Geodesic
When Is a Sum of Partial Reflections Equal to a Scalar Operator?
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 2, pp. 91-94

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We describe the set $\widetilde{W}_n$ of values of the parameter $\alpha\in\mathbb{R}$ for which there exists a Hilbert space $H$ and $n$ partial reflections $A_1,\dots,A_n$ (self-adjoint operators such that $A_k^3=A_k$ or, which is the same, self-adjoint operators whose spectra belong to the set $\{-1,0,1\}$) whose sum is equal to the scalar operator $\alpha I_H$.
Keywords: projection, reflection, partial reflection, self-adjoint operator, *-representation. 
@article{FAA_2004_38_2_a9,
     author = {A. S. Mellit and V. I. Rabanovich and Yu. S. Samoilenko},
     title = {When {Is} a {Sum} of {Partial} {Reflections} {Equal} to a {Scalar} {Operator?}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {91--94},
     publisher = {mathdoc},
     volume = {38},
     number = {2},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_2_a9/}
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A. S. Mellit; V. I. Rabanovich; Yu. S. Samoilenko. When Is a Sum of Partial Reflections Equal to a Scalar Operator?. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 2, pp. 91-94. http://geodesic.mathdoc.fr/item/FAA_2004_38_2_a9/