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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 [[#References|[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|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 $\mathcal{T}$ [[#References|[a2]]].

Existence and analytic properties of such factorizations have been established for various classes of matrix-functions $f$, [[#References|[a2]]]. For a rational $f$, the partial indices and factors $f _ { \pm }$ are effectively computable [[#References|[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 [[#References|[a3]]]. It is also of fundamental importance for the theory of singular integral equations [[#References|[a4]]], Riemann–Hilbert problems and other boundary value problems for holomorphic functions [[#References|[a5]]], as well as for the theories of Wiener–Hopf [[#References|[a6]]] and Toeplitz operators [[#References|[a7]]]; see also [[#References|[a13]]].

An extended concept of Birkhoff factorization has recently (1986) emerged in the geometric theory of loop groups [[#References|[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 [[#References|[a8]]] has stimulated the geometric study of loop groups [[#References|[a9]]] and has permitted one to generalize the classical theory of Riemann–Hilbert problems [[#References|[a10]]]. The geometric approach to Birkhoff factorization, developed in [[#References|[a8]]], has also interesting applications in the theory of completely integrable models [[#References|[a11]]] and $K$-theory [[#References|[a12]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> G.D. Birkhoff, "Singular points of ordinary linear differential equations" ''Trans. Amer. Math. Soc.'' , '''10''' (1909) pp. 436–470 {{MR|1500848}} {{ZBL|46.0695.02}} {{ZBL|45.0484.01}} {{ZBL|44.0373.01}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> K.F. Clancey, I.Z. Gohberg, "Factorization of matrix functions and singular integral operators" , Birkhäuser (1981) {{MR|0657762}} {{ZBL|0474.47023}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> A. Grothendieck, "Sur la classification des fibrés holomorphes sur la sphère de Riemann" ''Amer. J. Math.'' , '''79''' (1957) pp. 121–138 {{MR|0087176}} {{ZBL|0079.17001}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> N.P. Vekua, "Systems of singular integral equations" , Nauka (1970) (In Russian) {{MR|0270086}} {{ZBL|0166.09802}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> F.D. Gakhov, "Boundary value problems" , Nauka (1977) (Edition: Third) {{MR|0486547}} {{ZBL|0449.30030}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> 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 {{MR|113114}} {{ZBL|}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> A. Böttcher, B. Silbermann, "Analysis of Toeplitz operators" , Springer (1990) {{MR|1086453}} {{MR|1071374}} {{ZBL|0732.47029}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> A. Pressley, G. Segal, "Loop groups" , Clarendon Press (1986) {{MR|0900587}} {{MR|0849057}} {{ZBL|0618.22011}} {{ZBL|0603.17012}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> D. Freed, "The geometry of loop groups" ''J. Diff. Geom.'' , '''28''' (1988) pp. 223–276 {{MR|0961515}} {{ZBL|0619.58003}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> G. Khimshiashvili, "On the Riemann–Hilbert problem for a compact Lie group" ''Dokl. Akad. Nauk SSSR'' , '''310''' (1990) pp. 1055–1058 (In Russian) {{MR|1050159}} {{ZBL|0731.30033}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> G. Segal, G. Wilson, "Loop groups and equations of KdV type" ''Publ. Math. IHES'' , '''61''' (1985) pp. 5–65 {{MR|0783348}} {{ZBL|0592.35112}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> S. Zhang, "Factorizations of invertible operators and K-theory of $C ^ { * }$-algebras" ''Bull. Amer. Math. Soc.'' , '''28''' (1993) pp. 75–83</td></tr><tr><td valign="top">[a13]</td> <td valign="top"> M. Hazewinkel, C.F. Martin, "Representations of the symmetric groups, the specialization order, systems, and Grassmann manifolds" ''Enseign. Math.'' , '''29''' (1983) pp. 53–87 {{MR|0702734}} {{ZBL|}} </td></tr></table>