The group of L2-isometries on H01

Let Omega be an open subset of R-n. Let L-2 = L-2(Omega, dx) and H-0(1) = H-0(1)(Omega) 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-0(1) which preserve the L-2-inner product. When Omega is bounded and partial derivative Omega is smooth, this group acts as the intertwiner of the H-0(1) solutions of the non-homogeneous Helmholtz equation u - Delta u = f, u vertical bar(partial derivative Omega) = 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 boolean OR(I - B-p,(H-0(1))), where B-p (H-0(1),) is the 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 on L-2. We prove that any pair of operators G(1), G(2) is an element of G(p) can be joined by a minimal curve of the form delta(t) = G(1)e(itX), where X is a symmetrizable operator in B-p (H-0(1)).