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54.3. SECTORIAL OPERATORS AND ANALYTIC SEMIGROUPS 1735

Similarly using the boundedness of (b2I−A)β (b1I−A)−β , it follows∣∣∣∣∣∣(b2I−A)β x∣∣∣∣∣∣≤C′

∣∣∣∣∣∣(b1I−A)β x∣∣∣∣∣∣

Thus showing the two norms are equivalent. This proves the proposition.

54.3. SECTORIAL OPERATORS AND ANALYTIC SEMIGROUPS 1735Similarly using the boundedness of (bol —ayh (bil —aA)yF , it follows||(bar—ayPx|] <c(bu — A)? x||Thus showing the two norms are equivalent. This proves the proposition.