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, 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.