Linear operators on matrices: The invariance of the decomposable numerical radius (Q1821846)

Let \(1\leq m\leq n\). For \(A\in {\mathbb{C}}^{n\times n}\) one defines the \(m^{th}\) decomposable numerical range by \(W_ m^{\wedge}(A):=\{\det X*A|\) \(X\in C^{n\times m}\), det X*X\(=1\}\) and \(r_ m^{\wedge}(A):=\max \{| z| |\) \(z\in W_ m^{\wedge}(A)\}\). The linear operator \(T: {\mathbb{C}}^{n\times n}\to {\mathbb{C}}^{n\times n}\) has the property that (a) \(r_ m^{\wedge}(T(A))=r_ m^{\wedge}(A)\) and (b) \(W_ m^{\wedge}(T(I_ n))=W_ n^{\wedge}(I_ n)\) if and only if there exist a unitary \(U\in {\mathbb{C}}^{n\times n}\) and \(\xi\in {\mathbb{C}}\) with \(\xi^ m=1\) such that \(T(A)=\xi U*AU\) for all A or \(T(A)=\xi U*A^ TU\) for all A.