• Title, Summary, Keyword: ($\varepsilon$, p)-isometry

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STABILITY Of ISOMETRIES ON HILBERT SPACES

  • Jun, Kil-Woung;Park, Dal-Won
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.141-151
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    • 2002
  • Let X and Y be real Banach spaces and $\varepsilon$, p $\geq$ 0. A mapping T between X and Y is called an ($\varepsilon$, p)-isometry if |∥T(x)-T(y)∥-∥x-y∥|$\leq$ $\varepsilon$∥x-y∥$^{p}$ for x, y$\in$X. Let H be a real Hilbert space and T : H longrightarrow H an ($\varepsilon$, p)-isometry with T(0) = 0. If p$\neq$1 is a nonnegative number, then there exists a unique isometry I : H longrightarrow H such that ∥T(x)-I(y)∥$\leq$ C($\varepsilon$)(∥x∥$^{ 1+p)/2}$+∥x∥$^{p}$ ) for all x$\in$H, where C($\varepsilon$) longrightarrow 0 as $\varepsilon$ longrightarrow 0.

GENERALIZED STABILITY OF ISOMETRIES ON REAL BANACH SPACES

  • Lee, Eun-Hwi;Park, Dal-Won
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.2
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    • pp.309-318
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    • 2006
  • Let X and Y be real Banach spaces and ${\varepsilon}\;>\;0$, p > 1. Let f : $X\;{\to}\;Y$ be a bijective mapping with f(0) = 0 satisfying $$|\;{\parallel}f(x)-f(y){\parallel}-{\parallel}{x}-y{\parallel}\;|\;{\leq}{\varepsilon}{\parallel}{x}-y{\parallel}^p$$ for all $x\;{\in}\;X$ and, let $f^{-1}\;:\;Y\;{\to}\;X$ be uniformly continuous. Then there exist a constant ${\delta}\;>\;0$ and N(${\varepsilon},p$) such that lim N(${\varepsilon},p$)=0 and a unique surjective isometry I : X ${\to}$ Y satisfying ${\parallel}f(x)-I(x){\parallel}{\leq}N({\varepsilon,p}){\parallel}x{\parallel}^p$ for all $x\;{\in}\;X\;with\;{\parallel}x{\parallel}{\leq}{\delta}$.