Explicit counterexamples in Hilbert spaces show Cesàro means of firmly nonexpansive iterates need not converge strongly.
Reich: Almost convergence and nonlinear ergodic theorems
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Ces\`aro means of firmly nonexpansive iterates need not converge strongly
Explicit counterexamples in Hilbert spaces show Cesàro means of firmly nonexpansive iterates need not converge strongly.