A universal theorem for stability of epsilon-isometries of Banach spaces

作者：Cheng Lixin^*; Cheng Qingjin; Tu Kun; Zhang Jichao

来源：Journal of Functional Analysis, 2015, 269(1): 199-214.

DOI：10.1016/j.jfa.2015.04.015

• epsilon-isometry
• Weak stability and strong stability
• Banach space

Let X, Y be two Banach spaces, and f : X -> Y be a standard epsilon-isometry for some epsilon >= 0. In this paper, we show the following sharp weak stability inequality of f: for every x* is an element of X* there exists phi is an element of Y* with parallel to phi parallel to = parallel to x*parallel to equivalent to r such that
vertical bar < x* x > - vertical bar <= 2 epsilon r for all x is an element of X.
It is not only a sharp quantitative extension of Figiel's theorem, but it also unifies, generalizes and improves a series of known results about stability of epsilon-isometries. For example, if the mapping f satisfies C(f) equivalent to (co) over bar [f(X) boolean OR - f(X)] = Y, then it is equivalent to the following sharp stability theorem: There is a linear surjective operator T : Y -> X of norm one such that parallel to Tf (x) - x parallel to <= 2 epsilon, for all x is an element of X; When the epsilon-isometry f is surjective, it is equivalent to Omladic-Semrl's theorem: There is a surjective linear isometry U: X -> Y so that
parallel to f(x) - Ux parallel to <= 2 epsilon, for all x is an element of X.

• 出版日期2015-7-1
• 单位厦门大学