Almost homogeneous functions

Question

Let $X$ and $Y$ be Banach spaces and $T: X \to Y$. Working with large scale geometry of Banach spaces, I reached the following property:

Suppose that for every scalar $\alpha\in\mathbb K$ and every $u\in X$ there exists $v\in X$ such that $$ \max\{\|\alpha u-v\|, \|\alpha Tu-Tv\|\}\leq N. $$

Intuitively, this means that $T$ is "almost homogeneous" by a finite error $N$. Of course every homogeneous function satisfies this property (just take $v=\alpha u$).

This property is the result of a study on least hypothesis for a technique I've been working with. However, I couldn't find any other examples of $T$ aside from the class of homogeneous functions. Also, I'm not sure if there is already such a definition in the theory. Any hints on this would be highly appreciated.

Answer 1

1) "Other examples of $T$ aside from the class of homogeneous functions".

Consider a map $T:X\to Y$ of the form $$T(u):=p(u)H(u),$$ where $H:X\to Y$ is a Lipschitz homogeneous map (here always meaning $H(\alpha u)=\alpha H(u)$ for all $u\in X$ and $\alpha\in\mathbb{K}$), and $p:X\to\mathbb{R}$ is a bounded function that has the property that the whole image of $p$ is already reached on any ball of some fixed radius $R$. In formulas, $p(X)=p(B(x,R))$ for any $x\in X$. For instance, any bounded $R$-periodic function of the norm does.

So for any $u\in X$ and $\alpha\in\mathbb{K}$ there is $v\in X$ such that $\| \alpha u -v\|\le R$ and $p(v)=p(u)$. Then $\alpha T(u)-T(v)=\alpha p(u) H(u)-p(v)H(v)=p(u)(H(\alpha u) -H(v)) $ so that $\|\alpha T(u)-T(v)\|\le\|p\|_\infty\operatorname{Lip}(H)R$. Thus $T$ satisfies your hypothesis with $N:=\max\big(1, \|p\|_\infty\operatorname{Lip}(H)\big)R.$

2) "$T$ is almost homogeneous".

On the other hand, if a map $T$ satisfies your hypothesis, and also has the following mild property: for any $u\in X$
$$\sup_{\|\alpha u-w\|\le N} \|T(w)-T(\alpha u)\|=o(\alpha)\qquad (\text{for}\; |\alpha|\to+\infty),$$ (for instance, it is Lipschitz), then it is a homogeneous map. For, if $v$ is as in your notation, $$\alpha T(u)= T(\alpha u)+ \big( T(v)-T(\alpha u)\big)+ \big(\alpha T(u)-T(v)\big)= T(\alpha u)+o(\alpha)$$ so that $T(0)=0$, and for all $u\in X$ $$T(u)=\lim_{|\alpha|\to +\infty}{1\over\alpha}T(\alpha u)$$ which implies it is a homogeneous map.