Parallel sums and norm convergence
type: Item
instance of: scholarly article
title: Parallel sums and norm convergence
zbMATH Open document ID: 0709.47019
DOI: 10.1007/BF01236453
author: William L. Green
author: T. D. Morley
published in: Circuits, Systems, and Signal Processing
publication date: 1990-01-01T00:00:00Z
review text: For positive operators A, B in a Hilbert space, the parallel sum A:B is defined by \(A:B=(A^{-1}+B^{-1})^{-1}\) for A and B invertible, and by \(A:B=\lim_{\epsilon \searrow 0}(A+\epsilon I):(B+\epsilon I)\) otherwise, where the convergence is understood in the strong topology. When A and B represent the impedance operator of two resistive parts, A:B represents the impedance of the pair, connected in parallel. This paper surveys some results about the norm convergence of the limit in the definition of A:B, and presents two examples disproving several conjectures about this norm convergence.
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 94C05
zbMATH DE Number: 4165781
zbMATH Keywords: positive operators
zbMATH Keywords: parallel sum
zbMATH Keywords: impedance
reviewed by: M. González
MaRDI profile type: Publication
cites work: Shorted Operators
cites work: Series and parallel addition of matrices
cites work: Q5673924
cites work: Shorted Operators. II
cites work: Q4169818
cites work: Q5670526
cites work: On operator ranges
cites work: Operator means, fixed points, and the norm convergence of monotone approximants.
cites work: Q3667432
cites work: Hermitian semidefinite matrix means and related matrix inequalities—an introduction
full work available at URL: https://doi.org/10.1007/bf01236453
OpenAlex ID: W2020275547
description: scientific article; zbMATH DE number 4165781