The group of L²-isometries on H¹₀

• Autorzy: Esteban Andruchow , Eduardo Chiumiento , Gabriel Larotonda
• Języki publikacji: EN

Abstrakty

EN

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 𝔾 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 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 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 $𝔾_{p}: = 𝔾 ∩ (I - ℬ_{p}(H¹₀))$, 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 $G₁, G₂ ∈ 𝔾_{p}$ can be joined by a minimal curve of the form $δ(t) = G₁ e^{itX}$, where X is a symmetrizable operator in $ℬ_{p}(H¹₀)$.

Kategorie tematyczne

• 58B20: Riemannian, Finsler and other geometric structures
• 47D03: Groups and semigroups of linear operators
• 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2013
• Tom: 217
• Numer: 3
• Strony: 193-217

Daty

wydano

2013

Twórcy

autor

Esteban Andruchow

• Instituto de Ciencias, Universidad Nacional de General Sarmiento
• Instituto Argentino de Matemática, 'Alberto P. Calderón', CONICET, J.M. Gutierrez 1150 (B1613GSX), Los Polvorines, Argentina

autor

Eduardo Chiumiento

• Departamento de Matemática, FCE-Universidad Nacional de La Plata
• Instituto Argentino de Matemática, 'Alberto P. Calderón', CONICET, Calles 50 y 115 (1900), La Plata, Argentina

autor

Gabriel Larotonda

• Instituto de Ciencias, Universidad Nacional de General Sarmiento
• Instituto Argentino de Matemática, `Alberto P. Calderón', CONICET, J.M. Gutierrez 1150 (B1613GSX), Los Polvorines, Argentina

Identyfikatory

DOI

10.4064/sm217-3-1