Optimal extensions for $p$-th power factorable operators

Let $X(\mu)$ be a function space related to a measure space $(\Omega,\Sigma,\mu)$ with $\chi_\Omega\in X(\mu)$ and let $T\colon X(\mu)\to E$ be a Banach space valued operator. It is known that if $T$ is $p$-th power factorable then the largest function space to which $T$ can be extended preserving $p$-th power factorability is given by the space $L^p(m_T)$ of $p$-integrable functions with respect to $m_T$, where $m_T\colon\Sigma\to E$ is the vector measure associated to $T$ via $m_T(A)=T(\chi_A)$. In this paper we extend this result by removing the restriction $\chi_\Omega\in X(\mu)$. In this general case, by considering $m_T$ defined on a certain $\delta$-ring, we show that the optimal domain for $T$ is the space $L^p(m_T)\cap L^1(m_T)$. We apply the obtained results to the particular case when $T$ is a map between sequence spaces defined by an infinite matrix.