Birkhoff factorization

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 $f = f _ { - } \cdot \delta \cdot f _ { + }$.

The Birkhoff theorem is closely related to a number of fundamental topics in algebraic geometry, complex analysis, the theory of differential equations, and operator theory. In particular, it is equivalent to Grothendieck's theorem on decomposition of holomorphic vector bundles over the Riemann sphere [a3]. It is also of fundamental importance for the theory of singular integral equations [a4], Riemann–Hilbert problems and other boundary value problems for holomorphic functions [a5], as well as for the theories of Wiener–Hopf [a6] and Toeplitz operators [a7]; see also [a13].

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].

References

[a1]	G.D. Birkhoff, "Singular points of ordinary linear differential equations" Trans. Amer. Math. Soc. , 10 (1909) pp. 436–470 MR1500848 Zbl 46.0695.02 Zbl 45.0484.01 Zbl 44.0373.01
[a2]	K.F. Clancey, I.Z. Gohberg, "Factorization of matrix functions and singular integral operators" , Birkhäuser (1981) MR0657762 Zbl 0474.47023
[a3]	A. Grothendieck, "Sur la classification des fibrés holomorphes sur la sphère de Riemann" Amer. J. Math. , 79 (1957) pp. 121–138 MR0087176 Zbl 0079.17001
[a4]	N.P. Vekua, "Systems of singular integral equations" , Nauka (1970) (In Russian) MR0270086 Zbl 0166.09802
[a5]	F.D. Gakhov, "Boundary value problems" , Nauka (1977) (Edition: Third) MR0486547 Zbl 0449.30030
[a6]	I.Z. Gohberg, M.G. Krein, "Systems of integral equations on a half-line with kernels depending on the difference of the arguments" Transl. Amer. Math. Soc. , 14 (1960) pp. 217–284 MR113114
[a7]	A. Böttcher, B. Silbermann, "Analysis of Toeplitz operators" , Springer (1990) MR1086453 MR1071374 Zbl 0732.47029
[a8]	A. Pressley, G. Segal, "Loop groups" , Clarendon Press (1986) MR0900587 MR0849057 Zbl 0618.22011 Zbl 0603.17012
[a9]	D. Freed, "The geometry of loop groups" J. Diff. Geom. , 28 (1988) pp. 223–276 MR0961515 Zbl 0619.58003
[a10]	G. Khimshiashvili, "On the Riemann–Hilbert problem for a compact Lie group" Dokl. Akad. Nauk SSSR , 310 (1990) pp. 1055–1058 (In Russian) MR1050159 Zbl 0731.30033
[a11]	G. Segal, G. Wilson, "Loop groups and equations of KdV type" Publ. Math. IHES , 61 (1985) pp. 5–65 MR0783348 Zbl 0592.35112
[a12]	S. Zhang, "Factorizations of invertible operators and K-theory of $C ^ { * }$-algebras" Bull. Amer. Math. Soc. , 28 (1993) pp. 75–83
[a13]	M. Hazewinkel, C.F. Martin, "Representations of the symmetric groups, the specialization order, systems, and Grassmann manifolds" Enseign. Math. , 29 (1983) pp. 53–87 MR0702734