On operator ideals related to (p,σ)-absolutely continuous operators

We study tensor norms and operator ideals related to the ideal ${\displaystyle P_{p,\sigma }}$, 1 < p < ∞, 0 < σ < 1, of (p,σ)-absolutely continuous operators of Matter. If α is the tensor norm associated with ${\displaystyle P_{p,\sigma }}$ (in the sense of Defant and Floret), we characterize the ${\displaystyle {(\alpha \prime )}^t}$-nuclear and ${\displaystyle {(\alpha \prime )}^t}$- integral operators by factorizations by means of the composition of the inclusion map ${\displaystyle L^r(\mu )\to L^1(\mu )+L^p(\mu )}$ with a diagonal operator ${\displaystyle B_w:L^{\infty }(\mu )\to L^r(\mu )}$, where r is the conjugate exponent of p'/(1-σ). As an application we study the reflexivity of the components of the ideal ${\displaystyle P_{p,\sigma }}$.