Abstract: Let \((\mathcal C_E, \|\cdot\|_{\mathcal C_E})\) be a Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space \(\mathcal H\). Let \(\mathcal C_E^h=\{x\in \mathcal C_E : x=x^*\}\) be the real Banach subspace of self-adjoint operators in \((\mathcal C_E, \|\cdot\|_{\mathcal C_E})\). We show that in the case when \((\mathcal C_E, \|\cdot\|_{\mathcal C_E})\) is a separable or perfect Banach symmetric ideal (\(\mathcal C_E \neq \mathcal C_2\)) any skew-Hermitian operator \(H: \mathcal C_E^h\to \mathcal C_E^h\) has the following form \(H(x)=i(xa - ax)\) for same \(a^*=a\in \mathcal B(\mathcal H)\) and for all \(x\in \mathcal C_E^h\). Using this description of skew-Hermitian operators, we obtain the following general form of surjective linear isometries \(V:\mathcal C_E^h \to \mathcal C_E^h\). Let \((\mathcal C_E, \|\cdot\|_{\mathcal C_E})\) be a separable or a perfect Banach symmetric ideal with not uniform norm, that is \(\|p\|_{\mathcal C_E}> 1\) for any finite dimensional projection \(p \in\mathcal C_E\) with \(\dim p(\mathcal H)>1\), let \(\mathcal C_E \neq \mathcal C_2\), and let \(V: \mathcal C_E^h \to \mathcal C_E^h\) be a surjective linear isometry. Then there exists unitary or anti-unitary operator \(u\) on \(\mathcal H\) such that \(V(x)=uxu^*\) or \(V(x)=-uxu^*\) for all \(x \in \mathcal C_E^h\).
Keywords: symmetric ideal of compact operators, skew-Hermitian operator, isometry.
For citation: Aminov, B. R. and Chilin, V. I. Isometries of Real Subspaces of Self-Adjoint Operators in Banach Symmetric Ideals, Vladikavkaz Math. J., 2019, vol. 21, pp. 11-24. DOI 10.23671/VNC.2019.21.44607
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