The group of L²-isometries on H¹₀

Esteban Andruchow ; Eduardo Chiumiento ; Gabriel Larotonda

Studia Mathematica, Tome 215 (2013), p. 193-217 / Harvested from The Polish Digital Mathematics Library

Résumé

Let Ω be an open subset of ℝⁿ. Let L² = L²(Ω,dx) and H¹₀ = H¹₀(Ω) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group of invertible operators on H¹₀ which preserve the L²-inner product. When Ω is bounded and ∂Ω is smooth, this group acts as the intertwiner of the H¹₀ solutions of the non-homogeneous Helmholtz equation u - Δu = f, $u|_{∂Ω} = 0$. We show that 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 by means of examples. In particular, we give an example of an operator in whose spectrum is not contained in the unit circle. We also study the one-parameter subgroups of . Curves of minimal length in are considered. We introduce the subgroups $_{p}: = ∩ (I - ℬ_{p}(H¹₀))$, where $ℬ_{p}(H₀¹)$ is the Schatten ideal of operators on H₀¹. An invariant (weak) Finsler metric is defined by the p-norm of the Schatten ideal of operators on L². We prove that any pair of operators $G₁, G₂ ∈ _{p}$ can be joined by a minimal curve of the form $δ(t) = G₁ e^{itX}$, where X is a symmetrizable operator in $ℬ_{p}(H¹₀)$.

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Publié le : 2013-01-01

Zbl 1290.47042

EUDML-ID : urn:eudml:doc:285483