Commutators with powers of an unbounded operator in Hilbert space

Abstract

Let H be a Hilbert space, B a bounded linear operator on H, A a closed, densely defined, unbounded positive self-adjoint linear operator in H. It is the object of the present paper to prove the following theorem: THEOREM. Suppose that for a given positive integer n, (1) BAnAnB C BiAn-1 (2) BlAn AnBB C B2Awhere B1 and B2 are bounded linear operators on H. Then BA -AB is bounded. The original context in which this theorem arose was one in which A2 was the differential operator (I-A) on a compact Riemannian manifold M, B a singular elliptic operator on M, n = 2. The theorem was proved by the writer in connection with a program begun by R. Palais for giving an intrinsic treatment of singular integral operators on manifolds without the use of localization arguments. We publish it here since it may have other interesting applications. LEMMA 1. There exists a constant cn such that for u in D(A),