All unitary operators are tricycles, and most are bicycles

Abstract Let be a complex, separable Hilbert space (of finite or infinite dimension), and let denote the group of unitary operators on . In the finite‐dimensional setting, every unitary operator of determinant one can be expressed as the product of two operators, each unitarily equivalent to the unitary cycle. In the infinite‐dimensional setting, we prove that every unitary operator is a product of three operators, each unitarily equivalent to the bilateral shift, and if the spectrum of has non‐zero Lebesgue measure, then is a product of two operators, each unitarily equivalent to the bilateral shift.