By Bart H., Gohberg I., Kaashoek M. A.

The current booklet bargains with factorization difficulties for matrix and operator features. the issues originate from, or are inspired by means of, the speculation of non-selfadjoint operators, the idea of matrix polynomials, mathematical platforms and keep an eye on thought, the speculation of Riccati equations, inversion of convolution operators, concept of activity scheduling in operations study. The booklet systematically employs a geometrical precept of factorization which has its origins within the kingdom house concept of linear input-output structures and within the thought of attribute operator capabilities. This precept permits one to accommodate diversified factorizations from one viewpoint. lined are canonical factorization, minimum and non-minimal factorizations, pseudo-canonical factorization, and numerous varieties of measure one factorization.Considerable consciousness is given to the problem of balance of factorization which when it comes to the nation area process contains balance of invariant subspaces.invariant subspaces.

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**Example text**

30], page 6). , [30], page 143). 3. By definition, a system Θ = (A, R, −JK −∗ R∗ A, K; H, G) is a Kreˇın J-system if J is a signature operator, I − AA∗ = RJR∗ , J − R∗ R = K ∗ JK, and the operators A and K are invertible. Since A is invertible, I − RJR∗ is invertible. 5) and J = K ∗ JK +R∗ R to obtain that I + RK −1 JK −∗ R∗ = (I − RJR∗ )−1 . 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.

26) in the previous section. Notes The material in this chapter is standard, and can be found in much more detail in various monographs, books, and papers. For the theory of time invariant systems we refer to the books [84], [114], and the more recent [36]. Further information on the theory of characteristic operator functions can be found in the books [30] and [108]; see also the survey paper [7] and the references therein. For the general theory of matrix polynomials, including the monic case, we refer to the book [69].

The spaces U, X and Y are called the input space, state space and output space of the system, respectively. The operator A is referred to as the state space operator or main operator of the system Θ. When A is given or can be viewed as a matrix, the terms main matrix and state matrix will be used too. We call D the external operator of Θ. In the situation where U = Y , we denote Θ by (A, B, C, D; X, Y ). When, in addition, D is the identity operator I = IY on Y , we simply write (A, B, C; X, Y ) instead of (A, B, C, I; X, Y ), and in that case we refer to Θ as a unital system.