Elliptic operators A = ∑ ¦α¦ ⩽ m b α(x) D α , α a multi-index, with leading term positive and constant coefficient, and with lower order coefficients b α(x) ϵ L r α + L α ( with ( n r α ) + ¦α¦ < m) defined on R n or a quotient space R n R n U α , U α⊂ R n are considered. It is shown that the L p -spectrum of A is contained in a “parabolic region” Ω of the complex plane enclosing the positive real axis, uniformly in p. Outside Ω, the kernel of the resolvent of A is shown to be uniformly bounded by an L 1 radial convolution kernel. Some consequences are: A can be closed in all L p (1 ⩽ p ⩽ ∞), and is essentially self-adjoint in L 2 if it is symmetric; A generates an analytic semigroup e − tA in the right half plane, strongly L p and pointwise continuous at t = 0. A priori estimates relating the leading term and remainder are obtained, and summability φ(εA)ƒ→ ε → 0φ(0) ƒ , with φ analytic, is proved for ƒ ϵ L p , with convergence in L p and on the Lebesgue set of ƒ. More comprehensive summability results are obtained when A has constant coefficients.