The cone of nonnegative c-numerical range and its preservers

Abstract

Let H be a complex Hilbert space and B(H) the Banach space of all bounded linear operators on H. Let c=(c1,…,ck), where c1≥⋯≥ck>0. For each A∈B(H), the c-numerical range of A is the setWc(A)={∑j=1kcj〈Axj,xj〉:{x1,…,xk}is an orthonormal set in H}. In this note, the conePc={A∈B(H):Wc(A)⊆[0,∞)} of operators with nonnegative c-numerical ranges is considered. We first determine the extreme directions of Pc. Then we show that all bijective linear mappings T:B(H)→B(H) such that T(Pc)=Pc are positive multiples of Jordan isomorphisms. As an application, the result is used to obtain a description of surjective mappings on B(H) that preserve the c-numerical radius distance.