Additive preservers of numerical range
Let \({\mathbf C}\) be the field of complex numbers, \(M_n\) be the set of all \(n\times n\)-matrices, \(T_n\) be the set of all upper triangular \(n\times n\)-matrices with complex coefficients, \(\langle x,y \rangle\) denotes the scalar product of two vectors, \(\|x\|^2=\langle x,x\rangle\). The numerical range of a matrix \(A\in M_n\) is the set \(W(A)=\{\langle Ax,x\rangle \mid x\in {\mathbf C}^n\), \(\|x\|=1\}\). In the past two decades linear transformations on matrix algebras that preserve numerical range and various generalizations were the objects of intensive study. For the details one may visit the homepage of Chi-Kwong Li: \url{http://www.math.wm.edu/~ckli}. In this paper additive transformations preserving a numerical range on \(M_n\), \(T_n\), diagonal, block-diagonal, and block upper-triangle subalgebras of \(M_n\) are characterized. It is proved that in the case of \(M_n\) and its diagonal and block-diagonal subalgebras all such transformations are necessarily linear and exhausted by transposition and diagonal unitary similarity (similarity factors are unitary and have the same block-diagonal structure as matrices from subalgebra under consideration). For \(T_n\) and block-triangular subalgebras of \(M_n\) nonlinear additive numerical range preservers are found. The author establishes a complete classification of additive numerical range preservers over \(T_n\) and block upper-triangular subalgebras of \(M_n\). In the case of \(T_n\) the transformations under consideration are unitary similarity, transposition over the anti-diagonal, and the nonlinear transformation that maps \(a_{11}\) into itself, \(a_{1i}\) into its complex conjugate for all \(i=2,\ldots,n\), and the remaining block into its transposed over the anti-diagonal.
scientific article; zbMATH DE number 1741581
scientific article