Example of quasianalytic contraction whose spectrum is a proper subarc of the unit circle

Abstract

A partial answer on [KS2, Question 2] is given. Namely, an operator $R$ similar to a quasianalytic contraction whose quasianalytic spectral set is equal to its spectrum and is a proper subarc of the unit circle is constructed, but no estimates of $\|R^{-1}\|$ is given.