Abstract
We study sets \(\Sigma _n = \{ \alpha \in \mathbb{R}^1 |\) there exist n projectors P1,...,Pn such that \(\sum\nolimits_{k = 1}^n {P_k = \alpha I} \}\). We prove that if n ≥ 6, then \(\left\{ {0,1,1 + \frac{1}{{n - 1}},\left[ {1 + \frac{1}{{n - 2}},n - 1 - \frac{1}{{n - 2}}} \right],n - 1 - \frac{1}{{n - 1}},n - 1,n} \right\} \supset\) \(\Sigma _n \supset \left\{ {0,1,1 + \frac{k}{{k\left( {n - 3} \right) + 2}},k \in \mathbb{N},\left[ {1 + \frac{1}{{n - 3}},n - 1 - \frac{1}{{n - 3}}} \right],n - 1 - \frac{k}{{k\left( {n - 3} \right) + 2}},k \in \mathbb{N},n - 1,n} \right\}\).
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Rabanovich, V.I., Samoilenko, Y.S. Scalar Operators Representable as a Sum of Projectors. Ukrainian Mathematical Journal 53, 1116–1133 (2001). https://doi.org/10.1023/A:1013329232230
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DOI: https://doi.org/10.1023/A:1013329232230