# The set of collaboratively compact operators is compact in $B(X, Y)$?

I'm wondering if the set of collaboratively compact operators is compact in $B(X, Y)$?

A family of bounded operators $\mathscr{A} = \{A: X\to Y\}$ of linear operators, where $X,Y$ are banach spaces are called collectively compact, if for each bounded set $U \subset X$, the image set $\mathscr{A}(U) = \{ Af: f\in U, A \in \mathscr{A}\}$ is relatively compact in $Y$.

But in the Banach space $B(X,Y)$, does $\mathscr{A}$ necessarily form a compact set?

• 4 questions within these last 8 hours ! Take time to breath ! – Jean Marie Mar 25 '18 at 21:22
• @JeanMarie Can't breathe without math! – nekodesu Mar 25 '18 at 21:46
• Nice answer ! I appreciate your humor... – Jean Marie Mar 25 '18 at 22:25