The Grothendieck property of the sum and the intersection of Banach spaces

The main objective of the present paper is to study the Grothendieck property of the sum and the intersection of two Banach function spaces over σ-finite measure spaces. In particular, we show that for a reflexive symmetric function space E(0,∞)[jls-end-space/], the spaces (E∩L∞)(0,∞) and (E+L∞)(0,∞) are Grothendieck spaces. As a consequence, we fully characterize those symmetric function spaces (Lp∩Lq)(0,∞) and (Lp+Lq)(0,∞)[jls-end-space/], 1≤p,q≤∞[jls-end-space/], possessing the Grothendieck property. We also show that the sum of a noncommutative Lp[jls-end-space/]-space, 1<p<∞[jls-end-space/], and a noncommutative L∞[jls-end-space/]-space has the Grothendieck property.