Vol. 50, No. 2, 1974

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A counter example to the Blum Hanson theorem in general spaces

Mustafa Agah Akcoglu, John Philip Huneke and Hermann Rost

Vol. 50 (1974), No. 2, 305–308

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 T1 is disproved by constructing a counterexample in a Banach space of the type B = 𝒞(X),X a compact Hausdorff space.

Mathematical Subject Classification 2000
Primary: 47A35
Secondary: 28A65
Received: 12 October 1972
Revised: 25 June 1973
Published: 1 February 1974
Mustafa Agah Akcoglu
John Philip Huneke
Hermann Rost