Schur product of matrices and numerical radius (range) preserving maps (Q884406)

Let \(M_n\) (resp., \(H_n)\) be the algebra of \(n\)-by-\(n\) complex matrices (resp., Hermitian matrices). The numerical range \(W(A)\) and numerical radius \(w(A)\) of \(A\) in \(M_n\) are given by \[ W(A)=\{x^*Ax: x\in\mathbb{C}^n,x^*x=1\}\quad\text{and} \quad w(A)=\max\{|\lambda|: \lambda\in W(A)\}, \] respectively. The main results of this paper are characterizations of mappings on \(M_n\) and \(H_n\) which preserve the numerical ranges or numerical radii of Schur products of matrices. More specifically, it is shown that, for \(V=M\) or \(H_n\), (1) \(\varphi:V \to V\) is such that \(w(A\circ B)=w(\varphi(A)\circ\varphi(B))\) for all \(A\) and \(B\) in \(V\) if and only if it is given by \(\varphi(X)=R\circ(P^tD_X X^\tau E_XP)\) for \(x\in V\), where \(R\) in \(V\) is such that \(R\circ R= [\overline x_ix_j]\) with \(|x_1|=\cdots=|x_n|=1\), \(P\) is a permutation matrix, \(D_X\) and \(E_X\) are diagonal unitary matrices in \(V\) which depend on \(X\) and satisfy \(D_XE_XX=XD_X E_X\), and \(X^\tau\) denotes \(X,\overline X,X^t\) or \(X^*\), and (2) \(\varphi: M_n\to M_n\) is such that \(W(A\circ B)=W(\varphi(A)\circ\varphi(B))\) for all \(A\) and \(B\) in \(V\) if and only if \(\varphi(X)=R\circ(P^tD_X^*XD_XP)\) or \(\varphi(X) =R\circ(P^tD_X^*X^tD_XP)\) for \(X\) in \(V\), where \(R,P\) and \(D_X\) are as in (1).