A State Space Approach to Canonical Factorization with by Harm Bart, Israel Gohberg, Marinus A. Kaashoek, André C.M.

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

The current ebook offers with canonical factorization difficulties for di?erent periods of matrix and operator capabilities. Such difficulties seem in numerous parts of ma- ematics and its purposes. The capabilities we give some thought to havein universal that they seem within the kingdom house shape or should be represented in the sort of shape. the most effects are all expressed by way of the matrices or operators showing within the kingdom area illustration. This comprises precious and su?cient stipulations for canonical factorizations to exist and specific formulation for the corresponding f- tors. additionally, within the purposes the entries within the country house illustration play a vital position. Thetheorydevelopedinthebookisbasedonageometricapproachwhichhas its origins in di?erent ?elds. one of many preliminary steps are available in mathematical platforms concept and electric community thought, the place a cascade decomposition of an input-output method or a community is expounded to a factorization of the linked move functionality. Canonical factorization has an extended and engaging heritage which starts off within the thought of convolution equations. fixing Wiener-Hopf essential equations is heavily regarding canonical factorization. the matter of canonical factorization additionally seems to be in different branches of utilized research and in mathematical structures concept, in H -control concept in particular.

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Indeed, we would then have a realization of W on an open subset of C strictly larger than ΩW and such a subset would contain a pole of W , contradicting the fact that W has to be analytic on it. It is not difficult to construct realizations of W having a main matrix A with spectrum strictly larger than C \ ΩW and where certain eigenvalues of A (namely those belonging to ΩW ) do not correspond with poles of W . 1 enjoys a certain minimality property. However, it does this only in a weak sense. This one sees, for instance, by looking at the pole orders.

Let Ω ⊂ C be an open punctured neighborhood of ∞ in the Riemann sphere C∞ , let U and Y be complex Banach spaces, and let W : Ω → L(U, Y ) be analytic and proper. Then W admits a realization on Ω with external operator D = limλ→∞ W (λ). 3. Realization of analytic operator functions 25 Proof. First assume Ω is the full complex plane. Then, by Liouville’s theorem, the function W has the constant value D = limλ→∞ W (λ). Now take for the state space X the zero space {0}, and the desired realization for W on C is obtained trivially.

8) where P and P × are the Riesz projections of A and A× , respectively, corresponding to the spectra in the upper half plane. 7) be given by f (t) = Ce−itA x, t ≥ 0. 7) is given by × φ(t) = Ce−itA Πx, t ≥ 0. Here Π is the projection of Cn onto Ker P × along Im P . Proof. Since x ∈ Ker P , the vector e−itA x is exponentially decaying in norm when t → ∞, and thus the function f belongs to Lm p [0, ∞). 7) has a unique solution φ ∈ Lm p [0, ∞). 3 we know that φ is given by φ(t) = f (t) + iCe−itA t × × ΠeisA BCe−isA x ds 0 −iCe−itA ∞ × t × (I − Π)eisA BCe−isA x ds .

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