Decomposition of a unitary scalar operator into a product of roots of the identity. (English) Zbl 1289.47035

The author proves that, for any \(m_1,m_2,m_3\in \mathbb N\) such that \(m_1^{-1}+m_2^{-1}+m_3^{-1}\leq 1\), every unitary scalar operator \(\gamma I\), \(| \gamma | =1\), on a complex infinite-dimensional Hilbert space is a product \(\gamma I=U_1U_2U_3\), where \(U_i\) is a unitary operator such that \(U_i^{m_i}=I\). The proof is based on the theory of rewriting systems; see R. V. Book and F. Otto [String-rewriting systems. New York, NY: Springer-Verlag (1993; Zbl 0832.68061)].
For related problems involving additive decompositions see, for example, S. A. Kruglyak et al. [Funct. Anal. Appl. 36, No. 3, 182–195 (2002; Zbl 1038.47001)].


47A62 Equations involving linear operators, with operator unknowns
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
68Q42 Grammars and rewriting systems