Constructive proof of Carpenter's theorem
type: Item
instance of: scholarly article
zbMATH Open document ID: 1314.42032
author: Marcin Bownik
author: John Jasper
published in: Canadian Mathematical Bulletin
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.6632
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 6359636
zbMATH Keywords: diagonals of projections
zbMATH Keywords: Schur-Horn theorem
zbMATH Keywords: Pythagorean theorem
zbMATH Keywords: Carpenter's theorem
zbMATH Keywords: spectral theory
MaRDI profile type: Publication
arXiv ID: 1302.6632
DOI: 10.4153/CMB-2013-037-X
title: Constructive proof of Carpenter's theorem
review text: Elementary variants of the Pythagorean theorem were examined by Kadison, a converse of which is referred to as Carpenter's theorem. The authors considered the following result due to Kadison.NEWLINENEWLINE``Let \(\{d_i\}_{i\in I}\) be a sequence in \([0,1]\). Define NEWLINE\[NEWLINE a= \sum\limits_{d_i < 1/2} d_i \;\;\text{and} \;\;b=\sum_{d_i\geq 1/2} (1-d_i). NEWLINE\]NEWLINE There exists a projection with diagonals \(\{d_i\}\) if and only if one of the following holds:NEWLINENEWLINE(i) \(a,\;b < \infty\) and \(a-b\in \mathbb{Z}\).NEWLINENEWLINE(ii) \(a = \infty\) or \(b=\infty\).''NEWLINENEWLINEIn this paper the authors give a constructive proof of Carpenter's theorem which also yields the real case. They also introduce an algorithm for constructing a projection with prescribed diagonal that is reminiscent of the spectral tetris construction introduced by Casazza et al. while studying tight fusion frames.
reviewed by: Shiv K. Kaushik
description: scientific article; zbMATH DE number 6359636