Isometries of real subspaces of self-adjoint operators in Banach symmetric ideals

Автор: Aminov Behzod R., Chilin Vladimir I.

Журнал: Владикавказский математический журнал @vmj-ru

Статья в выпуске: 4 т.21, 2019 года.

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Let (CE,∥⋅∥CE) be a Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space H. Let ChE={x∈CE:x=x∗} be the real Banach subspace of self-adjoint operators in (CE,∥⋅∥CE). We show that in the case when (CE,∥⋅∥CE) is a separable or perfect Banach symmetric ideal (CE≠C2) any skew-Hermitian operator H:ChE→ChE has the following form H(x)=i(xa-ax) for same a∗=a∈B(H) and for all x∈ChE. Using this description of skew-Hermitian operators, we obtain the following general form of surjective linear isometries V:ChE→ChE. Let (CE,∥⋅∥CE) be a separable or a perfect Banach symmetric ideal with not uniform norm, that is ∥p∥CE>1 for any finite dimensional projection p∈CE with dimp(H)>1, let CE≠C2, and let V:ChE→ChE be a surjective linear isometry. Then there exists unitary or anti-unitary operator u on H such that V(x)=uxu∗ or V(x)=-uxu∗ for all x∈ChE.

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Symmetric ideal of compact operators, skew-hermitian operator, isometry

Короткий адрес: https://sciup.org/143168810

IDR: 143168810   |   DOI: 10.23671/VNC.2019.21.44607

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