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Linear transformations on \(M_ n({\mathbb{R}})\) that preserve the Ky Fan k- norm and a remarkable special case when \((n,k)=(4,2)\). (English) Zbl 0658.15029

The Ky Fan k-norm of an \(n\times n\) complex (real) matrix A is the sum of square roots of k largest eigenvalues of \(A^*A\) (respectively, \(A^ tA.)\) It is known that a linear map \(\phi\) on \(M_ n({\mathbb{C}})\) preserves the Ky Fan k-norm if and only if there exist unitary matrices U and V such that \(\phi (A)=UA^+V\), for all \(A\in M_ n({\mathbb{C}})\), where \(A^+\) is either A or \(A^ t\). In this article the authors study real linear maps \(\phi\) on \(M_ n({\mathbb{R}})\) preserving the Ky Fan k-norm and show that except for the case \(n=4\) and \(k=2\), the real version of the above result is true. For \((n,k)=(4,2)\), \(\phi\) is either of the mentioned form or the composition of such form with an interesting particular operator.
Reviewer: A.A.Jafarian

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A04 Linear transformations, semilinear transformations
15A21 Canonical forms, reductions, classification
15A18 Eigenvalues, singular values, and eigenvectors
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