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

Mikkola, Kalle and Sasane, Amol J. (2007) Tolokonnikov’s Lemma for Real H ∞ and the Real Disc Algebra. Complex Analysis and Operator Theory, 1 (3). pp. 439-446. ISSN 1661-8254

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Identification Number: 10.1007/s11785-007-0016-z


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.

Item Type: Article
Official URL:
Additional Information: © 2007 Birkhaeuser Verlag AG
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 27 Jul 2011 15:05
Last Modified: 16 May 2024 00:38

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