An operator equation characterizing the Laplacian
Hermann König, Vitali Davidovich Milman · St Petersburg Mathematical Journal · 2013
The Laplace operator on R n satisfies the equationfor all f, g ∈ C 2 (R n , R) and x ∈ R n .In the paper, an operator equation generalizing this product formula is considered.Suppose T : C 2 (R n , R) → C(R n , R) and A :for all f, g ∈ C 2 (R n , R) and x ∈ R n .Assume, in addition, that T is O(n)-invariant and annihilates the affine functions, and that A is nondegenerate.Then T is a multiple of the Laplacian on R n , and A a multiple of the derivative,where d ∈ C(R + , R) is a continuous function.The solutions are also described if T is not O(n)-invariant or does not annihilate the affine functions.For this, all operators (T, A) satisfying (1) for scalar operators A : C 2 (R n , R) → C(R n , R) are determined.The map A, both in the vector and the scalar case, is closely related to T and there are precisely three different types of solution operators (T, A).No continuity or linearity requirement is imposed on T or A.