Hereditarily finitely decomposable Banach spaces
Volume 123 / 1997
Studia Mathematica 123 (1997), 135-149
DOI: 10.4064/sm-123-2-135-149
Abstract
A Banach space is said to be $HD_n$ if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of $HD_n$ spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex $HD_n$, then dim $(ℒ(X)/S(X)) ≤ n^2$, 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.
