The present monograph concerns with the invariant subspace problem within the context of families of linear bounded operators on Banach spaces as well as families of positive operators on Banach lattices. Although it is primarily based on the PhD thesis of the author, the monograph provides further results and information on recent advances about the topic. The general invariant subspace problem concerns linear and bounded operators on infinite-dimensional complex Hilbert spaces, and asks whether there exists a closed subspace, different than the trivial ones, that is mapped to...