J. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5

Vol. 37 ( 2017 ) No. 5 J. of Math. (PRC) 1,2, 1, 1 (1., 225002) (2., 225009) :. I +AT +, T + = T + (I +AT + ) 1, T +. Banach Hilbert Moore-Penrose.. : ; ; Moore-Penrose ; ; MR(2010) : 47L05; 46A32 : O177.2 : A : 0255-7797(2017)05-1013-09 1 X, Y, L(X, Y ) X Y. T L(X, Y ), D(T ), N(T ) R(T ) T,. I. M, N X, M +N M N. X, Y Banach, M N X, M N M N. B(X, Y ), C(X, Y ) X Y Banach. 1.1 [1,2] X, Y, T L(X, Y ). S L(Y, X) R(T ) D(S), R(ST ) D(T ), T ST = T, S T ; S R(S) D(T ), R(T S) D(S), ST S = S, S T. S T, T, S T, T +. [3],. Banach Hilbert,,. 1.2 [3] X, Y Banach, T B(X, Y ). S B(Y, X) T ST = T, S T ; S ST S = S, S T. S T, T, S T, T +. Banach, 1.3 [4] T + B(Y, X) T B(X, Y ) X, Y, X = N(T ) R(T + ), Y = R(T ) N(T + ). : 2017-03-10 : 2017-06-02 (11771378; 11271316); (BK20141271); (2016zqn03). : (1970 ),,,, :. :.

1014 Vol. 37 1.4 [3] X, Y Hilbert, 1.3, T Moore-Penrose, T. 1.5 [5] X, Y Banach, T C(X, Y ). S C(Y, X) R(S) D(T ), R(T ) D(S), D(T ) T ST = T ; D(S) ST S = S, S T ( ), T +. 1.6 [3] X, Y Banach, T C(X, Y ), N(T ) R(T ) R(T ) X, Y N(T ) c R(T ) c, X = N(T ) N(T ) c, Y = R(T ) R(T ) c. (1.1) P, Q X N(T ) c N(T ) Y R(T ) c R(T ), S C(Y, X) : 1) D(T ) T ST = T ; 2) D(S) ST S = S; 3) D(T ) ST = I P 4) D(S) T S = Q, D(S) = R(T ) + R(T ) c., S R(T ) Y. 1.6, (1.1), T. X, Y Hilbert, (1.1), T Moore-Penrose, T., [3,4,6 8].,, ;, ( ). Banach Hilbert Moore-Penrose [3,6,7,9 18], 1.7 [7] X, Y Banach, T B(X, Y ) T + B(Y, X), δt B(X, Y ) δt T + < 1. (1) B = T + (I + δt T + ) 1 T = T + δt ; (2) R(T ) N(T + ) = {0}; (3) (I T + T )N(T ) = N(T ); (4) X = N(T ) R(T + ) X = N(T ) + R(T + ); (5) Y = R(T ) N(T + ); (6) (I + δt T + ) 1 T N(T ) R(T ). 1.8 [9] X, Y Hilbert, T B(X, Y ) Moore-Penrose T B(Y, X). δt B(X, Y ) δt T < 1, B = T (I +δt T ) 1 T = T +δt Moore-Penrose R(T ) = R(T ), N(T ) = N(T ). Banach Hilbert, Moore-Penrose,.,. 1974, Nashed Votruba [1,2,3]. Banach,., [3].?

No. 5 : 1015?, [6, 7, 9, 12 16, 18]. 2 2.1 T L(X, Y ), S L(Y, X) R(S) D(T ), R(T ) D(S). S T, (1) S T ; (2) R(T ) = R(T S); (3) D(S) = R(T ) +N(S); (4) R(T ) N(S) = {0}; (5) N(ST ) = N(T ); (6) D(T ) = R(S) +N(T ) D(T ) = R(S) + N(T ). S T, T S, ST, R(S) = R(ST ), N(T S) = N(S), D(T ) = R(ST ) +N(ST ) = R(S) +N(ST ), D(S) = R(T S) +N(T S) = R(T S) +N(S). (1) (2) S T, T ST = T, R(T ) = R(T ST ) R(T S) R(T ). (2) (3) R(T ) = R(T S), D(S) = R(T S) +N(S) = R(T ) +N(S). (3) (4). (4) (1) x D(T ), S(T ST x T x) = ST ST x ST x = ST x ST x = 0, T ST x T x N(S). T ST x T x R(T ), (4), T ST x = T x. T ST = T., S T. (1) (5) S T, T ST = T. ST x = 0, T x = T ST x = 0, N(ST ) N(T ), N(T ) N(ST ). N(ST ) = N(T ). (5) (6) N(ST ) = N(T ), D(T ) = R(S) +N(ST ) = R(S) +N(T ). (6) (1) x D(T ), (6), y D(S) x 1 N(T ), x = Sy + x 1. T ST x = T ST (Sy + x 1 ) = T ST Sy = T Sy = T (Sy + x 1 ) = T x. T ST = T, S T.. 2.2 T + L(Y, X) T L(X, Y ), A L(X, Y ) D(T ) D(A), R(T + ) D(A), R(A) D(T + ). T = T + A, (1) I + AT + : D(T + ) D(T + ) ; (2) I + T + A : D(T ) D(T ) ; (3) N(T ) R(T + ) = {0} D(T + ) = T R(T + ) +N(T + ). (1) (2) I + T + A : D(T ) D(T ). x D(T ), (I + T + A)x = 0, A(I + T + A)x = 0, (I + AT + )Ax = 0. (1), Ax = 0, x = T + Ax = 0. I + T + A : D(T ) D(T ), y D(T ), x D(T ) (I +T + A)x = y., Ay D(T + ), (1), h D(T + ), (I +AT + )h = Ay. x = y T + h, x D(T ), (I + T + A)x = (I + T + A)(y T + h) = y + T + Ay (I + T + A)T + h = y + T + Ay T + (I + AT + )h = y + T + Ay T + Ay = y.

1016 Vol. 37 (2) (3) I+T + A : D(T ) D(T ). y N(T ) R(T + ), x D(T + ), y = T + x, T T + x = T y = 0. (I + T + A)T + x = T + x + T + AT + x = T + T T + x + T + AT + x = T + T T + x = 0. T + x = 0, y = T + x = 0. N(T ) R(T + ) = {0}. T R(T + ) N(T + ) = {0}. v T R(T + ) N(T + ), u D(T + ), v = T T + u. 0 = T + v = T + T T + u = T + (T + A)T + u = T + T T + u + T + AT + u = T + u + T + AT + u = (I + T + A)T + u. T + u = 0, v = T T + u = 0. D(T + ) = T R(T + ) + N(T + )., T R(T + ) + N(T + ) D(T + )., x D(T + ), T + x D(T ). (2), y D(T ), T + x = (I + T + A)y, T + x = (I T + T )y + T + T y. T + (x T y) = (I T + T )y R(T + ) N(T ) = {0}. y = T + T y R(T + ), T + x = T + T y, x T y N(T + ). x = T y + (x T y) = T T + T y + (x T y) T R(T + ) + N(T + ), D(T + ) = T R(T + ) + N(T + ). (3) (1) I + AT + : D(T + ) D(T + )., x D(T + ) (I +AT + )x = 0, T T + x = T T + x x T R(T + ) N(T + ) = {0}, x = T T + x, T T + x = 0. T + x N(T ) R(T + ), (3), T + x = 0. x = T T + x = 0., I + AT + : D(T + ) D(T + ). y D(T + ), D(T + ) = T R(T + ) + N(T + ), y y = T T + y 1 + y 2, y 1 D(T + ), y 2 N(T + ). (I + AT + )(T T + y 1 + y 2 ) = (T T + y 1 + y 2 ) + (T T )T + T T + y 1 = T T + y 1 + y 2 + T T + y 1 T T + y 1 = y 2 + T T + y 1 = y. I + AT + : D(T + ) D(T + ).. T T + T T +,, T +,. 2.3 T + L(Y, X) T L(X, Y ), A L(X, Y ) D(T ) D(A), R(T + ) D(A), R(A) D(T + ). T = T + A T + L(Y, X), D(T + ) = D(T + ), N(T + ) = N(T + ), R(T + ) = R(T + ), I + AT + : D(T + ) D(T + ), T + = T + (I + AT + ) 1 = (I + T + A) 1 T +.

No. 5 : 1017 T + T, N(T ) R(T + ) = {0} D(T + ) = R(T T + ) +N(T T + ) = T R(T + ) +N(T + )., N(T ) R(T + ) = {0}, D(T + ) = T R(T + ) +N(T + ). 2.2, I + AT + : D(T + ) D(T + ) I + T + A : D(T ) D(T ). T + (I + AT + ) 1 (I + T + A) 1 T +, T + (I + AT + ) 1 = (I + T + A) 1 T +. N(T + ) = N(T + ) T + (I T T + ) = 0 R(T + ) = R(T + ) (I T + T )T + = 0, T + = T + T T + T + T T + = T +. T + +T + T T + T + T T + = T +, T + (I +AT + ) = T +., T + = T + (I + AT + ) 1.. T + T + = T + (I + AT + ) 1. 2.4 T + L(Y, X) T L(X, Y ), A L(X, Y ) D(T ) D(A), R(T + ) D(A), R(A) D(T + ). I + AT + : D(T + ) D(T + ), (1) B = T + (I + AT + ) 1 = (I + T + A) 1 T + T = T + A ; (2) R(T ) = R(T T + ); (3) D(T + ) = R(T ) +N(T + ); (4) R(T ) N(T + ) = {0}; (5) N(T + T ) = N(T ); (6) D(T ) = N(T ) +R(T + ) D(T ) = N(T ) + R(T + ); (7) (I + AT + ) 1 R(T ) = R(T ); (8) (I + T + A) 1 N(T ) = N(T ); (9) (I + AT + ) 1 T N(T ) R(T ). B = T + (I + AT + ) 1 = (I + T + A) 1 T +, R(B) = R(T + ), N(B) = N(T + ). BT B = (I + T + A) 1 T + (T + A)T + (I + AT + ) 1 = (I + T + A) 1 (T + + T + AT + )(I + AT + ) 1 = (I + T + A) 1 (I + T + A)T + (I + AT + ) 1 = T + (I + AT + ) 1 = B, B T. R(T B) = R(T T + ) N(BT ) = N(T + T ), 2.1, (1) (6). (1) (7) B T, R(T ) = R(T B) = T R(B) = T R(T + ) = T R(T + T ) = T T + R(T ) = (T T + + I T T + )R(T ) = (I + AT + )R(T ). (7) (8), (I + T + A)N(T ) = [I + T + (T T )]N(T ) = (I T + T )N(T ) N(T )., (7), x N(T ), T x R(T ) (I + AT + )R(T ) = T T + R(T ) = T R(T + T ) = T R(T + ).

1018 Vol. 37 y R(T + ), T y = T x. x y N(T ), (I + T + A)(x y) = (I T + T )(x y) = (I T + T )x = x. N(T ) (I + T + A)N(T ). (8) (4) y R(T ) N(T + ), x D(T ), y = T x T + T x = 0. T (I + T + A)x = T x + T T + Ax = T x + T T + T x T x = 0, (I + T + A)x N(T ). (8), x N(T ), y = T x = 0. (7) (9). (9) (4) y R(T ) N(T + ), x D(T ) = D(T ) y = T x T + T x = 0. D(T ) = N(T ) +R(T + ), x = x 1 + x 2, x 1 N(T ), x 2 R(T + ). (I + AT + )T x 2 = [I + (T T )T + ]T x 2 = T T + T x 2 = T x 2. (I + AT + ) 1 T x 2 = T x 2 R(T ). (9), (I + AT + ) 1 T x 1 R(T ). y N(T + ), (I + AT + )y = y = T x, y = (I + AT + ) 1 T x = (I + AT + ) 1 T (x 1 + x 2 ) R(T ). y R(T ) N(T + ). R(T ) N(T + ) = {0}, y = 0.. 2.5 T L(X, Y ), T + L(Y, X) T, A L(X, Y ) D(T ) D(A), R(T + ) D(A), R(A) D(T + ). I+AT + : D(T + ) D(T + ), B = T + (I + AT + ) 1 = (I + T + A) 1 T + T = T + A Rank T = Rank T < +. 2.4 (1) (7).. (I + AT + )T = T + AT + T = T T + T, T = (I + AT + ) 1 T T + T. Rank T = Rank T, dim R(T ) = dim R(T ) = dim R(T T + T ). R(T ) = R(T T + T ) R(T T + ) R(T ). 2.4 (1) (2), B T.., 2.6 T + L(Y, X) T L(X, Y ), A L(X, Y ) D(T ) D(A), R(T + ) D(A), R(A) D(T + ). dim N(T ) < +, I + AT + : D(T + ) D(T + ), B = T + (I + AT + ) 1 = (I + T + A) 1 T + T = T + A, dim N(T ) = dim N(T ) < +. 3, Banach Hilbert Moore-Penrose. 2.4,

No. 5 : 1019 3.1 X, Y Banach, T C(X, Y ) T + C(Y, X), δt B(X, Y ) R(δT ) D(T + ). T + δt < 1, (1) B = T + (I + δt T + ) 1 = (I + T + δt ) 1 T + T = T + δt ; (2) R(T ) = R(T T + ); (3) R(T ) N(T + ) = {0}; (4) N(T + T ) = N(T ); (5) (I + δt T + ) 1 R(T ) = R(T ); (6) (I + T + δt ) 1 N(T ) = N(T ); (7) (I + δt T + ) 1 T N(T ) R(T ). T C(X, Y ) δt B(X, Y ), T = T + δt. T + δt < 1, Banach, I + T + δt : X X, (I + T + δt ) 1. T +, B = (I + T + δt ) 1 T +. 2.4,.. 3.2 X, Y Banach, T C(X, Y ) T + B(Y, X). δt B(X, Y ) δt T + < 1, (1) B = T + (I + δt T + ) 1 = (I + T + δt ) 1 T + : Y D(T ) T = T + δt ; (2) R(T ) = R(T T + ); (3) R(T ) N(T + ) = {0}; (4) N(T + T ) = N(T ); (5) (I + δt T + ) 1 R(T ) = R(T ); (6) (I + T + δt ) 1 N(T ) = N(T ); (7) (I + δt T + ) 1 T N(T ) R(T ); (8) X = N(T ) R(T + ) X = N(T ) + R(T + ); (9) Y = R(T ) N(T + ). δt T + < 1, I + δt T + : Y Y, (I + δt T + ) 1. B = T + (I + δt T + ) 1. B T, 1.6 2.4, (2) (9)., (2) (7), (1). (9), (3). X = N(T ) + R(T + ), x N(T ), x 1 N(T ), x 2 R(T + ), x = x 1 + x 2. x 2 = x x 1 D(T ), x 2 T + T x 2 R(T + ) N(T ) = {0}, x 2 = T + T x 2. (I + δt T + ) 1 T x = (I + δt T + ) 1 T x 2 = (I + δt T + ) 1 T T + T x 2 = (I + δt T + ) 1 (I + δt T + )T T + T x 2 = T x 2 R(T ). (7).. T B(X, Y ), T + B(Y, X), R(T + ). 3.2, 3.3 X, Y Banach, T B(X, Y ) T + B(Y, X). δt B(X, Y ) δt T + < 1. (1) B = (I + T + δt ) 1 T + = T + (I + δt T + ) 1 T = T + δt ; (2) R(T ) = R(T T + ); (3) R(T ) N(T + ) = {0};

1020 Vol. 37 (4) N(T + T ) = N(T ); (5) X = N(T ) R(T + ) X = N(T ) + R(T + ); (6) (I + δt T + ) 1 R(T ) = R(T ); (7) (I + T + δt ) 1 N(T ) = N(T ); (8) (I + δt T + ) 1 T N(T ) R(T ); (9) Y = R(T ) N(T + ). 3.2 3.3 [6, 7, 13 16, 18]. 3.4 X, Y Hilbert, T C(X, Y ) Moore-Penrose T B(Y, X). δt B(X, Y ) δt T < 1, B = T (I +δt T ) 1 = (I +T δt ) 1 T T = T +δt Moore-Penrose R(T ) = R(T ), N(T ) = N(T ). R(T ) = R(T ), N(T ) = N(T ), 3.2, B T. X = N(T ) R(B), Y = R(T ) N(B). (3.1) Moore-Penrose, X Y X = N(T ) R(T ), Y = R(T ) N(T ). R(T ) = R(B), N(T ) = N(B), X = N(T ) R(B), Y = R(T ) N(B), (3.1). B T Moore-Penrose., B T Moore-Penrose, X = N(T ) R(B), Y = R(T ) N(B). N(T ) = R(B), R(T ) = N(B). N(T ) = R(T ), R(T ) = N(T ), R(T ) = R(B) N(T ) = N(B), R(T ) = R(T ) N(T ) = N(T ).. 3.4 [9, 14 16]. [1] Nashed M Z, Votruba G F. A unified approach to generalized inverses of linear operators: I. Algebraic, topological, and projectional properties [J]. Bull. Amer. Math. Soc., 1974, 80(5): 825 830. [2] Nashed M Z, Votruba G F. A unified approach to generalized inverses of linear operators: II. Extremal and proximal properties [J]. Bull. Amer. Math. Soc., 1974, 80(5): 831 834. [3] Nashed M Z. Generalized inverses and applications [M]. New York: Academic Press, 1976. [4]. [M]. :, 2005. [5] Ben-Israel A, Greville T N E. Generalized inverses: theory and applications (2nd ed.)[m]. New York: Wiley, 2003. [6] Chen G L, Xue Y F. Perturbation analysis for the operator equation T x = b in Banach spaces [J]. J. Math. Anal. Appl., 1997, 212 (1): 107 125. [7] Ma J P. Complete rank theorem of advanced calculus and singularities of bounded linear operators [J]. Front. Math. China, 2008, 3(2): 305 316. [8] Nashed M Z. Inner, outer, and generalized inverses in Banach and Hilbert spaces [J]. Numer. Funct. Anal. Optim., 1987, 9(3): 261 325.

No. 5 : 1021 [9] Ding J. On the expression of generalized inverses of perturbed bounded linear operators [J]. Missouri J. Math. Sci., 2003, 15(1): 40 47. [10] Du N L. The basic principles for stable approximations to orthogonal generalized inverses of linear operators in Hilbert spaces [J]. Numer. Funct. Anal. Optim., 2005, 26(6): 675 708. [11] Du N L. Finite-dimensional approximation settings for infinite-dimensional Moore-Penrose inverses [J]. SIAM J. Numer. Anal., 2008, 46(3): 1454 1482. [12],,. Banach [J]., 2011, 32A(5): 635 646. [13] Huang Q L, Zhai W X. Perturbations and expressions for generalized inverses in Banach spaces and Moore-Penrose inverses in Hilbert spaces of closed linear operators [J]. Linear Algebra Appl., 2011, 435(1): 117 127. [14] Huang Q L, Zhu L P, Geng W H, Yu J N. Perturbation and expression for inner inverses in Banach spaces and its applications [J]. Linear Algebra Appl., 2012, 436(9): 3715 3729. [15] Huang Q L, Zhu L P, Jiang Y Y. On the stable perturbation of outer inverses of linear operators in Banach spaces [J]. Linear Algebra Appl., 2012, 437(7): 1942 1954. [16] Wang Y W, Zhang H. Perturbation analysis for oblique projection generalized inverses of closed linear operators in Banach spaces [J]. Linear Algebra Appl., 2007, 426(1): 1 11. [17] Xu Q X, Song C N, Wei Y M. The stable perturbation of the Drazin inverse of the square matrices [J]. SIAM J. Matrix Anal. Appl., 2009, 31(3): 1507 1520. [18] Yang X D, Wang Y W. Some new perturbation theorems for generalized inverses of linear operators in Banach spaces [J]. Linear Algebra Appl., 2010, 433(11): 1939 1949. THE SIMPLEST EXPRESSION OF THE ALGEBRAIC GENERALIZED INVERSES IN LINEAR SPACES GUO Zhi-rong 1,2, HUANG Qiang-lian 1, ZHANG Li 1 (1. School of Mathematical Sciences, Yangzhou University, Yangzhou 225002 China) (2. College of Mathematics, Yangzhou Vocational University, Yangzhou 225009 China) Abstract: In this paper, the authors study the additivity and expression of algebraic generalized inverses from the view of pure algebra in the framework of linear space. Utilizing the algebraic direct sum decomposition of linear space, we first give the necessary and sufficient condition of the invertibility of I + AT + and T + = T + (I + AT + ) 1. We also provide some necessary and sufficient conditions for T + to have the simplest expression. As applications, we discuss the perturbation problem of generalized inverse in Banach space and Moore-Penrose inverse in Hilbert space, which extend and improve many recent results in this topic. Keywords: algebraic generalized inverse; generalized inverse; Moore-Penrose inverse; the simplest expression; linear space 2010 MR Subject Classification: 47L05; 46A32