By Harm Bart, Israel Gohberg, Marinus Kaashoek, André C.M. Ran
This booklet delineates a few of the varieties of factorization difficulties for matrix and operator capabilities. the issues originate from, or are inspired via, the speculation of non-selfadjoint operators, the speculation of matrix polynomials, mathematical platforms and keep an eye on concept, the idea of Riccati equations, inversion of convolution operators, and the speculation of activity scheduling in operations learn. The ebook offers a geometrical precept of factorization which has its origins within the nation area thought of linear input-output structures and within the thought of attribute operator functions.
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Extra info for Factorization of Matrix and Operator Functions - The State Space Method
Hence (AA∗ )−1 = (I − RJR∗ )−1 = I + RK −1 JK −∗ R∗ . It follows that Θ× = A−∗ , RK −1 , JR∗ A−∗ , K −1 ; H, G . From this we see that Θ× is a Kreˇın (−J)-system. Observe that in this case the associate main operator A× depends again exclusively on A and coincides with A−∗ . Let Π be an orthogonal projection of H. With respect to the decomposition H = Ker Π ⊕ Im Π, we write A= A11 A12 A21 A22 , R= R1 R2 . , Ker Π is an invariant subspace for A) and A11 is invertible. Then A22 is invertible too and Im Π is an invariant subspace for A× = A−∗ .
27) in the form ∞ aj−k ξk = ηj , k=−∞ j = 0, ±1, ±2, . . 28) for j < 0. 29) where the functions a, η+ , η− , ξ+ and ξ+ are given by a(λ) = ξ+ (λ) = ∞ j=−∞ ∞ j=0 λj aj , λj ξj , η+ (λ) = ∞ j=0 η− (λ) = −1 j=−∞ λj ηj , λj ηj . 29) holds, while moreover, ξ+ and η− must be as above ∞ m with (ξj )∞ j=0 and (η−j−1 )j=0 from ℓp . 22 Chapter 1. 29) is again by factorizing the symbol a of the given block Toeplitz equation. Assume that a admits a right canonical factorization with respect to the unit circle.
On the other hand, if two Brodskii J-systems are similar, say with system similarity S, then one can prove that there exists a unitary operator U that provides the similarity too. In fact for √ U one may take the unitary operator appearing in the polar decomposition S = U S ∗ S of S. Let Θ = (A, B, C; H, G) be a Brodskii J-system. As the external operator is equal to the identity operator on G, the associate system Θ× = (A× , B, −C; H, G) is well defined. Note that A× = A − BC = A∗ . So in this case the associate main operator of Θ depends exclusively on A and coincides with the adjoint of A.