Suppose X is a reflexive Banach
space and V is a continuous linear operator in X such that ∥Vn∥≦ M for some
M(n = 0,1,2,…). If N is the null space of I −V and R is the closure of the range of
I − V , then the mean-ergodic theorem states that
where P is the projection associated with N and R; the convergence is in the norm
of X. This is pointwise C1-summability of the sequence {Vk}k=0∞ to P, and it
suggests a similar theorem for more general Hausdorff summability methods. The
purpose of this note is to demonstrate a wide class of operatorvalued Hausdorff
summability methods which contain the sequence {Vk}k=0∞ in their wirkfelder and
sum it to certain transforms of the projection operator P. This result shows much
more clearly the sense in which convergence actually has meaning for such a sequence
{Vk}k=0∞.
