Constructive Methods of Wiener-Hopf Factorization

The main part of this paper concerns Toeplitz operators of which the symbol W is an m x m matrix function defined on a disconnected curve r. The curve r is assumed to be the union of s + 1 nonintersecting simple smooth closed contours rOo r •. . . • rs which form the positively l oriented boundary of a finitely connected bounded domain in t. Our main requirement on the symbol W is that on each contour rj the function W is the restriction of a rational matrix function Wj which does not have poles and zeros on rj and at infinity. Using the realization theorem from system theory (see. e. g . • [1]. Chapter 2) the rational matrix function Wj (which differs from contour to contour) may be written in the form 1 (0. 1) W . (A) = I + C. (A - A. f B. A E r· J J J J J where Aj is a square matrix of size nj x n• say. B and C are j j j matrices of sizes n. x m and m x n . • respectively. and the matrices A. J x J J and Aj = Aj - BjC have no eigenvalues on r . (In (0. 1) the functions j j Wj are normalized to I at infinity.

I: Canonical and Minimal Factorization.- Editorial introduction.- Left Versus Right Canonical Factorization.- 1. Introduction.- 2. Left and right canonical Wiener-Hopf factorization.- 3. Application to singular integral operators.- 4. Spectral and antispectral factorization on the unit circle.- 5. Symmetrized left and right canonical spectral factorization on the imaginary axis.- References.- Wiener-Hopf Equations With Symbols Analytic In A Strip.- 0. Introduction.- I. Realization.- 1. Preliminaries.- 2. Realization triples.- 3. The realization theorem.- 4. Construction of realization triples.- 5. Basic properties of realization triples.- II. Applications.- 1. Inverse Fourier transforms.- 2. Coupling.- 3. Inversion and Fredholm properties.- 4. Canonical Wiener-Hopf factorization.- 5. The Riemann-Hilbert boundary value problem.- References.- On Toeplitz and Wiener-Hopf Operators with Contour-Wise Rational Matrix and Operator Symbols.- 0. Introduction.- 1. Indicator.- 2. Toeplitz operators on compounded contours.- 3. Proof of the main theorems.- 4. The barrier problem.- 5. Canonical factorization.- 6. Unbounded domains.- 7. The pair equation.- 8. Wiener-Hopf equation with two kernels.- 9. The discrete case.- References.- Canonical Pseudo-Spectral Factorization and Wiener-Hopf Integral Equations.- 0. Introduction.- 1. Canonical pseudo-spectral factorizations.- 2. Pseudo-?-spectral subspaces.- 3. Description of all canonical pseudo-?-spectral factorizations.- 4. Non-negative rational matrix functions.- 5. Wiener-Hopf integral equations of non-normal type.- 6. Pairs of function spaces of unique solvability.- References.- Minimal Factorization of Integral operators and Cascade Decompositions of Systems.- 0. Introduction.- I. Main results.- 1. Minimal representation and degree.- 2. Minimal factorization (1).- 3. Minimal factorization of Volterra integral operators (1).- 4. Stationary causal operators and transfer functions.- 5. SB-minimal factorization (1).- 6. SB-minimal factorization in the class (USB)..- 7. Analytic semi-separable kernels.- 8. LU- and UL-factorizations (1).- II. Cascade decomposition of systems.- 1. Preliminaries about systems with boundary conditions.- 2. Cascade decompositions.- 3. Decomposing projections.- 4. Main decomposition theorems.- 5. Proof of Theorem II.4.1.- 6. Proof of Theorem II.4.2.- 7. Proof of Theorem II.4.3.- 8. Decomposing projections for inverse systems..- III. Proofs of the main theorems.- 1. A factorization lemma.- 2. Minimal factorization (2).- 3. SB-minimal factorization (2).- 4. Proof of Theorem I.6.1.- 5. Minimal factorization of Volterra integral operators (2).- 6. Proof of Theorem I.4.1.- 7. A remark about minimal factorization and inversion.- 8. LU- and UL-f actorizations (2).- 9. Causal/anticausal decompositions.- References.- II: Non-Canonical Wiener-Hopf Factorization.- Editorial introduction.- Explicit Wiener-Hopf Factorization and Realization.- 0. Introduction.- 1. Preliminaries.- 1. Peliminaries about transfer functions.- 2. Preliminaries about Wiener-Hopf factorization.- 3. Reduction of factorization to nodes with centralized singularities.- II. Incoming characteristics.- 1. Incoming bases.- 2. Feedback operators related to incoming bases.- 3. Factorization with non-negative indices.- III. Outgoing characteristics.- 1. Outgoing bases.- 2. Output injection operators related to outgoing bases.- 3. Factorization with non-positive indices.- IV. Main results.- 1. Intertwining relations for incoming and outgoing data.- 2. Dilation to a node with centralized singularities.- 3. Main theorem and corollaries.- References,.- Invariants for Wiener-Hopf Equivalence of Analytic Operator Functions.- 1. Introduction and main result.- 2. Simple nodes with centralized singularities.- 3. Multiplication by plus and minus terms.- 4. Dilation.- 5. Spectral characteristics of transfer functions: outgoing spaces.- 6. Spectral characteristics of transfer functions: incoming spaces.- 7. Spectral characteristics and Wiener-Hopf equivalence.- References.- Multiplication by Diagonals and Reduction to Canonical Factorization.- 1. Introduction.- 2. Spectral pairs associated with products of nodes.- 3. Multiplication by diagonals.- References.- Symmetric Wiener-Hopf Factorization of Self-Adjoint Rational Matrix Functions and Realization.- 0. Introduction and summary.- 1. Introduction.- 2. Summary.- I. Wiener-Hopf factorization.- 1. Realizations with centralized singularities..- 2. Incoming data and related feedback operators.- 3. Outgoing data and related output injection operators.- 4. Dilation to realizations with centralized singularities.- 5. The final formulas.- II. Symmetric Wiener-Hopf factorization.- 1. Duality between incoming and outgoing operators.- 2. The basis in (C and duality between the feedback operators and the output injection operators.- 3. Proof of the main theorems.- References.