Linear operators preserving the (p,q)-numerical range

Abstract

Let Cn×n and Hn denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p≤q≤n. For Aε{lunate}Cn×n, the (p,q)-numerical range of A is the set Wp,q(A)={trCp(JqUAU*):U unitary}, where Cp(X) is the pth compound matrix of X, and Jq is the matrix Iq{equivalent to}On-q. Let L denote Hn or Cn×n. The problem of determining all linear operators T: L→L such that Wp,q(T(A))=Wp,q(A) for all Aε{lunate}L is treated in this paper. © 1988.