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Isometries of the special orthogonal group. (English) Zbl 1297.47038

Let \(c=(c_1,\dots,c_n)\in\mathbb R^n\) satisfy \(c_1\geq\dots\geq c_n\geq 0\) and \(c_1>0\). The \(c\)-spectral norm of \(A\in\mathbb R^{n\times n}\) is defined by \(\|A\|_c=c_1\sigma_1(A)+\dots+c_n\sigma_n(A)\), where \(\sigma_1(A)\geq\dots\geq\sigma_n(A)\) are the singular values of \(A\). The authors give a complete characterization of all isometries on the special orthogonal group \(SO(n)\) in \(\mathbb R^{n\times n}\) with respect to \(\|\cdot\|_c\). As an application, they characterize all spectrally multiplicative mappings on \(SO(n)\).

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15B10 Orthogonal matrices
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References:

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