Some approximation properties and nuclear operators in spaces of analytical functions

Abstract

‎‎We introduce and investigate a new notion of the approximation property $AP_{[c]},$‎ ‎where $c= (c_n)$ is an arbitrary positive real sequence‎, ‎tending to infinity‎. ‎Also‎, ‎we study the corresponding notion of $[c]$-nuclear operators in Banach spaces‎. ‎Some characterization of the $AP_{[c]}$ in terms of tensor products‎, ‎as well as‎ ‎sufficient conditions for a Banach space to have the $AP_{[c]},$ are given‎. ‎We give also sufficient conditions for a positive answer to the question‎: ‎When it follows from the $[c]$-nuclearity of an adjoint operator the nuclearity of‎ ‎the operator itself‎. ‎Obtained results are applied then to the study of properties‎ ‎of nuclear operators in some spaces of analytical functions‎. ‎Many examples are given‎.