Israel Journal of Mathematics

, Volume 12, Issue 1, pp 79–93 | Cite as

On semigroups generated by restrictions of elliptic operators to invariant subspaces

  • Tamar Burak


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<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2θ<θ 23π/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)=ELp (G) ⊕E+Lp (G) and each of the operatorsAE and—AE + is the infinitestimal generator of an analytic semigroup.


Elliptic Operator Differential System Elliptic System Elliptic Boundary Unbounded Operator 
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Copyright information

© Hebrew University 1972

Authors and Affiliations

  • Tamar Burak
    • 1
  1. 1.Department of Mathematical SciencesTel Aviv UniversityTel Aviv

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