Translation invariant subspaces of some quasi-analytic weighted Hilbert spaces of sequences

Abstract

We use the theory of asymptotically holomorphic functions in the disc to study the lattice of left-invariant subspaces of ℓ2ω(ℤ), where ω is nonincreasing, in the quasianalytic case ∑n<0 log ω(n)/n2 = +∞When (ω(-n))n≥0 satisfies suitable growth and regularity conditions, we show in particular that all bilaterally invariant subspaces of ℓ2ω(ℤ) are generated by their intersection with ℓ2ω(ℤ+). When ω(n) = 1 for n ≥ 0 and ω(n) = e |n|/1+log|n| for n < 0 this shows that all nontrivial bi-invariant subspaces of ℓ2ω(ℤ) are generated by the Fourier transform of a nonconstant singular inner function.