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

Li, Chi-Kwong; Sze, Nung-Sing

doi:10.1090/s0002-9939-08-09536-1

Cited by 67 publications

(62 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Analogues of Theorem 1.2 in the context of pairs of complex or quaternionic hermitian matrices A and B, where µA + iλB of Theorem 1.2 is replaced by µA+λB, have been obtained in [9,11]. We mention in passing that an analogue of Theorem 1.2 for pairs of real symmetric matrices fails, see [11] for more details.…”

Section: Proofmentioning

confidence: 93%

Neutral subspaces of pairs of symmetric/skewsymmetric real matrices

Rodman¹,

Šemrl²

2011

ELA

View full text Add to dashboard Cite

Abstract. Let A and B be n × n real matrices with A symmetric and B skewsymmetric. Obviously, every simultaneously neutral subspace for the pair (A, B) is neutral for each Hermitian matrix X of the form X = µA + iλB, where µ and λ are arbitrary real numbers. It is well-known that the dimension of each neutral subspace of X is at most In + (X) + In 0 (X), and similarly, the dimension of each neutral subspace of X is at most In − (X) + In 0 (X). These simple observations yield that the maximal possible dimension of an (A, B)-neutral subspace is no larger thanwhere the outer minimum is taken over all pairs of real numbers (λ, µ). In this paper, it is proven that the maximal possible dimension of an (A, B)-neutral subspace actually coincides with the above expression.

show abstract

Section: Proofmentioning

confidence: 93%

Neutral subspaces of pairs of symmetric/skewsymmetric real matrices

Rodman¹,

Šemrl²

2011

ELA

View full text Add to dashboard Cite

show abstract

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…It turns out that even for a single matrix A, determining Λ k (A) is highly non-trivial, and the results are useful in quantum computing, say, in constructing binary unitary channels; see [5]. In a sequence of papers [4][5][6][7]9,12,13,16], researchers studied the set Λ k (A) for A ∈ B(H). Many interesting results (see (P1)-(P8) below) were obtained.…”

Section: Introductionmentioning

confidence: 99%

Higher rank numerical ranges and low rank perturbations of quantum channels

Poon

Sze

2008

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 67 publications

References 12 publications

Neutral subspaces of pairs of symmetric/skewsymmetric real matrices

Neutral subspaces of pairs of symmetric/skewsymmetric real matrices

Spectra, norms and numerical ranges of generalized quadratic operators

Higher rank numerical ranges and low rank perturbations of quantum channels

Contact Info

Product

Resources

About