Constructive proof of Carpenter's theorem (Q2925363)

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.