Inverses of generators

Abstract

Let A A be a (possibly unbounded) linear operator on a Banach space X X that generates a bounded holomorphic semigroup of angle θ ( 0 > θ ≤ π / 2 ) \theta (0 > \theta \leq \pi /2) . We show that, if the range of A A is dense, then A A is one-to-one, and A − 1 {A^{ - 1}} (defined on the range of A A ) generates a bounded holomorphic semigroup of angle θ \theta , given by \[ e z A − 1 = ∫ e − w ( w A + z ) − 1 d w 2 π i , {e^{z{A^{ - 1}}}} = \int {{e^{ - w}}{{(wA + z)}^{ - 1}}\frac {{dw}}{{2\pi i}},} \] over an appropriate curve. When X X is reflexive, it is sufficient that A A be one-to-one.