Invariant dissipative operators on locally compact groups (Q1086488)

Let G be a Lie group and \(C_ 0(G)\) the Banach space of all continuous functions on G vanishing at infinity. In this note we show that if the Lie algebra of G is not compact, then the closure \=D of a left invariant elliptic differential operator D on G has a restriction T such that T is also a left invariant closed dissipative linear operator in \(C_ 0(G)\) satisfying the inequality \(Range(I-T)\subsetneqq c_ 0(G)\).