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The group of L²-isometries on H¹₀

Esteban Andruchow;  Eduardo Chiumiento;  Gabriel Larotonda — 2013

Studia Mathematica

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....

Proper subspaces and compatibility

Esteban Andruchow;  Eduardo Chiumiento;  María Eugenia Di Iorio y Lucero — 2015

Studia Mathematica

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,...

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