Linear operators preserving the generalized decomposable numerical range
For a fixed subgroup \(G\) of the symmetric group \(S_m\) and for a fixed positive semidefinite \(f:G\to \mathbb{C}\), the \(f\)-decomposable numerical range \(W_f(A)\) of a given complex \(n\times n\) matrix \((n\geq m)\) is defined. It is noted that if \(m=1\) and \(f=1\), then \(W_f(A)\) is simply the classical numerical range of \(A\). Let \(\mathbb{C}_{n\times n}\) be the set of all complex \(n\times n\) matrices. From the author's abstract: ``The linear operator \(T:\mathbb{C}_{n\times n}\to \mathbb{C}_{n\times n}\) is said to preserve the \(f\)-decomposable numerical range if \(W_f(T(A))= W_f(A)\) for all \(A\in \mathbb{C}_{n\times n}\). Pellegrini determined the linear operators preserving the classical numerical range. Marcus and Filippenko, Tam determined the linear operators preserving the \(f\)-decomposable numerical range for some characters \(f\) of degree \(1\) satisfying certain conditions. However, for a general irreducible character, this linear preserver problem remained open. In this note, we determine the linear operators preserving the \(f\)-decomposable numerical range so that the problem is completely solved.'' The set \(W_f(A)\) depends on the subgroup \(G\). From the presentation the role of \(G\) is not completely clear.
scientific article; zbMATH DE number 1054517