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 this [paper][1], the authors defined the $\phi-$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 a relatively weakly compact, then $C$ is compact.**

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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