中科院数学与系统科学研究院

数学研究所

 

学术报告会 

 

 

报告人Prof. Philippe Thieullen(University of Bordeaux)

 A uniform bound from below of the angle between the fast and slow spaces for two-sided sequences of bounded operators in a Banach spaces

  2017.07.24(星期一),10 :30-11:30

  点:数学院南楼N913

  要:

We consider a two-sided sequence of bounded operators in a Banach space which are not necessarily injective and which satisfy the following two properties (SVG) and (FI). The "singular value gap" (SVG) property says that  two successive singular values of the cocycle at some index d admit a uniform exponential gap, the "fast invertibility" (FI) property says that the cocycle is uniformly invertible on the fastest d-dimensional direction. We prove the existence of a uniform equivariant splitting of the Banach space into a fast space of dimension d and a slow space of codimension d. We compute an explicit constant of the bound from below of the angle between these two spaces using solely the constants defining the properties (SVG) and (FI). We extend the results obtained by Bochi-Gourmelon in finite dimension for bijective operators and the results obtained by Blumenthal-Morris in inifinite dimension for injective norm-continuous cocycles, in the direction that no dynamical system is involved, no compactness of the underlying system, no smoothness of the cocycle is required. Moreover we give quantitative estimates of the angle between the fast and slow spaces that are also new in finite dimension for bijective operators. A joint work with Anthony Quas and Mohamed Zarrabi.

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