I'm looking for results concerning the following questions. If those have been already addressed in the literature, it would be nice to know proper citations:
Let $(E, \tau_E)$ and $(F, \tau_F)$ be two locally convex topological vector spaces. We denote by $\mathcal{L}(E, F) = \mathcal{L}((E, \tau_E), (F, \tau_F))$ the linear space of all linear Operators $T: E \longrightarrow F$ that are continuous w.r.t. the topologies $\tau_E$ and $\tau_F$. Then $\mathcal{L}(E, F)$ becomes a locally convex topological vector space, if equipped with one of the topologies $\tau_{UB}$, $\tau_{SOT}$ and $\tau_{WOT}$, respectively, which denote the topology of uniform convergence on bounded subsets of $(E, \tau_E)$, the strong-operator-topology and the weak-operator-topology, respectively.
Do there exist any sufficient conditions (e.g. quasi-complete, complete, bornological, barrelled, ...) on $(E, \tau_E)$ and/or $(F, \tau_F)$ that ensure $(\mathcal{L}(E, F), \tau)$ to be bornological or barrelled for $\tau \in \{\tau_{UB}, \tau_{SOT}, \tau_{WOT}\}$?
Answers for special cases (e.g. $(E, \tau_E)$ and/or $(F, \tau_F)$ (separable/reflexive) Banach-spaces, Fréchet-spaces, ...) would also be appreciated.