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In this form it was proved by G.-C. Rota, On models for linear operators, Comm. Pure Appl. Math., 13 (1960), 469-472. Related questions and generalizations are considered by B. Sz.-Nagy, Completely continuous operators with uniformly bounded iterates, Publ. Math. Inst. Hung. Acad. Sci., 4 (1959), 89-92; F. Gilfeather, Norm conditions on resolvents.., Proc. Amer. Math. Soc., 68 (1978), 44-48; and P. Halmos, A Hilbert Space Problem Book, Cor. 4 to Problem 153. The infimum in the last formula is attained for spaces of dimension at most one, but need not be attained in any higher dimension. In fact, the matrix operator