Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximate isometries of Hilbert spaces

Peter Semrl

TL;DR

This work addresses the stability of approximate isometries between real Hilbert spaces without assuming surjectivity. It develops a refined stability result: for any $\varepsilon$-isometry $f$ with $f(0)=0$, there exists a linear isometry $U:H\to K$ such that, with $K=\mathrm{Im}\,U \oplus (\mathrm{Im}\,U)^{\perp}$ and $P$ the orthogonal projection onto $\mathrm{Im}\,U$, one has $\| P f(x) - Ux \| \le 2\varepsilon$ and $\| (I-P) f(x) \| \le \sqrt{6\varepsilon \|x\| + \varepsilon^2}$ for all $x$. The authors analyze optimality, showing the first bound is sharp and providing near-sharp insights into the second bound (with $B=\varepsilon^2$ and $A$ between $2\varepsilon$ and $6\varepsilon$). The proof combines the Hyers–Ulam limiting construction with a decomposition of the target space and a geometric two-ball estimate to control the non-imaged component. Overall, the results sharpen our understanding of stability for non-surjective maps in Hilbert spaces and tighten constants in the corresponding stability bounds.

Abstract

We improve the Hyers-Ulam stability result for isometries of real Hilbert spaces by removing the surjectivity assumption.

Approximate isometries of Hilbert spaces

TL;DR

This work addresses the stability of approximate isometries between real Hilbert spaces without assuming surjectivity. It develops a refined stability result: for any -isometry with , there exists a linear isometry such that, with and the orthogonal projection onto , one has and for all . The authors analyze optimality, showing the first bound is sharp and providing near-sharp insights into the second bound (with and between and ). The proof combines the Hyers–Ulam limiting construction with a decomposition of the target space and a geometric two-ball estimate to control the non-imaged component. Overall, the results sharpen our understanding of stability for non-surjective maps in Hilbert spaces and tighten constants in the corresponding stability bounds.

Abstract

We improve the Hyers-Ulam stability result for isometries of real Hilbert spaces by removing the surjectivity assumption.

Paper Structure

This paper contains 3 sections, 3 theorems, 37 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. The optimality of the main theorem
  3. Proof of the stability result for nonsurjective approximate isometries of Hilbert spaces

Key Result

Theorem 1.1

HyU Let $H$ be a Hilbert space and $f : H \to H$ a surjective $\varepsilon$-isometry satisfying $f(0) = 0$. Then the map $U : H \to H$ defined by is a linear bijective isometry (an orthogonal operator) and we have for every $x \in H$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['tooe']}