The joint $k$-numerical range of operators

Abstract. Let H be an infinite dimensional complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H. For A = (A1, . . . , Am) ∈ B(H), the joint knumerical range Wk(A) of A is the set of (α1, . . . , αm) ∈ C such that αi = ∑k j=1〈Aixj , xj〉 for an orthonormal set {x1, . . . , xk} in H. In this paper, the condition for the convex hull of Wk(A) to be closed in terms of the joint essential numerical range of A is determined. Moreover, it is shown that if A1, . . . , Am are self-adjoint and m ≤ 3 then Wk(A) is closed whenever Wk+1(A) is closed. It is also shown that if the convex hull of Wk+1(A) is closed, then so is the convex hull of Wk(A). Examples are given to show that there is no implication between the convexity of Wk(A) and Wk+1(A).