Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space generated by $I$.

In [Measures of Weak Compactness and Fixed Point Theory](https://arxiv.org/abs/math/0310422), Barroso and O'Regan defined a $\phi$-space. Inspired by their definition we introduce the concept:

>Let $\psi\in \mathcal{L}(X)$ be such that $\psi\notin \operatorname{span}\{I\}$. We will say that $X$ is a $\psi$-space if the following condition is satisfied:
>
>**if $C \in \mathcal{F}$ and $\psi(C)$ is relatively weakly compact, then $C$ is compact.**

---------------------------------------------------
I want to study this class of functions and describe it, I'm looking for more characterization/examples of the $\psi$-space.

I will appreciate your help.

  [1]: https://arxiv.org/pdf/math/0310422.pdf