There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the Invariant Subspace Problem, he says (paraphrasing) "…this question is still open. It is also an open question whether or not this question is interesting."
To avoid lengthy discussions involving subjective views about what makes math interesting, I'd simply like to know if there are examples of math papers out there that begin with something like, "Suppose the invariant subspace problem has a positive answer…"
Of course, papers that are about the ISP itself don't count!
Best Answer
The invariant subspace problem for Banach spaces was solved in the negative for Banach spaces by Per Enflo and counterexamples for many classical spaces were constructed by Charles Read. The problem is open for reflexive Banach spaces. On the other hand, S. Argyros and R. Haydon recently constructed a Banach space $X$ s.t. $X^*$ is isomorphic to $\ell_1$ and every bounded linear operator on $X$ is the sum of a scalar times the identity plus a compact operator, hence the invariant subspace problem has a positive solution on $X$.
The invariant subspace problem has spurred quite a lot of interesting mathematics. Usually when a positive result is proved, much more comes out, such as a functional calculus for operators. See, e.g., recent papers by my colleague C. Pearcy and his collaborators.
In cases where the ISP has a positive solution for a class of operators, there may be a structure theory for the operators. There is, for example, J. Ringrose's classical structure theorem for compact operators on a Banach space. This is a beautiful and useful theorem, which, BTW, I am using currently with T. Figiel and A. Szankowski to relate the Lidskii trace formula to the J. Erdos theorem in Banach spaces.
Why is the twin prime conjecture interesting?