Inclusion relations for thec-numerical range
M. S. Jones, H. P. Rogosinski · Linear and Multilinear Algebra · 1993
Let , and let be a complex Hilbert space of dimension greater than or equal to k. Let be a bounded linear operator. The c-numerical range of T, denoted by Wc (T), is defined to be the set of scalars obtained by letting (e 1,…,ek ) vary over the set of all orthonormal k-tuples in In this paper we show that, for vectors , the inclusion relation Wc (T)⊆convW c'(T) holds for all bounded linear operators T if, and only if, there exists a doubly stochastic k×k matrix S such that c=c'S. Let (e 1,…e k−1) be a given orthonormal (k−1)-tuple in . In the case when the coordinates of c' are collinear in we also show that if the inclusion relation Wc (Er )⊆W c'(Er ) hold for r=1,…,k−1, and if Wc (I)⊆W c'(I) then Wc (T)⊆W c'(T) holds for all bounded linear operators T on Moreover, when dim we show that in general, no one of these k inclusion relations is implied by the remaining ones, and in this sense the result is best possible.