Some convexity theorems for the generalized numerical ranges

Abstract

Let M_n be the algebra n × n complex matrices, where n ≥ 2. Given a nonscalar matrix C ∊ M_n , the C-numerical range of A ∊ M_n is defined by

If rank (C − γ^I) for some γ ∊ C(which is always true when n = 2) then W(C:A) be written as a + bW (q:A) for some a,b ∊ C and q ∊ C and q ∊[0,1], where

is the q-numerical range of A. We give short proofs for the faets that W(q :A) is convex for all A ∊ M_n , and that W(C : A) is an elliptical disk if A,C ∊ M₂ . These results have been proved by Tsing and Nakazato, respectively, by some very involved computational methods.

¹Research partially supported by a NATO grant. This paper was done while the author was visiting the Mathematics Department of the University of Hong Kong. He would like to think the staff of the department for their warm hosnitality.

¹Research partially supported by a NATO grant. This paper was done while the author was visiting the Mathematics Department of the University of Hong Kong. He would like to think the staff of the department for their warm hosnitality.

Notes

¹Research partially supported by a NATO grant. This paper was done while the author was visiting the Mathematics Department of the University of Hong Kong. He would like to think the staff of the department for their warm hosnitality.