In Banach Space Has Available Generalized Prediction Solution Of Operator Beam Type

Theory of generalized inverse emerges from analysis. The concept of a generalized inverse seems to have been first mentioned in print in1903by Fredholm, where a particular generalized inverse (called by him "pseudoinverse") of an integral operator was given. Generalized inverses of differential operators were already implicit in Hilbert’s discussion in1904of generalized Green functions.In term of operator theory, when an operator is not bijective, its inverse doesn’t exist, we can discuss its generalized inverse. Theory of generalized inverse has wide and important applications in many applied subjects such as numerical analysis, statistics, economics and optimization. One core of the theory of generalized inverse is its perturbation theory which studies the problem that whether the operator, after a minor perturbation, still has generalized inverse and whether the generalized inverse (in some sense) convergece to the original one. As early as1955, R. Penrose discussed the continuity of the Moore-Penrose generalized inverse of matrix.In the70s of last century, the international leader of the research of generalized inverse, American famous mathematician, the one-hour reporter of Finland international congress of mathematicians, Professor M. Z. Nashed made a profound exposition for generalized inverse of linear operators in Banach spaces. After that, M. Z. Nashed, X. Chen, G. L. Chen, J. Ding, M. S. Wei, Y. F. Xue, Y. M. Wei, J. P. Ma, Q. L. Huang, G. Z. Song, W. P. Cao and Y. W. Wang had systemamtically studied the continuity of the Moore-Penrose generalized inverse in Hilbert spaces and the perturbed problem for the generalized inverse in Banach spaces.As to the resolvent, we can investigate the generalized inverse and generalized resolvent. The generalized resolvent has important applications in the rearch of spectral theory and of Fredholm operators. M. A. Shubin showed that there exists a continuous generalized inverse function which satisfies the generalized inverse but not possibly the generalized resolvent identity. And he pointed out that it remains an open problem whether or not this can be done while also satisfying the generalized resolvent identity, i.e., it is not known whether generalized resolvents always exist. This problem has received great concern form many scholars such as A. Hoefer, C. Badea, M. Mbekhta, S. Christoph. For example, M. Mbekhta proved that there exists an analytic generalized resolvent for linear pencil λâ†'T-λS on a neighborhood of0if and only if T has a generalized inverse and N(T) c R(Tm), m∈N. After that, utilizing the reduced minimum modulus and the gap between closed subspaces, C. Badea and M. Mbekhta proved that there exists an analytic generalized resolvent for λâ†'T-λS onU(0) if and only if N(T-AS) and R(T-AS) have fixed complements in X and Y, respectively.In this paper, we utilize the stability characterizations of generalized inverses of linear operator to investigate the existence of generalized resolvents of linear pencils in Banach spaces. Firstly, we obtain some stability characterizations of generalized inverse of linear operator under weaker perturbation conditions. Secondly, we give a necessary and sufficient condition for the existence of the generalized resolvent of linear pencils. Furthermore, we can conclude that some spectrum points come down to the generalized regular points which have a certain regularity. Generalized regular points have been widely used in the research of local conjugacy theorem in nonlinear analysis and the generalized transversility in global analysis. The main results in this paper generalize and improve some well-known results, and our characterizations can be easy to be verified and calculated which can promote the study of generalized regular points.Theorem Let X and Y be two Banach spaces and T,T e B{X,Y).If T has a generalized inverse T+e∈(Y,X) and||(T-T)T+y||≤λ1||y||+λ2||[I+(T-T)T+]y||,(?)y∈Y,where λ1,λ2∈[0,1), then the following statements are equivalent:(1) B=T+[I+(T-T)T+]-1is a generalized inverse of T;(2) R(T)∩N(T+)={0};(3) N(T)(?)R(T+)=X;(4) R(T)(?)N(T+)=YTheorem Let X and Y be two Banach spaces and T, S e B(X, Y).(1) If the linear pencil λâ†'T-λS has an analytic generalized resolvent on a neighborhood of0, then for any generalized inverse T+of T, there exists a neighborhood U(0) of0such that R(T-λS)∩N(T+)={0},(?)λ∈U(0);(2) If T has a generalized inverse and for any generalized inverse T+of T, there exists a neighborhood U of0such that R(T-λS)∩N(T+)={0},(?)λ∈U, then the linear pencil λâ†'T-λS has an analytic generalized resolvent on a neighborhood of0. In fact, G{λ)=T+(I-λST+)-1: Yâ†'X is a generalized resolvent of λâ†'T-λS on a neighborhood of0.As applications, we provide the following statements:1) If the linear pencil λâ†'T-λS has a continuous generalized inverse function, then there is an analytic generalized resolvent for linear pencil λâ†'T-λS;2) The Moore-Penrose inverse of the pencil λâ†'T-λS is its analytic generalized resolve-nt on a neighborhood U of0if and only if λâ†'T-λS perserves the null space and the range on a neighborhood U of0.3) The existence characterizations for generalized resolvent of the finite rank operators, Fredholm operators and semi-Fredholm operators are consistent with their stability character-izations of generalized inverse.