The group of \(L^{2}\)-isometries on \(H^{1}_{0}\) (Q2854204)

Let \(\Omega\) be an open subset in \({\mathbb R}^n\). Let \(H_0^1=H^1_0(\Omega)\) be a Sobolev space of functions vanishing on the border \(\partial\Omega\). Let \(\mathbb G\subset L(H_0^1)\) be a group of bounded invertible operators which are isometric with respect to \(L_2\)-inner product. It is proved that \(\mathbb G\) is a real Banach-Lie group and its Lie algebra is a subspace of antihermitian operators. It is shown that the spectrum of operators from \(\mathbb G\) may not be contained in the unit circle and may have an arbitrary large norm. An analogue of Stone's theorem for one-parameter groups of unitary operators is obtained. Curves of minimal length are considered. Let \(\mathbb G_p\subset\mathbb G\) be a subgroup of operators of the form \(I+G\), \(G\in\sigma_p(H_0^1)\) (\(\sigma_p(H_0^1)\) is the Schatten ideal). It is proved that any \(G_1,G_1\in\mathbb G_p\) can be joined by a minimal curve of the form \(G(t)=\exp(itX)\), where \(X\in\sigma_p(H_0^1)\) is a symmetrizable operator.