IN HILBERT SPACE MARIO KRNIĆ NEDA LOVRIČEVIĆ AND JOSIP PEČARIĆ Abstract. In this paper we consider Jensen s operator which includes
More informationPrincipal pivot transforms of range symmetric matrices in indefinite inner product space
Functional Analysis, Approximation and Computation 8 (2) (2016), 37 43 Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac Principal pivot
More informationThe singular value of A + B and αa + βb
An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. (N.S.) Tomul LXII, 2016, f. 2, vol. 3 The singular value of A + B and αa + βb Bogdan D. Djordjević Received: 16.II.2015 / Revised: 3.IV.2015 / Accepted: 9.IV.2015
More informationSome basic properties of idempotent matrices
J. Edu. & Sci., Vol. (21), No. (4) 2008 Some basic properties of idempotent matrices Alaa A. Hammodat Ali A. Bilal Akram S. Mohammed Dept. of Math. Dept. of Math. Dept. of Math. College of Education College
More informationOn product of range symmetric matrices in an indefinite inner product space
Functional Analysis, Approximation and Computation 8 (2) (2016), 15 20 Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac On product
More informationMATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006
MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006 Sherod Eubanks HOMEWORK 2 2.1 : 2, 5, 9, 12 2.3 : 3, 6 2.4 : 2, 4, 5, 9, 11 Section 2.1: Unitary Matrices Problem 2 If λ σ(u) and U M n is unitary, show that
More informationJACOBSON S LEMMA FOR DRAZIN INVERSES
JACOBSON S LEMMA FOR DRAZIN INVERSES T.Y. LAM AND PACE P. NIELSEN Abstract. If a ring element α = 1 ab is Drazin invertible, so is β = 1 ba. We show that the Drazin inverse of β is expressible by a simple
More informationInvertible Matrices over Idempotent Semirings
Chamchuri Journal of Mathematics Volume 1(2009) Number 2, 55 61 http://www.math.sc.chula.ac.th/cjm Invertible Matrices over Idempotent Semirings W. Mora, A. Wasanawichit and Y. Kemprasit Received 28 Sep
More informationAbel rings and super-strongly clean rings
An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. N.S. Tomul LXIII, 2017, f. 2 Abel rings and super-strongly clean rings Yinchun Qu Junchao Wei Received: 11.IV.2013 / Last revision: 10.XII.2013 / Accepted: 12.XII.2013
More informationBulletin of the. Iranian Mathematical Society
ISSN 1017-060X (Print ISSN 1735-8515 (Online Bulletin of the Iranian Mathematical Society Vol 42 (2016, No 1, pp 53 60 Title The reverse order law for Moore-Penrose inverses of operators on Hilbert C*-modules
More informationACCEPTS HUGE FLORAL KEY TO LOWELL. Mrs, Walter Laid to Rest Yesterday
$ j < < < > XXX Y 928 23 Y Y 4% Y 6 -- Q 5 9 2 5 Z 48 25 )»-- [ Y Y Y & 4 j q - Y & Y 7 - -- - j \ -2 -- j j -2 - - - - [ - - / - ) ) - - / j Y 72 - ) 85 88 - / X - j ) \ 7 9 Y Y 2 3» - ««> Y 2 5 35 Y
More informationSome bounds for the spectral radius of the Hadamard product of matrices
Some bounds for the spectral radius of the Hadamard product of matrices Guang-Hui Cheng, Xiao-Yu Cheng, Ting-Zhu Huang, Tin-Yau Tam. June 1, 2004 Abstract Some bounds for the spectral radius of the Hadamard
More information