arXiv:1203.1292v1 [math.FA] 6 Mar 2012 The group of L 2 -isometries on H 1 0 ∗ E. Andruchow, E. Chiumiento and G. Larotonda † Abstract Let Ω be an open subset of R n . Let L 2 = L 2 (Ω,dx) and H 1 0 = H 1 0 (Ω) 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 Ω is bounded and ∂ Ω is smooth, this group acts as the intertwiner of the H 1 0 solutions of the non-homogeneous Helmholtz equation u − Δu = f , u| ∂Ω = 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 ∩ (I −B p (H 1 0 )), where B p (H 1 0 ) is a Schatten ideal of operators on H 1 0 . 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 ∈ G p can be joined by a minimal curve of the form δ(t)= G 1 e itX , where X is a symmetrizable operator in B p (H 1 0 ). 1 1 Introduction Let Ω ⊂ R n be an open subset. Denote by L 2 = L 2 (Ω,dx) the Lebesgue space of square- integrable functions endowed with its usual inner product 〈· , ·〉. Let H 1 0 = H 1 0 (Ω) be the closure in the Sobolev norm of the C ∞ functions with compact support contained in Ω. In this paper, we study the group G of invertible operators on H 1 0 that preserve the L 2 -inner product: G = { G ∈ Gl(H 1 0 ): 〈Gf,Gg〉 = 〈f,g〉}. In the case where Ω = R n , the group G was introduced in [5] in relation with the geometry of the variational spaces arising in the many-particle Hartree-Fock theory. One could give an abstract definition of G, involving a complex Hilbert space H and a dense and continuously included subspace E ⊂ H with their respective (non equivalent) inner products. However, we preferred this concrete setting given by the inclusion H 1 0 ⊂ L 2 because we shall deal mainly with examples. From the definition of the group G, it is clear that the theory of operators on spaces with two norms will play a central role in the study of this group. This theory was independently initiated by M. G. Krein [10] and P. D. Lax [14]. In Section 2 we recall the most useful results for our purposes. * 2010 Mathematical Subject Classification: Primary 47D03; Secondary 22E65, 58B20. † All authors are partially supported by Instituto Argentino de Matem´ atica Alberto P. Calder´ on and CONICET 1 Keywords and phrases: Group of isometries of a positive form, Sobolev space, symmetrizable operator, one parameter subgroup, minimal curve 1