Weak compactness of almost limited operators

The paper is devoted to the relationship between almost limited operators and weakly compacts operators. We show that if $F$ is a $\sigma$-Dedekind complete Banach lattice then, every almost limited operator $T:E\rightarrow F$ is weakly compact if and only if $E$ is reflexive or the norm of $F$ is order continuous. Also, we show that if $E$ is a $\sigma$-Dedekind complete Banach lattice then the square of every positive almost limited operator $T:E\rightarrow E$ is weakly compact if and only if the norm of $E$ is order continuous.