Operators factorable through Lorentz sequence spaces

Weestablishkeyresultsandproperties of the Banach space of vector-valued weakly Lorentz sequence, denoted $\ell_{p,q}^{w}(\cdot)$. These results extend a classical result due to Grothendieck for $\ell_{p}$-spaces to the Lorentz setting. Additionally, we introduce and study a new concept, namely$\ell_{p,q}$-factorable operators. We establish fundamental properties of this class and provide a characterization of $\ell_{p,q}$-factorable operators. Specifically, we show that an operator $T$ between Banach spaces is $\ell_{p,q}$-factorable if and only if it admits a continuous linear factorization through the Lorentz space $\ell_{p,q}$.