Weakly compact operator

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

A bounded linear operator $T:X\to Y$ between Banach spaces is a weakly compact operator exactly when its second adjoint takes values in the canonical copy of $Y$ inside $Y^{\ast \ast }$. The Goldstine theorem proves necessity; the Banach-Alaoglu theorem, continuity of the second adjoint, and the Mazur theorem prove sufficiency.

A bounded linear operator between Banach spaces is weakly compact exactly when its Banach-space adjoint is weakly compact. Equivalently, that adjoint is continuous from its domain's weak-star topology to its range's weak topology. For the reverse direction, the bidual condition applied to $T^{\ast }$ forces $T^{\ast \ast \ast }$ to annihilate every functional on $Y^{\ast \ast }$ vanishing on $J_{Y}Y$; the Hahn-Banach theorem then places $T^{\ast \ast }(X^{\ast \ast })$ inside $J_{Y}Y$.

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.