On Kalton’s theorem for regular compact operators and Grothendieck property for positive projective tensor products

Abstract:

For Banach lattices $E, F$, and $T_n \in \mathcal {K}^r_+(E;F^\ast )$ for $n \in \mathbb {N}$, we prove that $\{T_n\}$ is a weakly null sequence in $\mathcal {K}^r(E;F^\ast )$ if and only if $\lim _n\langle x^{\prime \prime }, T_n^\ast (y^{\prime \prime }) \rangle = 0$ for each $x^{\prime \prime } \in E^{\ast \ast }$ and each $y^{\prime \prime } \in F^{\ast \ast }$. By using this result, we prove that if both $E$ and $F$ have the positive Grothendick property and every positive linear operator from $E$ to $F^\ast$ is compact, then the positive projective tensor product $E\hat {\otimes }_{|\pi |}F$ has the positive Grothendick property. Moreover, if $F^\ast$ has the bounded regular approximation property, then these sufficient conditions for $E\hat {\otimes }_{|\pi |}F$ to have the positive Grothendick property are also necessary.