  Some questions are posted at the end of Chapter 16 of Huybrechts' book
'Lectures on K3 Surfaces', concerning the bounded derived category of a K3
surface $D^b(S)$. Let $E$ be a spherical object in $D^b(S)$. The first question
asks if there always exists a non-zero object $F$ satisfying RHom$(E,F)=0$.
Further, let $E$ be a spherical bundle. The second question is whether $E$ is
always semistable with respect to some polarization on $S$ and if there is a
way to `count' spherical bundles with a fixed Mukai vector. In this note, we
provide (partial) answers to these two questions. In the appendix, Genki Ouchi
shows that any spherical twist associated to an $n$-spherical object on a
smooth projective $n$-dimensional variety is not conjugate to a standard
autoequivalence.
