Ergodic projections of continuous and discrete semigroups

Abstract:

Let X be a Banach space. Let $\{ T(t);t > 0\}$ be a uniformly bounded semigroup of operators on X, which converges strongly to P, known to be a projection, as t goes to 0. If A is its generator and ${X_0}$ [resp., ${X_t},t > 0$] is the set of x for which \[ {P_0}x \equiv \lim \limits _{t \to \infty } {t^{ - 1}}\int _0^t {T(\tau )x\;d\tau \;\left [ {{\text {resp}}.,\;{P_t}x \equiv \lim \limits _{n \to \infty } {n^{ - 1}}\sum \limits _{i = 0}^{n - 1} {T(it)x} } \right ]} \] exists, then, for each $t \geqslant 0,{P_t}$ is a bounded projection in ${X_t}$; when $t = 0,{X_0} = N(A) \oplus \overline {R(A)} \oplus N(P),\;R({P_0}) = N(A)$ and $N({P_0}) = \overline {R(A)} \oplus N(P)$; when $t > 0$, then \[ {X_t} = N(T(t) - I) \oplus \overline {R(T(t) - I)} ,\] $R({P_t}) = N(T(t) - I)$ and $N({P_t}) = \overline {R(T(t) - I)} ;\;{X_t} = X$ for all $t \geqslant 0$ if X is reflexive. Some results on relations among the projections ${P_t},t \geqslant 0$, are obtained. In particular, we have ${P_t} = {P_0}$ for all sufficiently small t if A is bounded.