A Banach space is said to be $H{D}_n$ if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of $H{D}_n$ spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex $H{D}_n$, then dim $(ℒ\left(X\right)/S\left(X\right))\le n^2$, 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....

Extending previous results of H. Salas we obtain a characterisation of hypercyclic weighted shifts on an arbitrary F-sequence space in which the canonical unit vectors $\left(e_n\right)$ form a Schauder basis. If the basis is unconditional we give a characterisation of those hypercyclic weighted shifts that are even chaotic.

In this paper we give some sufficient conditions for the adjoint of a weighted composition operator on a Hilbert space of analytic functions to be hypercyclic.