Optimal extension of the Cesàro operator in $L^p ([0,1])$

Abstract

The Cesàro operator $C_p : L^p ([0,1]) \to L^p ([0,1]) $, a classical kernel operator, induces the vector measure $m_p : A \mapsto C_p (\big.\chi_A)$ which generates a factorization of $C_p$ through $L^1 (m_p) $ via the integration map $I_{m_p} : f \mapsto \int_0^1 f \; d m_p$, for $f \in L^1 (m_p)$. This provides a technique to investigate various operator theoretic properties of $C_p$. Even though the variation measure $|m_p|$ of $m_p$ is finite it turns out, atypically for a kernel operator, that the restriction of $I_{m_p} $ to $L^1(|m_p|) \subseteq L^1 (m_p)$ is not an extension of $C_p$, that is, $C_p $ fails to factorize through the more traditional space $L^1(|m_p|)$.