Decomposition of a scalar operator into a product of unitary operators with two points in spectrum

We consider products of unitary operators with at most two points in their spectra, 1 and eiα. We prove that the scalar operator eiγI is a product of k such operators if α(1+1/(k-3))⩽γ⩽α(k-1-1/(k-3)) for k⩾5. Also we prove that for eiα≠-1, only a countable number of scalar operators can be decomposed in a product of four operators from the mentioned class. As a corollary we show that every unitary operator on an infinite-dimensional space is a product of finitely many such operators.