For a positive integer k, the rank-k numerical range Λk(A) of an
operator A acting on a Hilbert space \cH of dimension at least k is the
set of scalars λ such that PAP=λP for some rank k
orthogonal projection P. In this paper, a close connection between low rank
perturbation of an operator A and Λk(A) is established. In
particular, for 1≤r<k it is shown that Λk(A)⊆Λk−r(A+F) for any operator F with \rank (F) \le r. In quantum
computing, this result implies that a quantum channel with a k-dimensional
error correcting code under a perturbation of rank ≤r will still have a
(k−r)-dimensional error correcting code. Moreover, it is shown that if A is
normal or if the dimension of A is finite, then Λk(A) can be
obtained as the intersection of Λk−r(A+F) for a collection of rank
r operators F. Examples are given to show that the result fails if A is a
general operator. The closure and the interior of the convex set Λk(A)
are completely determined. Analogous results are obtained for
Λ∞(A) defined as the set of scalars λ such that PAP=λP for an infinite rank orthogonal projection P. It is shown that
Λ∞(A) is the intersection of all Λk(A) for k=1,2,>.... If A−μI is not compact for any \mu \in \IC, then the closure
and the interior of Λ∞(A) coincide with those of the essential
numerical range of A. The situation for the special case when A−μI is
compact for some \mu \in \IC is also studied.Comment: 21 page