Absolutely summing operators on the Banach space of vector-valued essentially bounded measurable functions

Abstract

Let $(\Omega ,\Sigma ,\mu )$ be a finite complete measure space and $X$ and $Y$ Banach spaces. Let ${\gamma }_{\infty }$ be a natural mixed topology on the Lebesgue–Bochner space $L^{\infty }(\mu ,X)$. We study absolutely summing operators $T:L^{\infty }(\mu ,X)\to Y$ between the locally convex space $(L^{\infty }(\mu ,X),{\gamma }_{\infty })$ and the Banach space $Y$. It is shown that every $({\gamma }_{\infty },\Vert \cdot {\Vert }_Y)$-continuous nuclear operator $T:L^{\infty }(\mu ,X)\to Y$ is an absolutely summing operator between the locally convex space $(L^{\infty }(\mu ,X),{\gamma }_{\infty })$ and the Banach space $Y$.