Bilateral shorted operators and parallel sums

Antezana, Jorge; Corach, Gustavo; Stojanoff, Demetrio

doi:10.1016/j.laa.2005.10.039

Cited by 45 publications

(57 citation statements)

References 35 publications

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“…Following the ideas of [4], we extend the notion of Schur complement to selfadjoint operators in Krein spaces: Let W ∈ L(H) [s] and let S be a closed subspace of H. Then, applying Lemma 2.2 to B = JW, with J any signature operator, S can be decomposed as…”

Section: Proofmentioning

confidence: 99%

“…If S is a regular subspace of H (meaning that H = S [∔] S [⊥] ) then it is possible to give a characterization of the S-weak complementability of W in terms of the entries of the first row of the 2 × 2 block matrix representation of W with respect to S [∔] S [⊥] . Indeed if W = (w ij ) i,j=1,2 and w 11 = dd # is a Bognár-Krámli factorization of w 11 obtained as in [ [4]. Based on a formula given by Pekarev [22], Maestripieri and Martínez Pería [18] extended the concept of the Schur complement to bounded selfadjoint operators in Krein spaces with the so-called "unique factorization property".…”

Section: Introductionmentioning

confidence: 99%

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Schur complements of selfadjoint Krein space operators

Contino

Maestripieri

Marcantognini

2019

Linear Algebra and its Applications

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Given a bounded selfadjoint operator W on a Krein space H and a closed subspace S of H, the Schur complement of W to S is defined under the hypothesis of weak complementability. A variational characterization of the Schur complement is given and the set of selfadjoint operators W admitting a Schur complement with these variational properties is shown to coincide with the set of S-weakly complementable selfadjoint operators.

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Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Schur complements of selfadjoint Krein space operators

Contino

Maestripieri

Marcantognini

2019

Linear Algebra and its Applications

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“…While a short history of parallel sum related to electrical networks is provided in the introduction to [10], applications to network connections are explored in [9,196]. Parallel sum have also found applications in quantum effects [90].…”

Section: Network Flowmentioning

confidence: 99%

What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications

Lucet¹

2009

SIAM J. Optim.

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Abstract. Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of Computational Convex Analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolic-numeric algorithms. Our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing (computing the distance transform, the generalized distance transform, and mathematical morphology operators), partial differential equations (solving Hamilton-Jacobi equations, and using differential equations numerical schemes to compute the convex envelope), max-plus algebra (computing the equivalent of the Fast Fourier Transform), multi-fractal analysis, etc. The fields of applications include, among others, computer vision, robot navigation, thermodynamics, electrical networks, medical imaging, and network communication.

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“…Ever since the publication of [2], the parallel sum has been studied in the more general settings of non-square matrices under certain conditions of range inclusions [17], of positive operators A and B on a Hilbert space such that the range of A + B is closed [5] and furthermore, without any assumptions on the range of A + B [14,19]. As the generalizations of the parallel sum, shorted operators and the weakly parallel sum are also studied in [1,6,12,16,18] and [7,13], respectively. For many different equivalent definitions and the properties of the parallel sum, see a recent review paper [9] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Perturbation estimation for the parallel sum of Hermitian positive semi-definite matrices

Luo

Song

2018

Linear and Multilinear Algebra

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Let C n×n be the set of all n × n complex matrices. For any Hermitian positive semi-definite matrices A and B in C n×n , their new common upper bound lessn×n are any Hermitian positive semi-definite perturbations of A and B, respectively. Based on the derived factorization formula and the constructed common upper bound of X and Y , some new and sharp norm upper bounds of (A + X) : (B + Y ) − A : B are provided. Numerical examples are also provided to illustrate the sharpness of the obtained norm upper bounds.

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Bilateral shorted operators and parallel sums

Cited by 45 publications

References 35 publications

Schur complements of selfadjoint Krein space operators

Schur complements of selfadjoint Krein space operators

What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications

Perturbation estimation for the parallel sum of Hermitian positive semi-definite matrices

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