Optimal Extensions for pth Power Factorable Operators

Abstract

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 pth power factorable then the largest function space to which T can be extended preserving pth 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.