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The group of L²-isometries on H¹₀

100%

Andruchow E., Chiumiento E., Larotonda G.

Studia Mathematica

2013

tom 217

nr 3

193-217

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 $ℬ_{p}(H¹₀)$.

Proper subspaces and compatibility

100%

Andruchow E., Chiumiento E., Di Iorio y Lucero M.

Studia Mathematica

2015

tom 231

nr 3

195-218

Let 𝓔 be a Banach space contained in a Hilbert space 𝓛. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambickiĭ, we say that a bounded operator on 𝓔 is a proper operator if it admits an adjoint with respect to the inner product of 𝓛. A proper operator which is self-adjoint with respect to the inner product of 𝓛 is called symmetrizable. By a proper subspace 𝓢 we mean a closed subspace of 𝓔 which is the range of a proper projection. Furthermore, if there exists a symmetrizable projection onto 𝓢, then 𝓢 belongs to a well-known class of subspaces called compatible subspaces. We find equivalent conditions to describe proper subspaces. Then we prove a necessary and sufficient condition for a proper subspace to be compatible. The existence of non-compatible proper subspaces is related to spectral properties of symmetrizable operators. Each proper subspace 𝓢 has a supplement 𝒯 which is also a proper subspace. We give a characterization of the compatibility of both subspaces 𝓢 and 𝒯. Several examples are provided that illustrate different situations between proper and compatible subspaces.

Ograniczanie wyników

2 Studia Mathematica

2 Andruchow E.

2 Chiumiento E.

1 Di Iorio y Lucero M.

1 Larotonda G.

1 2015

1 2013

Wyniki wyszukiwania

The group of L²-isometries on H¹₀

Proper subspaces and compatibility