Weakly compact operator

A linear operator $T:X\to Y$ is weakly compact when $T(B_X)$ is relatively weakly compact in $Y$.

For Banach spaces $X,Y$ and bounded $T:X\to Y$, the following are equivalent: $T$ is weakly compact; $T^{\ast \ast }(X^{\ast \ast })\subseteq J_Y(Y)$; and $T^{\ast }$ is weakly compact.

Weakly compact operators form a norm-closed vector subspace of $\mathcal{B}(X,Y)$ and have the ideal property: bounded compositions on either side of a weakly compact operator remain weakly compact.

Pitt's theorem implies in particular that every bounded operator $c_0\to {\ell }^1$ is compact. Thus two nonreflexive Banach spaces can have only weakly compact operators between them.