Tolokonnikov’s Lemma for Real H ∞ and the Real Disc Algebra

We prove Tolokonnikov’s Lemma and the inner-outer factorization for the real Hardy space HR∞ , the space of bounded holomorphic (possibly operator-valued) functions on the unit disc all of whose matrix-entries (with respect to fixed orthonormal bases) are functions having real Fourier coefficients, or equivalently, each matrix entry f satisfies f(z)=f(z) for all z ∈ D. Tolokonnikov’s Lemma for H∞R means that if f is left-invertible, then f can be completed to an isomorphism; that is, there exists an F, invertible in HR∞ , such that F = [ f f c ] for some f c in HR∞ . In control theory, Tolokonnikov’s Lemma implies that if a function has a right coprime factorization over HR∞, then it has a doubly coprime factorization in HR∞ . We prove the lemma for the real disc algebra AR as well. In particular, HR∞ and AR are Hermite rings.