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Published online by Cambridge University Press: 14 November 2011
Let A and B be closed linear operators on a Banach space X. Assume that ε(εI – A)−1f→f as |ε|→ ∞ for all f in X, ζ∊∑ ⊂ℂ. Under what conditions on B − A does the same relationship hold for B? When does [ε(εI − A)−1 − ε(εI − B)−1 ] f→ 0 in some stronger norm than that of X? The questions are discussed in an abstract setting and the results are generalised to other analytic functions of A. Applications are given to second order elliptic operators.