Birkhoff factorization

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Birkhoff decomposition

Traditionally, Birkhoff factorization refers to representations of an invertible matrix-function $f$ on the unit circle $\mathcal{T}$ of the form $f = f_+ . \delta . f_-$, where $f _ { \pm }$ are the boundary values of invertible matrix-functions holomorphic inside (respectively, outside) $\mathcal{T}$, and $\delta = \operatorname { diag } ( z ^ { k _ { i } } )$ is a diagonal matrix-function with some integers $k_i$ as exponents.

More precisely, let $S = \overline { \mathbf{C} } = D _ { + } \cup \mathcal{T} \cup D _ { - }$ be the standard decomposition of the Riemann sphere $S$, where $D _ { + }$ is the unit disc and $D_{-}$ is the complementary domain containing the point $\{ \infty \}$. For any domains $U \subset \mathbf{C}$, $V \subset {\bf C} ^ { m }$, denote by $A ( \overline { U } , V )$ the set of all continuous mappings $g : \overline { U } \rightarrow V$ which are holomorphic inside $U$. A classical theorem of G. Birkhoff [a1] yields that, for any Hölder-continuous matrix-function $f : \mathcal{T} \rightarrow \operatorname {GL} ( n , \mathbf{C} )$, there exist integers $k _ { 1 } , \dots , k _ { n }$ and functions $f _ { \pm } \in A ( \overline { D } _ { \pm } , \operatorname{GL} ( n , \mathbf{C} ) )$, with $f_- ( \{ \infty \} )$ equal to the identity matrix, such that $f = f_+ . \delta . f_-$ on $\mathcal{T}$, where $\delta ( z ) = \operatorname { diag } ( z ^ { k _ { 1 } } , \ldots , z ^ { k _ { n } } )$. The integers $k_i$, called the partial indices of $f$, are uniquely determined up to the order and define an interesting decomposition of the space of matrix-functions (cf. Birkhoff stratification). Their sum is an important topological invariant, equal to $( 2 \pi ) ^ { - 1 }$ times the increment of the argument of $\operatorname{det} f$ along $\mathcal{T}$ [a2].

Existence and analytic properties of such factorizations have been established for various classes of matrix-functions $f$, [a2]. For a rational $f$, the partial indices and factors $f _ { \pm }$ are effectively computable [a2]. Similar results are available for factorizations of the form $f = f _ { - } \cdot \delta \cdot f _ { + }$.


An extended concept of Birkhoff factorization has recently (1986) emerged in the geometric theory of loop groups [a8]. It turned out that such factorizations are available for various classes of loops on any compact Lie group, with diagonal matrices $\delta$ replaced by homomorphisms of $\mathcal{T}$ into a maximal torus of $G$. The classical Birkhoff factorization is obtained by taking $G = U ( n )$, the unitary group. The approach of [a8] has stimulated the geometric study of loop groups [a9] and has permitted one to generalize the classical theory of Riemann–Hilbert problems [a10]. The geometric approach to Birkhoff factorization, developed in [a8], has also interesting applications in the theory of completely integrable models [a11] and $K$-theory [a12].


[a12] S. Zhang, "Factorizations of invertible operators and K-theory of $C ^ { * }$-algebras" Bull. Amer. Math. Soc. , 28 (1993) pp. 75–83
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