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- W1558040516 abstract "Let $X$, $Y$ be two real Banach spaces and $varepsilon>0$. A standard $varepsilon$-isometry $f:Xrightarrow Y$ is said to be $(alpha,gamma)$-stable (with respect to $T:L(f)equivoverline{{rm span}}f(X)rightarrow X$ for some $alpha, gamma>0$) if $T$ is a linear operator with $|T|leqalpha$ so that $Tf-Id$ is uniformly bounded by $gammavarepsilon$ on $X$. The pair $(X,Y)$ is said to be stable if every standard $varepsilon$-isometry $f:Xrightarrow Y$ is $(alpha,gamma)$-stable for some $alpha,gamma>0$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show that universal right-stability spaces are just Hilbert spaces; every injective space is universally left-stable; a Banach space $X$ isomorphic to a subspace of $ell_infty$ is universally left-stable if and only if it is isomorphic to $ell_infty$; and that a separable space $X$ satisfies the condition that $(X,Y)$ is left-stable for every separable $Y$ if and only if it is isomorphic to $c_0$." @default.
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- W1558040516 date "2013-01-15" @default.
- W1558040516 modified "2023-09-23" @default.
- W1558040516 title "Universal stability of Banach spaces for $varepsilon$-isometries" @default.
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- W1558040516 doi "https://doi.org/10.48550/arxiv.1301.3374" @default.
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