Condition for the higher rank numerical range to be non-empty

Li, Chi-Kwong; Poon, Yiu‐Tung; Sze, Nung-Sing

doi:10.1080/03081080701786384

Cited by 49 publications

(49 citation statements)

References 9 publications

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“…Results on Λ k (A) for infinite dimensional operators have been obtained in [10] An open question in [2] concerns the lower bound of dim H which ensures that Λ k (A) is nonempty for every bounded linear operator A acting on H. The following result was proved in [9] and answers the above question. …”

Section: Infinite Dimensional Operators and Related Resultsmentioning

confidence: 85%

Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations

Sze

2008

Proc. Amer. Math. Soc.

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Abstract. The results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of closed half planes (of complex numbers). As a result, it is always a convex set in C. Moreover, the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues. These two consequences confirm the conjectures of Choi et al. on the subject. In addition, the results are used to derive a formula for the optimal upper bound for the dimension of a totally isotropic subspace of a square matrix and to verify the solvability of certain matrix equations.

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Section: Infinite Dimensional Operators and Related Resultsmentioning

confidence: 85%

Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations

Sze

2008

Proc. Amer. Math. Soc.

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“…The first equality in (1.7) gives an affirmative answer to a question of MartinezAvendano [14]. We close this section by listing some basic properties for the higher rank numerical range; see [4][5][6][7]9,12,13,16].…”

Section: ) If K Is the Algebra Of Compact Operators In B(h) And If mentioning

confidence: 95%

“…First, we establish the inclusion Λ k (A) ⊆ Ω k (A). By [12,Corollary 4], Λ k (A) is always non-empty. Suppose μ ∈ Λ k (A).…”

Section: Theorem 21 Let a ∈ B(h) Be An Infinite Dimensional Operatomentioning

confidence: 96%

Higher rank numerical ranges and low rank perturbations of quantum channels

Poon

Sze

2008

Journal of Mathematical Analysis and Applications

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For a positive integer k, the rank-k numerical range Λ k (A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that P A P = λP for some rank k orthogonal projection P . In this paper, a close connection between low rank perturbation of an operator A and Λ k (A) is established. In particular, forIn quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most 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 Λ k (A) can be obtained as the intersection of Λ k−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 Λ k (A) are completely determined. Analogous results are obtained for Λ ∞ (A) defined as the set of scalars λ such that P A P = λP for an infinite rank orthogonal projection P . It is shown that Λ ∞ (A) is the intersection of all Λ k (A) for k = 1, 2, . . . . If A − μI is not compact for all μ ∈ C, 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 μ ∈ C is also studied.Published by Elsevier Inc.

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“…Recently, researchers have studied the higher rank numerical range in connection to quantum error correction; see [4,5,6,21,23] and Section 2.1. Each of these generalizations encodes certain specific information of the operator that leads to interesting applications.…”

Section: Generalized Numerical Rangesmentioning

confidence: 99%

Spectra, norms and numerical ranges of generalized quadratic operators

Poon

Tominaga

2011

Linear and Multilinear Algebra

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Abstract. A bounded linear operator acting on a Hilbert space is a generalized quadratic operator if it has an operator matrix of the form aI cT dT * bI .It reduces to a quadratic operator if d = 0. In this paper, spectra, norms, and various kinds of numerical ranges of generalized quadratic operators are determined. Some operator inequalities are also obtained. In particular, it is shown that for a given generalized quadratic operator, the rank-k numerical range, the essential numerical range, and the q-numerical range are elliptical disks; the c-numerical range is a sum of elliptical disks. The Davis-Wielandt shell is the convex hull of a family of ellipsoids unless the underlying Hilbert space has dimension 2.

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Condition for the higher rank numerical range to be non-empty

Cited by 49 publications

References 9 publications

Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations

Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations

Higher rank numerical ranges and low rank perturbations of quantum channels

Spectra, norms and numerical ranges of generalized quadratic operators

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