The group of $L^{2}$-isometries on $H^{1}_{0}$
Volume 217 / 2013
Abstract
Let $\varOmega $ be an open subset of $\mathbb R^n$. Let $L^2=L^2(\varOmega ,dx)$ and $H^1_0=H^1_0(\varOmega )$ be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group $\mathbb {G}$ of invertible operators on $H^1_0$ which preserve the $L^2$-inner product. When $\varOmega $ is bounded and $\partial \varOmega $ is smooth, this group acts as the intertwiner of the $H^1_0$ solutions of the non-homogeneous Helmholtz equation $u-\varDelta u=f$, $u|_{\partial \varOmega }=0$. We show that $\mathbb {G}$ 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 $\mathbb {G}$ by means of examples. In particular, we give an example of an operator in $\mathbb {G}$ whose spectrum is not contained in the unit circle. We also study the one-parameter subgroups of $\mathbb {G}$. Curves of minimal length in $\mathbb {G}$ are considered. We introduce the subgroups $\mathbb {G}_p:=\mathbb {G}\cap (I - {\mathcal B}_p(H^1_0))$, where ${\mathcal B}_p(H_0^1)$ 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 $G_1 , G_2 \in \mathbb {G}_p$ can be joined by a minimal curve of the form $\delta (t)=G_1 e^{itX}$, where $X$ is a symmetrizable operator in ${\mathcal B}_p(H^1_0)$.
