Linear operators preserving the (p,q)-numerical range
Let \(C_{m\times n},H_ n\) and \(U_ n\) denote respectively the space of complex \(m\times n\) matrices, the space of Hermitian \(n\times n\) matrices, and the group of unitary matrices in \(C_{n\times n}\). Let p,q,n be integers such that \(1\leq p\leq q\leq n\). For \(A\in C_{n\times n}\) the (p,q)-numerical range of A is defined to be the set \(W_{p,q}(A)=\{trC_ p(J_ qUAU^*):\) \(U\in U_ n\}\), where \(C_ p(A)\) denote the p-th compound matrix of A and \(J_ q=I_ q\oplus O_{n-q}\). The authors consider the problem of determining all linear operators \(T: C_{n\times n}\to C_{n\times n}\) (or \(H_ n\to H_ n)\) such that \(W_{p,q}(T(A))=W_{p,q}(A).\)
scientific article; zbMATH DE number 4069099
scientific article