Closedness of the k-numerical range

Let H be an infinite dimensional complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H. For every positive integer k and A is an element of B(H), the k-numerical range of A is the set W-k(A) = {Sigma(k)(j=1)< Ax(j), x(j)> : {x(1), ..., x(k)} is an orthonormal set in H}. In this note, we show that the closure of W-k(A) can be written as the convex hull of sets involving the essential numerical range of A and W-l(A) for 1 <= l <= k. We also show that if W-k(A) is closed, then W-l(A) is also closed for 1 <= l <= k.