Convexity of the Joint Numerical Range

Abstract

We consider linearly independent families of Hermitian matrices {A 1 ,. .. , Am} so that W k (A) is convex. It is shown that m can reach the upper bound 2k(n − k) + 1. A key idea in our study is relating the convexity of W k (A) to the problem of constructing rank k orthogonal projections under linear constraints determined by A. The techniques are extended to study the convexity of other generalized numerical ranges and the corresponding matrix construction problems.

The joint essential numerical range of operators: convexity and related results

“…, A m . One may see [1,5,12,14,15,16,19,23,28,31,33,35] and their references for the background and many applications of the joint numerical range.…”

“…Unfortunately, cl(W (A)) is not always convex. Here are some results concerning the convexity of W (A) and cl(W (A)), and related to W e (A) (for example, see [5,10,11,36,21,29,31] and their references).…”

“…(P2) [31] For any A 1 , A 2 , A 3 ∈ S(H) such that span{I, A 1 , A 2 , A 3 } has dimension 4, there is always an A 4 ∈ S(H) for which W (A 1 , . .…”

“…, A 4 ) is not convex. (P3) [31] If m ≥ 4 then there exists A ∈ S(H) m such that W (A) is non-convex. (P4) For any positive integer m and any A ∈ S(H) m , W e (A) is a compact set contained in W (A).…”

Maps preserving the joint numerical radius distance of operators

“…, (A m x, x)). These concepts are useful in studying the joint behaviors of several operators, and have been studied extensively; see for example [1,4,6,9,11] and their references.…”