Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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STABILITY Of ISOMETRIES ON HILBERT SPACES

  • Jun, Kil-Woung (Department of Mathematics, Chungnam National University) ;
  • Park, Dal-Won (Department of Mathematics Education, Konju National University)
  • Published : 2002.02.01
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Abstract

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.

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References

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