an analytic semigroup

A general C_0 semigroup is merely continuous in time — it solves the equation but need not be especially smooth or smoothing. Parabolic equations do far better than that, and an analytic semigroup is the upgrade that captures it. Not only does the flow e^{-tA} depend smoothly on t for t > 0; it extends to complex times t in a whole sector of the plane and is genuinely analytic (holomorphic) there. That extra rigidity is the abstract source of every nice feature of diffusion: instantaneous smoothing, strong decay estimates, and the ability to solve nonlinear problems by fixed point.

Concretely, a C_0 semigroup {T(t)} is analytic if t goes to T(t) extends to a holomorphic operator-valued function on a sector {complex z : |arg z| < theta} for some angle theta > 0, with the semigroup law and strong continuity preserved as z goes to 0 in the sector. The operators that generate them are exactly the sectorial operators (those whose spectrum sits in a sector opening to the left, with a resolvent bound outside it). The defining quantitative payoff is the smoothing estimate: for t > 0 the operator A T(t) is bounded with norm of A e^{-tA} <= C/t. In words, applying the generator costs only a factor 1/t — the semigroup maps rough data into the domain of A immediately, then into the domain of A^2, and so on, for every t > 0. Iterating, e^{-tA} u_0 lands in the intersection of all the domains, i.e. it is infinitely smooth.

This is the abstract face of parabolic regularity. The bound norm of A e^{-tA} <= C/t is exactly why rough initial data instantly becomes smooth, and it is exactly the estimate that powers Duhamel's formula for semilinear equations: a nonlinearity that would be too rough to handle directly becomes manageable because the semigroup smooths it at every step, paying only an integrable 1/t. The heat semigroup is the prototype; the wave semigroup is NOT analytic (it does not smooth, consistent with finite propagation speed).