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

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1 Vol. 37 ( 2017 ) No. 5 J. of Math. (PRC) 1,2, 1, 1 (1., ) (2., ) :. I +AT +, T + = T + (I +AT + ) 1, T +. Banach Hilbert Moore-Penrose.. : ; ; Moore-Penrose ; ; MR(2010) : 47L05; 46A32 : O177.2 : A : (2017) 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 + ). : : ( ; ); (BK ); (2016zqn03). : (1970 ),,,, :. :.

2 1014 Vol [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 [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].?

3 No. 5 : 1015?, [6, 7, 9, 12 16, 18] 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 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.

4 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 +.

5 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 + ) 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 + ).

6 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 = 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 < (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,

7 No. 5 : 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 , 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, , (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};

8 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 + ) [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 ) [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): [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): [3] Nashed M Z. Generalized inverses and applications [M]. New York: Academic Press, [4]. [M]. :, [5] Ben-Israel A, Greville T N E. Generalized inverses: theory and applications (2nd ed.)[m]. New York: Wiley, [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): [7] Ma J P. Complete rank theorem of advanced calculus and singularities of bounded linear operators [J]. Front. Math. China, 2008, 3(2): [8] Nashed M Z. Inner, outer, and generalized inverses in Banach and Hilbert spaces [J]. Numer. Funct. Anal. Optim., 1987, 9(3):

9 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): [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): [11] Du N L. Finite-dimensional approximation settings for infinite-dimensional Moore-Penrose inverses [J]. SIAM J. Numer. Anal., 2008, 46(3): [12],,. Banach [J]., 2011, 32A(5): [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): [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): [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): [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): [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): [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): 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 China) (2. College of Mathematics, Yangzhou Vocational University, Yangzhou 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