Definition of a $\psi$-Banach space [closed]

Question

Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. 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 relatively 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.

Answer 1

The definition makes no sense due to the mixing up of "relatively" (weakly) compact and (not relatively) compact.

I guess that what you mean is:

$\psi$ is strongly-weakly proper on closed balls (that is, the intersection of preimages of weakly compact sets with closed balls are compact). For strongly-weakly continuous maps and strong/weak measures of noncompactness $s,w$ on $X$, the latter can be rephrased as

$$w(\psi(C))=0\implies s(C)=0\quad\text{for every bounded set $C$.}$$

A sufficient condition for this is therefore the existence of a constant $c>0$ with

$$w(\psi(C))\ge c\,s(C)\quad\text{for every bounded set $C$.}$$

For the strong topology (and strong measures of noncompactness), both properties are well studied.

BTW, both are properties of the map $\psi$ and not of the Banach space $X$, so I find the notion $\psi$-space very strange.

To be honest, I doubt that there are any useful examples of maps satisfying any of these two properties for the strong-weak topology. It is not even clear whether such a map $\psi$ does exist at all (even if one considers nonlinear $\psi$).

For instance, in a reflexive infinite-dimensional Banach space the image of every bounded open set has to be unbounded which is a weird condition if one requires strong-weak continuity. Maybe in some exotic non-reflexive spaces, it is easier to find an example $\psi$, but to find any example at all is certainly non-trivial (if at all possible).