Let T and S be two bounded
linear operators on a Banach space B. One studies the question whether weak
convergence of the powers Tn to S implies convergence of the Cesaro averages
1∕n∑k=1nTi(k) to S for all subsequences 0 ≦ i(1) < i(2) <⋯ of the integers. It is
well known that this implication holds if B is the L2 of a finite measure space and T
is induced by a measure preserving transformation of that space (this is the Blum
Hanson theorem) or, more generally, if B is a Hilbert space and T of norm at
most 1, or if B is a L1 space and T a positive operator of norm at most
1. In the present paper the conjecture that the above implication holds in
general Banach spaces for all T with ∥T∥≦ 1 is disproved by constructing a
counterexample in a Banach space of the type B = 𝒞(X),X a compact Hausdorff
space.
