The group of L²-isometries on H¹₀

Esteban Andruchow, Eduardo Chiumiento, and Gabriel Larotonda. "The group of L²-isometries on H¹₀." Studia Mathematica 217.3 (2013): 193-217. <http://eudml.org/doc/285483>.

@article{EstebanAndruchow2013,
abstract = {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 $ℬ_\{p\}(H¹₀)$.},
author = {Esteban Andruchow, Eduardo Chiumiento, Gabriel Larotonda},
journal = {Studia Mathematica},
keywords = {group of isometries of a positive form; Sobolev space; symmetrizable operator; one-parameter subgroup; minimal curve},
language = {eng},
number = {3},
pages = {193-217},
title = {The group of L²-isometries on H¹₀},
url = {http://eudml.org/doc/285483},
volume = {217},
year = {2013},
}

TY - JOUR
AU - Esteban Andruchow
AU - Eduardo Chiumiento
AU - Gabriel Larotonda
TI - The group of L²-isometries on H¹₀
JO - Studia Mathematica
PY - 2013
VL - 217
IS - 3
SP - 193
EP - 217
AB - 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 $ℬ_{p}(H¹₀)$.
LA - eng
KW - group of isometries of a positive form; Sobolev space; symmetrizable operator; one-parameter subgroup; minimal curve
UR - http://eudml.org/doc/285483
ER -