The Birkhoff–James orthogonality in Hilbert C∗-modules

Arambašić, Ljiljana; Rajić, Rajna

doi:10.1016/j.laa.2012.05.011

Cited by 38 publications

(18 citation statements)

References 16 publications

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“…where H is a complex Hilbert space. Since then, many results on local symmetry of Birkhoff orthogonality have been published, especially, in the fields of Banach (or Hilbert) space operators; see [11,13,21,26] for results on spaces of bounded linear operators between Banach spaces, and [3,7,12,[22][23][24][25]27,28] for important techniques about Birkhoff orthogonality in such spaces. See also [18][19][20] for developments in the setting of operator algebras.…”

Section: Introductionmentioning

confidence: 99%

On symmetry of strong Birkhoff orthogonality in $$B(\mathcal {H},\mathcal {K})$$ and $$K(\mathcal {H},\mathcal {K})$$

Tanaka

Sain

2020

Ann. Funct. Anal.

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Section: Introductionmentioning

confidence: 99%

On symmetry of strong Birkhoff orthogonality in $$B(\mathcal {H},\mathcal {K})$$ and $$K(\mathcal {H},\mathcal {K})$$

Tanaka

Sain

2020

Ann. Funct. Anal.

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“…Recently Bhattacharyya and Grover [13] have investigated the Birkhoff-James orthogonality when the Banach space has structures. In particular, they consider Hilbert C * -modules (see also [4]). In [5] one presents some characterizations of the Birkhoff-James orthogonality of a pair of bounded operators and one considers a center of mass of a pair of operators.…”

Section: Orthogonality Of Matricesmentioning

confidence: 99%

“…For k = t and p = 2 we obtain the Frobenius norm. We define the nested sequence of minimizing sets S k , with 4) and the set S k of the minimizers of R k,2 over S k−1 . Then…”

Section: Examples Of Strict Spectral Approximants the Notion Of C P mentioning

confidence: 99%

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

Ziętak

2017

Banach Center Publ.

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The main aim of this review paper is approximation of a complex rectangular matrix, with respect to the unitarily invariant norms, by matrices from a linear subspace or from a convex closed subset of matrices. In particular, we focus on the properties and characterizations of the strict spectral approximant which is in some sense the best among all approximants to a given matrix with respect to the spectral norm. We formulate a conjecture that the strict spectral approximant to a matrix by matrices from a linear subspace is the limit of approximants with respect to the cp norms of Schatten when p tends to infinity. This conjecture corresponds to the conjecture stated by Rogers and Ward on approximation by positive semidefinite matrices and to the known analogous property of the strict Chebyshev approximation of a vector, proved by Descloux. We describe some special cases of an approximation of a matrix that confirm our conjecture and we discuss an attempt to prove the conjecture. Additionally, we focus on orthogonality of matrices in the sense of Birkhoff and James and approximation by matrices whose spectrum is in a strip. For the last case we recall a conjecture on characterization of the best approximation of a matrix in the spectral norm by matrices whose spectrum is in a strip. This kind of approximation is a generalization of the famous concept of Halmos of approximation by positive operators.

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“…More results in this direction have been obtained recently in [8,9]. Characterizations of orthogonality on Hilbert C * -modules have been studied in [2,[10][11][12].…”

Section: Introductionmentioning

confidence: 99%