a sectorial operator
What is it about an elliptic operator like -Laplacian that makes its diffusion smooth so violently? The answer is the geometry of its spectrum and resolvent in the complex plane. A sectorial operator is one whose spectrum is trapped inside a wedge (a sector) opening to one side, with its resolvent staying nicely bounded everywhere outside that wedge — including a region wrapping around to the left. That picture is precisely the condition needed for the operator to generate an analytic semigroup, so 'sectorial' is the spectral fingerprint of a parabolic problem.
Precisely: a closed densely defined operator A is sectorial if there is an angle theta in (pi/2, pi) and a real omega such that the spectrum of A lies in the closed sector {lambda : |arg(lambda - omega)| <= theta} (a wedge around the positive real axis, but wider than a half-plane), and outside that sector the resolvent obeys norm of (lambda I - A)^{-1} <= M/|lambda - omega|. The crucial part is 'wider than a half-plane' (theta > pi/2): the resolvent must remain bounded in a region that dips to the LEFT of the imaginary axis. That little overhang is what lets you draw a contour around the spectrum and define e^{-tA} by the integral (1/2 pi i) times the contour integral of e^{-t lambda} (lambda I - A)^{-1} d lambda — and the leftward dip is what makes that integral converge and gives the smoothing bound norm of A e^{-tA} <= C/t.
Why it matters: sectoriality is the working hypothesis under which all of parabolic semigroup theory runs. Self-adjoint elliptic operators bounded below are automatically sectorial (their spectrum is real and bounded above), and more generally any uniformly elliptic second-order operator with reasonable coefficients is sectorial on the right spaces. Once you know A is sectorial you get, for free, an analytic semigroup, instantaneous smoothing, fractional powers A^alpha and the interpolation spaces in between, and the framework for solving semilinear parabolic equations by Duhamel and fixed point.
A = -Laplacian on L^2(Omega) with Dirichlet conditions is self-adjoint with spectrum the positive Dirichlet eigenvalues {lambda_k}, all real and >= lambda_1 > 0. So its spectrum sits on a ray of the positive real axis, trivially inside a sector of angle just over pi/2, and the resolvent decays like 1/|lambda| off that ray. Hence -Laplacian is sectorial and generates the analytic heat semigroup e^{t Laplacian}.
Spectrum in a leftward-overhanging sector = generator of an analytic (smoothing) semigroup.
The angle must exceed pi/2 — a half-plane resolvent bound (angle exactly pi/2) only gives a plain C_0 semigroup, not an analytic one; the extra overhang to the left is precisely what buys analyticity and smoothing. So 'spectrum in the left half-plane' is necessary but not sufficient for the parabolic gains; you need the sector to be genuinely fat.