Abstract :
We consider linear unbounded operators, B, on a Banach space, X, whose spectrum is contained in an open set, V, such that the resolvent (w−B)−1 exists and is O((1 + |w|)α), for w ∉ V, for some α ≥ −1; we will call this an operator of α-type V.For such operators, we construct a functional calculus, ƒ↦ƒ(B), defined for functions holomorphic on V, whose real part is bounded above, such that ƒ(B) generates an exponentially bounded (λ − B)−m-semigroup, where m ≡ [α] + 2, λ ∉ V. This map is a continuous algebra homomorphism, in a sense that is made precise later. We give best possible spectral mapping theorems.Applications of this construction include the following. We characterize subsets, V, of the complex plane, with the property that, whenever A is densely defined and of α-type V, then the abstract Cauchy problem has a solution, for all initial data in a dense set. We show that, when B is of a-type V, then there exists a Ranach subspace, Y, such that [D(Bm)] ↷ Y ↷ X and there exists an H∞(V)-functional calculus for B|Y. On this interpolating space, the imaginary powers of B form a (C0)-group and we have spectral decompositions of B, when V is disconnected. Our construction produces solutions, for all initial data in dense sets, of wide classes of first and second order Cauchy problems, including the Schrödinger equation, the backwards heat equation and the Laplace equation, on Lp(Rn) or Lp(Ω), I ≤ p ≤ ∞, where Ω is a region in Rn with smooth boundary. We also consider incomplete Cauchy problems of arbitrarily high order.We emphasize that, in our construction of ƒ(B), we do not assume that ƒ is meromorphic at ∞. This allows us to define exponentials, cosines and logarithms of B directly, for large classes of operators, B; for example, we may define the exponential, e5, of an operator, B, with real spectrum.
