Hereditarily finitely decomposable Banach spaces

A Banach space is said to be ${\displaystyle H{D}_n}$ if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of ${\displaystyle H{D}_n}$ spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex ${\displaystyle H{D}_n}$, then dim ${\displaystyle (ℒ\frac{X}{S}\left(X\right))\le n^2}$, where S(X) denotes the space of strictly singular operators on X. It follows that if X is a real hereditarily indecomposable space, then ℒ(X)/S(X) is a division ring isomorphic either to ℝ, ℂ, or ℍ, the quaternionic division ring.