Tolokonnikov’s Lemma for Real H∞ and the Real Disc Algebra
Tóm tắt
We prove Tolokonnikov’s Lemma and the inner-outer factorization for the real Hardy space
$${{H^{\infty}_{\mathbb{R}}}}$$
, 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
$${\overline{f(\overline{z})}} = f(z)$$
for all z ∈
$${\mathbb{D}}$$
. Tolokonnikov’s Lemma for
$${{H^{\infty}_{\mathbb{R}}}}$$
means that if f is left-invertible, then f can be completed to an isomorphism; that is, there exists an F, invertible in
$${{H^{\infty}_{\mathbb{R}}}}$$
, such that F = [ f f
c
] for some f
c
in
$${{H^{\infty}_{\mathbb{R}}}}$$
. In control theory, Tolokonnikov’s Lemma implies that if a function has a right coprime factorization over
$${{H^{\infty}_{\mathbb{R}}}}$$
, then it has a doubly coprime factorization in
$${{H^{\infty}_{\mathbb{R}}}}$$
. We prove the lemma for the real disc algebra
$${A_{\mathbb{R}}}$$
as well. In particular,
$${{H^{\infty}_{\mathbb{R}}}}$$
and
$${A_{\mathbb{R}}}$$
are Hermite rings.