In the study of linear maps, an important and useful technique is to represent them in some sort of standard form. The analysis frequently focuses on the action of the maps on subspaces. My research follows this theme with operators on Hilbert space. In my research, I mainly follow the second course. For example, one might consider the set of all operators that leave invariant a given a family of subspaces.
More general families produce more complicated spaces of operators.
K.R. Davidson, Nest algebras, Pitman Research Notes in Mathematics No. 191, Longman 1988. J.A. Erdos, Reflexivity for subspace maps and linear spaces of operators, Proc. P.R. Halmos, Reflexive lattices of subspaces, J. London Math. Canada 17 193-196, 1995.