Let A be the closed unbounded operator inL p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ 1≦argλ≦θ 2} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1 p(A) ofA;π/2≦θ<θ 2≦3π/2 ; and there exists a constantc such that (I)││(λ-A)-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE − andE + such thatA is completely reduced by the direct sum decompositionL p(G)=E−Lp (G) ⊕E+Lp (G) and each of the operatorsAE − and—AE + is the infinitestimal generator of an analytic semigroup. |