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 H-R(infinity), 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)over bar>))over bar> = f(z) for all is an element of D. Tolokonnikov's Lemma for H-R(infinity) means that if f is left-invertible, then f can be completed to an isomorphism; that is, there exists an F, invertible in H-R(infinity), such that F = [f f(c)] for some f(c) in H-R(infinity). In control theory, Tolokonnikov's Lemma implies that if a function has a right coprime factorization over H-R(infinity), then it has a doubly coprime factorization in H-R(infinity). We prove the lemma for the real disc algebra A(R) as well. In particular, H-R(infinity) and A(R) are Hermite rings.