Let U be a continuous representation of a Lie group G on a Banach space X and a 1 , ..., a d $ an algebraic basis of the Lie algebra g of G, i.e., the a 1 , ..., a d $ together with their multi-commutators span g. Let A i =dU(a i ) denote the infinitesimal generator of the continuous one-parameter group t [ U(exp(&ta i )) and set
We analyze properties of m th order differential operators dU(C)= : :; |:| m c : A : with coefficients c : # C.
If L denotes the left regular representation of G in L 2 (G) then dL(C) satisfies a Ga# rding inequality on L 2 (G) if, and only if, the closure of each dU(C) generates a holomorphic semigroup S on X, the action of S z is determined by a smooth, representation independent, kernel K z which, together with its derivatives A : K z , satisfies mth order Gaussian bounds and, if U is unitary, S is quasi-contractive in an open representation independent subsector of the sector of holomorphy. Alternatively, dL(C) satisfies a Ga# rding inequality on L 2 (G) if, and only if, the closure of dL(C) generates a holomorphic, quasi-contractive, semigroup satisfying bounds
These results extend to operators for which the directions a 1 , ..., a d $ are given different weights. The unweighted Ga# rding inequality is a stability condition on the principal part, i.e., the highest-order part, of dL(C) but in the weighted case the condition is on the part of dL(C) with the highest weighted order. 1998 Academic Press 1. INTRODUCTION The theory of partial differential operators extends naturally from the Euclidean space R d to a general d-dimensional Lie group. The operators are defined in any continuous Banach space representation U of G as polynomials in the associated representatives of the Lie algebra g of G.