| 摘要: | Let A be a densely defined closed (linear) operator, and {A(o)}, {B(alpha)} be two nets of bounded operators on a Banach space X such that parallel-to A(alpha) parallel-to = O(1), A(o)A subset-of AA(alpha), parallel-to AA(alpha) parallel-to = o(1), and B(o)A subset-of AB(o) = I - A(alpha). Denote the domain, range, and null space of an operator T by D(T), R(T), and N(T), respectively, and let P (resp. B) be the operator defined by Px = lim(alpha)A(alpha)x (resp. By = lim(alpha) B(alpha)y) for all those x epsilon X (resp. y epsilon R(A))BAR for which the limit exists. It is shown in a previous paper that D(P) = N(A) + R(A)BAR, R(P) = N(A), D(B) = A(D(A) intersection-of R(A))BAR, R(B) = D(A) intersection-of R(A)BAR, and that B sends each y epsilon D(B) to the unique solution of Ax = y in R(A)BAR. In this paper, we prove that D(P) = X and parallel-to A(alpha) - P paralle-to --> 0 if and only if parallel-to B(alpha)\D(B) - B parallel-to --> 0, if and only if parallel-to B(alpha)\D(B) parallel-to = O(1), if and only if R(A) is closed. Moreover, when X is a Grothendiock space with the Dunford-Pettis property, all these conditions are equivalent to the mere condition that D(P) = X. The general result is then used to deduce uniform ergodic theorems for n-times integrated semigroups, (Y)-semigroups, and cosine operator functions. |
