A Remark on the Sharp Error Region for Approximate Isometries of Hilbert Spaces
Year: 2025
Type: article
Abstract: Motivated by a recent theorem of Semrl, we consider approximate isometries $f:H\to K$ between real Hilbert spaces that fix the origin and have separate nonnegative contraction and expansion errors $\alpha$ and $\beta$. Let $U$ be the canonical linear isometry obtained by rescaling $f$ at infinity, and let $P$ be the orthogonal projection onto $U(H)$. We determine a sharp joint error region for the tangential and transverse errors at every positive source distance. A two-parameter family ofcontinuous maps from $\mathbb{R}$ to $\mathbb{R}^3$ attains every point of its upper boundary. The joint region yields exact radial tangential, transverse, and total-error envelopes. In the symmetric case $\alpha=\beta=\varepsilon$, these results answer Semrl's question about the remaining optimal coefficient in his affine transverse estimate. More precisely, for $A,B\geq0$,$\|f(x)-Pf(x)\|^2\leq A\|x\|+B$ holds universally if and only if $A\geq6\varepsilon$ and $B\geq\varepsilon^2$.
Author: Jérôme Nicolas
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212372377