Abstract Cesàro spaces. Duality (Q482734)

Let \(I=[0,1]\) or \(I=[0,\infty)\). For a Banach ideal space \(X\) of measurable functions on \(I\), the abstract Cesàro space \(CX\) is defined as the space of all measurable functions \(f\) on \(I\) such that \(C|f|\in X\), equipped with the norm \(\|f\|_{CX}:=\|C|f|\|_X\). Here, \(C\) denotes the Cesàro operator, i.e., \[ (Cf)(x):=\frac{1}{x}\int_0^xf(t)\,\text{d}t \;\;\;\text{for} \;x\in I. \] This paper deals with descriptions of the Köthe dual of \(CX\). The cases \(I=[0,1]\) and \(I=[0,\infty)\) are treated separately, as there is a great difference between them.