[Paper Review] An Analytic Proof of the Matrix Spectral Factorization Theorem

Lasha Ephremidze, G. Janshia+1 more|ArXiv.org|Oct 21, 2007

Algebraic and Geometric Analysis16 references3 citations

TL;DR

This paper presents a novel analytic proof of the matrix spectral factorization theorem using Hardy space theory and unitary matrix functions, establishing the existence of a canonical spectral factorization $ S(z) = \chi^+(z)(\chi^+(z))^* $ for positive definite matrix-functions on the unit circle under the integrability condition $ \log\det S(z) \in L_1(\mathbb{T}) $. The proof leverages iterative unitary transformations to reduce the problem to outer functions, yielding a constructive and conceptually simple method with direct algorithmic implications.

ABSTRACT

An analytic proof is proposed of Wiener's theorem on factorization of positive definite matrix-functions.

Motivation & Objective

• To provide a new analytic proof of the matrix spectral factorization theorem within the framework of Hardy spaces, avoiding functional analysis or invariant subspace theory.
• To demonstrate that the existence of a spectral factor $ \chi^+(z) \in H_2^r $ with outer determinant follows directly from the theory of $ H_p $ spaces and unitary matrix functions.
• To unify the existence proof with an effective computational method by revealing the decisive role of unitary matrix functions in absorbing technical complexity.
• To establish the canonical factorization $ S(z) = \chi^+(z)(\chi^+(z))^* $ with $ \chi^+(0) $ positive definite, unique up to unitary multiplier.

Proposed method

• Perform a lower-triangular Cholesky-like factorization $ S(z) = A(z)A(z)^* $, where $ A(z) $ has entries in $ L_2(\mathbb{T}) $, ensuring positivity on the diagonal.
• Define outer analytic functions $ f_j^+(z) \in H_2^O $ via the Poisson-Jensen formula: $ f_j^+(z) = \exp\left(\frac{1}{2\pi}\int_0^{2\pi}\frac{e^{it}+z}{e^{it}-z}\log f_{jj}(e^{it})\,dt\right) $, so that $ |f_j^+(z)| = f_{jj}(z) $ a.e. on $ \mathbb{T} $.
• Construct unitary matrix functions $ U_j(z) $ such that $ f_j^+(z)/f_{jj}(z) = u_j(z) $ with $ |u_j(z)| = 1 $, forming a diagonal unitary matrix $ U(z) $.
• Define a sequence of matrix functions $ M_m(z) $ recursively via $ M_m(z) = M_{m-1}(z)V_m(z) $, where $ V_m(z) $ are block unitary matrices constructed using a recursive lemma to preserve analyticity and boundary behavior.
• Use the identity $ M_r(z) = M(z)U_2(z)\cdots U_r(z) $ to show that $ \det M_r(z) = \prod_{j=1}^r f_j^+(z) $, and since each $ f_j^+ $ is outer, their product is outer.
• Conclude that $ M_r(z) \in H_2^r $, $ M_r(z)M_r(z)^* = S(z) $ a.e. on $ \mathbb{T} $, and $ \det M_r(z) $ is outer, proving the canonical factorization.

Experimental results

Research questions

• RQ1 Can the matrix spectral factorization theorem be proven using only the theory of Hardy spaces and unitary matrix functions, without invoking invariant subspaces or functional analysis?
• RQ2 How can the canonical spectral factor $ \chi^+(z) $ be constructed explicitly from the matrix-function $ S(z) $ under the log-determinant integrability condition?
• RQ3 What is the role of unitary matrix functions in simplifying the spectral factorization problem and absorbing its technical complexity?
• RQ4 Is there a direct analytic proof of the existence of the spectral factor that also yields an effective computational algorithm?
• RQ5 Does the product of outer functions remain outer, and how does this property ensure the outerness of the determinant of the spectral factor?

Key findings

• The matrix spectral factorization $ S(z) = \chi^+(z)(\chi^+(z))^* $ exists for any positive definite $ r\times r $ matrix-function $ S(z) \in L_1(\mathbb{T}) $ with $ \log\det S(z) \in L_1(\mathbb{T}) $, and $ \chi^+(z) \in H_2^r $ with outer determinant.
• The canonical spectral factor $ \chi^+(z) $ is uniquely determined by the condition that $ \chi^+(0) $ is positive definite, and it is constructed via a sequence of unitary transformations applied to the lower-triangular factor $ A(z) $.
• The determinant $ \det \chi^+(z) $ is outer because it is the product of outer functions $ f_j^+(z) $, each defined via the Poisson integral formula on $ \log f_{jj}(z) \in L_1(\mathbb{T}) $.
• The proof is fully analytic and constructive, relying only on Hardy space theory and unitary matrix functions, with no use of invariant subspaces or advanced functional analysis.
• The method naturally leads to an effective algorithm for computing $ \chi^+(z) $, as unitary matrices absorb the complexity and reduce the problem to solving for outer functions and simple matrix multiplications.
• The final spectral factor $ \chi^+(z) = M_r(z) $ is analytic in the unit disk and satisfies $ M_r(z) \in L_2^+(\mathbb{T}) $, ensuring its analyticity and $ H_2 $-integrability.