Geometrical properties of the space \(\phi\) (X,Y) (Q801514)

In this note geometric properties of the space \(\phi\) (X,Y) are investigated, where X and Y are ideal Banach spaces and \(\phi\) is a concave function \((\phi: R^ 2_+\to R^ 1_+)\). The author supposes that the space X has absolutely continuous norm (Th. 1), is reflexive (Th. 2) and satisfies Banach-Saks condition (Th. 3). In the following the author has investigated under which assumptions about the space Y and the function \(\phi\), the ideal Banach spaces \(\phi\) (X,Y) possess the same properties as the space X (Theorem 1,2 and 3, respectively).