Inclusion regions for numerical ranges and Linear Preservers

There has been considerable interest in studying inclusion regions for numerical ranges. It is in fact very useful in knowing inclusion regions for W (A). For example, it is well known (see [4, Chapter 1]) that W (A) ⊆ IR if and only if A = A∗; W (A) ⊆ [0,∞) if and only if A is positive semidefinite; andW (A) ⊆ (0,∞) if and only if A is positive definite. Moreover, Ando [1] (see also [2]) showed that W (A) is contained in the unit disk if and only if A = X∗CX with a 2m × n matrix X such that X∗X = In and C = ( 0m 2Im 0m 0m ) for some integer m;