Uniform convergence of ergodic limits and approximate solutions

Abstract:

Let $A$ be a densely defined closed (linear) operator, and $\{ {A_\alpha }\}$, $\{ {B_\alpha }\}$ be two nets of bounded operators on a Banach space $X$ such that $||{A_\alpha }|| = O(1),{A_\alpha }A \subset A{A_\alpha },||A{A_\alpha }|| = o(1)$, and ${B_\alpha }A \subset A{B_\alpha } = I - {A_\alpha }$. Denote the domain, range, and null space of an operator $T$ by $D(T)$, $R(T)$, and $N(T)$, respectively, and let $P(\operatorname {resp} .B)$ be the operator defined by $Px = {\lim _\alpha }{A_\alpha }x(resp. By = {\lim _\alpha }{B_\alpha }y)$ for all those $x \in X(\operatorname {resp} .y \in \overline {R(A)} )$ for which the limit exists. It is shown in a previous paper that $D(P) = N(A) \oplus \overline {R(A)} ,R(P) = N(A),D(B) = A(D(A) \cap \overline {R(A)} ),R(B) = D(A) \cap \overline {R(A)}$, and that $B$ sends each $y \in D(B)$ to the unique solution of $Ax = y{\text { in }}\overline {R(A)}$. In this paper, we prove that $D(P) = X$ and $||{A_\alpha } - P|| \to 0$ if and only if $||{B_\alpha }|D(B) - B|| \to 0$, if and only if $||{B_\alpha }|D(B)|| = O(1)$, if and only if $R(A)$ is closed. Moreover, when $X$ is a Grothendieck 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.