The group of L^2 - isometries on H_0^1

Abstract

Let be an open subset of Rn. Let L2 = L2( ; dx) and H1 0 = H1 0 ( ) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H1 0 which preserve the L2-inner product. When is bounded and @ is smooth, this group acts as the intertwiner of the H1 0 solutions of the non-homogeneous Helmholtz equation u u = f, uj@ = 0. We show that G is a real Banach{Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belonging to G by means of examples. In particular, we give an example of an operator in G whose spectrum is not contained in the unit circle. We also study the one-parameter subgroups of G. Curves of minimal length in G are considered. We introduce the subgroups Gp := G(I Bp(H1 0 )), where Bp(H1 0 ) is the Schatten ideal of operators on H1 0 . An invariant (weak) Finsler metric is dened by the p-norm of the Schatten ideal of operators on L2. We prove that any pair of operators G1;G2 2 Gp can be joined by a minimal curve of the form (t) = G1eitX , where X is a symmetrizable operator in Bp(H1 0 ).