Vol. 27, No. 3, 1968

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An extended form of the mean-ergodic theorem

Lynn Clifford Kurtz and Don Harrell Tucker

Vol. 27 (1968), No. 3, 539–545

Suppose X is a reflexive Banach space and V is a continuous linear operator in X such that V nM 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

nli→m∞          n          = P x,

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 {V k}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 {V k}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 {V k}k=0.

Mathematical Subject Classification
Primary: 40.50
Received: 12 July 1967
Published: 1 December 1968
Lynn Clifford Kurtz
Don Harrell Tucker