Singular Inner Functions and Biinvariant Subspaces for Dissymetric Weighted Shifts

We consider dissymetric weights on Z , i.e., nonincreasing unbounded weightsω: Z→[1, ∞) such thatω(n)=1 forn⩾0 and such that limn→−∞ ω(n)1/|n|=1 whereω(n)=supp∈Z ω(n+p)/ω(p). A dissymetric weight is said to be quasianalytic if ∑n<0 log ω(n)/n2=+∞. Denote byΓthe unit circle and byDthe open unit disc. SetL2ω(Γ)={f∈L2(Γ) ∣ ‖f‖ω=(∑n∈Z |f(n)|2·ω2(n))1/2<+∞}.We identify the usual Hardy spaceH2(D) to the set {f∈L2ω(Γ) ∣ f(n)=0 (n<0)} and we denote bySω: f(eit)→eit·f(eit) the usual shift onL2ω(Γ). IfUis a singular inner function denote byEUthe closure of Span(Snω·U)n∈ZinL2ω(Γ). We show that ifωis a dissymetric weight such thatlog ω(−n)n→n→∞∞thenEUis a proper subspace ofL2ω(Γ) for every singular inner functionU. Also ifωis any dissymetric weight then there exists some singular inner functionUsuch thatEUis a proper subspace ofL2ω, so thatSωalways has nontrivial biinvariant subspaces (no nontrivial biinvariant subspaces ofSωwere known so far in the quasianalytic case).