The group of L^2 isometries on H^1_0

Abstract

Let U be an open subset of R^n. Let L^2=L^2(U,dx) and H^1_0=H^1_0(U) 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 H^1_0 which preserve the L^2-inner product. When U is bounded and the border $\partial U$ is smooth, this group acts as the intertwiner of the H^1_0 solutions of the non-homogeneous Helmholtz equation $u-\Delta u=f$, $u|_{\partial U}=0$. 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 G_p:=G \cap (I - B_p(H^1_0)), where B_p(H_0^1) is a Schatten ideal of operators on H_0^1. An invariant (weak) Finsler metric is defined by the p-norm of the Schatten ideal of operators of L^2. We prove that any pair of operators g_1,g_2 in G_p can be joined by a minimal curve of the form $a(t)=g_1 e^{itX}$, where X is a symmetrizable operator in B_p(H^1_0).