Compression of quasianalytic spectral sets of cyclic contractions

The class L0(H) of cyclic quasianalytic contractions was studied in Kérchy (2011) [12]. The subclass L1(H) consists of those operators T in L0(H) whose quasianalytic spectral set π(T) covers the unit circle T. The contractions in L1(H) have rich invariant subspace lattices. In this paper it is shown that for every operator T∈L0(H) there exists an operator T1∈L1(H) commuting with T. Thus, the hyperinvariant subspace problems for the two classes are equivalent. The operator T1 is found as an H∞-function of T. The existence of an appropriate function, compressing π(T) to the whole circle, is proved using potential theoretic tools by constructing a suitable regular compact set on T with absolutely continuous equilibrium measure.