Linear operators preserving generalized numerical ranges and radii on certain triangular algebras of matrices (Q2764785)

For real numbers \(c_1\geq \cdots\geq c_n\) the \(c\)-numerical range of an \(n\)-by-\(n\) complex matrix \(A\) is by definition NEWLINE\[NEWLINEW_c(A)= \left\{ \sum^n_{j= 1} c_j\langle Ax_j,x_j \rangle: \{x_1,\dots, x_n\}\text{ orthogonal basis of }\mathbb{C}^n \right\}NEWLINE\]NEWLINE and the \(c\)-numerical radius of \(A\) is \(w_c(A)=\max\{|z |: z\in W_c(A)\}\). When \(c_1=1\) and \(c_2=\cdots=c_n=0\), these reduce to the classical numerical range \(W(A)\) and numerical radius \(w(A)\), respectively.NEWLINENEWLINENEWLINELinear preservers of \(W(A)\) (resp., \(w(A))\), that is, linear operators \(\varphi\) on \(M_n\), the algebra of \(n\)-by-\(n\) matrices, satisfying \(W(\varphi(A)) =W(A)\) (resp., \(w(\varphi (A))=w(A))\) for all \(A\) in \(M_n\) have been characterized before: those for \(w(A)\) must have the form \(\varphi(A) =U^*AU\) or \(\varphi(A) =U^* A^tU\) for some unitary \(U\) in \(M_n\), while those for \(w(A)\) must be a unit multiple of either of the above.NEWLINENEWLINENEWLINEThe authors study linear preservers of the \(c\)-numerical range and the \(c\)-numerical radius on algebras of block upper-triangular matrices. They show that there is a duality between such linear preservers and other preservers on the triangular algebra, and such linear preservers can be extended to \(M_n\). These facts enable the authors to determine their structure. In the case of \(W_c(A)\), there is a complete characterization of the linear preservers while for \(w_c(A)\) with \(c_1+\cdots +c_n\neq 0\), the linear preservers are unit multiples of the ones for \(W_c(A)\).