We show that a bounded linear operator A ∈ B(H) is a multiple of a unitary operator if and only if AZ and ZA always have the same numerical radius or the same numerical range for all (rank one) Z ∈ B(H). More generally, for any bounded linear operators A,B ∈ B(H), we show that AZ and ZB always have the same numerical radius (resp., the same numerical range) for all (rank one) Z ∈ B(H) if and on...

The higher rank numerical range is useful for constructing quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ∈ Mn has eigenvalues a1, . . . , an, then its rank-k numerical range Λk(A) is the intersection of convex polygons with vertices aj1 , . . . , ajn−k+1 , where 1 ≤ j1 < · · · < jn−k+1 ≤ n. In this paper, it is shown that the higher rank numeri...

Here we give a closure free description of the higher rank numerical range normal operator acting on separable Hilbert space. This generalizes result Avendaño for self-adjoint operators. It has several interesting applications. We show using Durszt's example that there exists contraction T which intersection ranges all unitary dilations contains as proper subset. strengthen and generalize Wu by...

For a noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the joint rank-k numerical range associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint rank k-numerical range are obtained and their implications to quantum computing are discussed. It is shown that for a given k if the dimension of the un...