A simplified proof of the Gohberg-Sigal-Rouché theorem
نویسنده
چکیده
The GSR theorem generalizes the classical Rouché theorem of complex analysis, concerning the stability of the total multiplicity of “roots minus poles” for finitely meromorphic functions taking values in the space of Fredholm operators on a Hilbert space. This theorem has been a very useful tool in the stability analysis of nonlinear waves. Proofs based on logarithmic integrals are usually considerably complicated by the discontinuous nature of the trace operator in infinite dimensions. We describe how these proofs can be significantly simplified, by establishing continuity and analyticiy of trace based only on pointwise bounds for rank. This is a preliminary report on joint work with Zachary Stone and Jürgen Leiterer.
منابع مشابه
A new proof for the Banach-Zarecki theorem: A light on integrability and continuity
To demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the Banach-Zareckitheorem is presented on the basis of the Radon-Nikodym theoremwhich emphasizes on measure-type properties of the Lebesgueintegral. The Banach-Zarecki theorem says that a real-valuedfunction $F$ is absolutely continuous on a finite closed intervalif and only if it is continuo...
متن کاملThe Basic Theorem and its Consequences
Let T be a compact Hausdorff topological space and let M denote an n–dimensional subspace of the space C(T ), the space of real–valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii [7] attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel’man, which uses Helly’s Theorem, now the paper obtains a...
متن کاملAnother proof of Banaschewski's surjection theorem
We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform subl...
متن کاملStability Proof of Gain-Scheduling Controller for Skid-to-Turn Missile Using Kharitonov Theorem
Gain scheduling is one of the most popular nonlinear control design approaches which has been widely and successfully applied in fields ranging from aerospace to process control. Despite the wide application of gain scheduling controllers, there is a notable lack of analysis on the stability of these controllers. The most common application of these kinds of controllers is in the field of fligh...
متن کاملA SHORT PROOF FOR THE EXISTENCE OF HAAR MEASURE ON COMMUTATIVE HYPERGROUPS
In this short note, we have given a short proof for the existence of the Haar measure on commutative locally compact hypergroups based on functional analysis methods by using Markov-Kakutani fixed point theorem.
متن کامل