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

E. ANDRUCHOW, E. CHIUMIENTO , G. LAROTONDA

STUDIA MATHEMATICA

POLISH ACAD SCIENCES INST MATHEMATICS

Lugar: VARSOVIA; Año: 2013 vol. 217 p. 193 - 217

0039-3223

Let O be an open subset of R. Let L^2=L^2(O,dx) and  H^1_0=H^1_0(O) 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 O is bounded and its boundary 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 Omega}=0. We show thatG 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 d(t)=G_1 e^{itX}, where X is a symmetrizable operator in B_p(H^1_0).