TITLE: Is the Invariant Subspace Problem interesting?
QUESTION [29 upvotes]: 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!

REPLY [4 votes]: If the invariant subspace problem has a positive answer then every bounded operator $A \in B(H)$ can be put in upper triangular form, in the sense that there is a maximal chain $(E_\lambda)$ of closed subspaces of $H$ such that every $E_\lambda$ is invariant for $A$.
In $\mathbb{C}^n$, a maximal chain of subspaces looks like $$\{0\} = E_0 \subset E_1 \subset \cdots \subset E_n = \mathbb{C}^n,$$ where the dimension of $E_i$ is $i$, and any operator for which all the $E_i$ are invariant is literally upper triangular for an orthonormal basis whose first $i$ elements belong to $E_i$, for all $i$. The infinite dimensional version is a natural generalization and seems to say rather a lot about the structure of $A$.
No doubt this result would be considered "known" by experts, but I could not find it explicitly stated anywhere, so I included it near the end of this paper. Also, Matt Kennedy answered this question with a reference to a result that easily implies it.