By Harm Bart, Israel Gohberg, Marinus A. Kaashoek, André C.M. Ran

The current publication bargains with canonical factorization difficulties for di?erent periods of matrix and operator capabilities. Such difficulties look in a variety of parts of ma- ematics and its functions. The services we reflect on havein universal that they seem within the nation house shape or will be represented in this type of shape. the most effects are all expressed by way of the matrices or operators showing within the nation house illustration. This contains important and su?cient stipulations for canonical factorizations to exist and particular formulation for the corresponding f- tors. additionally, within the functions the entries within the country area illustration play an important position. Thetheorydevelopedinthebookisbasedonageometricapproachwhichhas its origins in di?erent ?elds. one of many preliminary steps are available in mathematical structures concept and electric community idea, the place a cascade decomposition of an input-output procedure or a community is said to a factorization of the linked move functionality. Canonical factorization has an extended and fascinating background which begins within the thought of convolution equations. fixing Wiener-Hopf fundamental equations is heavily with regards to canonical factorization. the matter of canonical factorization additionally appears to be like in different branches of utilized research and in mathematical platforms idea, in H -control conception in particular.

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2) holds. Next assume that Y1 = Im Q, where Q is a Riesz projection for S. Put Π = I − Q, and let S0 be the restriction of S to Im Π. Then σ(S11 )∩ σ(S0 ) = ∅. By the ﬁrst part of the proof, the operators S0 and S22 are similar. So σ(S0 ) = σ(S22 ), and hence we have shown that σ(S11 ) ∩ σ(S22 ) = ∅. Let Γ be a Cauchy contour. As before (see the one but last paragraph in Chapter 0) we denote by F+ and F− the interior and exterior domain of Γ, respectively. Note that ∞ ∈ F− . Let W be a rational m × m matrix function, with W (∞) = I, analytic on an open neighborhood of Γ, whose values on Γ are invertible matrices.

Here are the details. 5. Products 29 A2 : X2 → X2 , B2 : U2 → X2 , C2 : X2 → Y2 , D2 : U2 → Y2 . Now assume Y1 = U2 . Put U = U1 , Y = Y2 , and introduce A B = = A1 B1 C2 0 A2 B1 D2 ˙ 2, : Y → X1 +X B2 C1 ˙ 2 → X1 +X ˙ 2, : X1 +X ˙ 2 → Y, : X1 +X D1 C2 C = D = D1 D2 : U → Y. Then the following result holds true. 5. 6), respectively. Then, with A, B, C and D as above, W1 (λ)W2 (λ) = D + C(λ − A)−1 B, λ ∈ ρ(A1 ) ∩ ρ(A2 ) ⊂ ρ(A). Proof. Take λ ∈ ρ(A1 ) ∩ ρ(A2 ). Then λ ∈ ρ(A). Indeed, λ − A is invertible with inverse given by ⎤ ⎡ −1 (λ − A1 ) H(λ) ⎦ : X1 +X ˙ 2 → X1 +X ˙ 2, (λ − A)−1 = ⎣ −1 0 (λ − A2 ) −1 −1 where H(λ) = − (λ − A1 ) B1 C2 (λ − A2 ) .

Throughout this section, p will be ﬁxed and 1 ≤ p < ∞. 7) that also belongs to the space Lm p [0, ∞). 5). 5). Therefore, in the sequel we assume that the symbol is a rational m × m matrix function. As K(λ) is the Fourier (−∞, ∞)–function, the symbol is continuous on the real transform of an Lm×m 1 line. In particular, Im − K(λ) has no poles on the real line. Furthermore, by the Riemann-Lebesgue lemma, lim λ∈R, |λ|→∞ K(λ) = 0, which implies that the symbol Im − K(λ) has the value In at ∞. 2. Wiener-Hopf integral operators 43 where p(t) and q(t) are matrix polynomials in t with coeﬃcients in Cm×m , and α and β are complex numbers with α > 0 and β < 0.