Optimal extensions for $p$-th power factorable operators

Let $X(μ)$ be a function space related to a measure space $(Ω,Σ,μ)$ with $χ_Ω\in X(μ)$ and let $T\colon X(μ)\to E$ 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 $m_T\colonΣ\to E$ is the vector measure associated to $T$ via $m_T(A)=T(χ_A)$. In this paper we extend this result by removing the restriction $χ_Ω\in X(μ)$. In this general case, by considering $m_T$ defined on a certain $δ$-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.

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