The Invariant Subspace Problem
This chapter is about the most vexing problem in the theory of linear operators on Hilbert space:
The invariant subspace problem. Does every operator on Hilbert space have a nontrivial invariant subspace?
Here “operator” means “continuous linear transformation,” and “invariant subspace” means “closed (linear) subspace that the operator takes into itself.” To say that a subspace is “nontrivial” means that it is neither the zero subspace nor the whole space. Examples constructed toward the end of the last century show that in the generality of Banach spaces there do exist operators with only trivial invariant subspaces. For Hilbert space, however, the Invariant Subspace Problem remains open, and is the subject of much research. In this chapter we’ll see why invariant subspaces are of interest and then will prove one of the subject’s landmark theorems: Victor Lomonosov’s 1973 result, a special case of which states:
If an operator T on a Banach space commutes with a non-zero compact operator, then T has a nontrivial invariant subspace.
This result, which far surpassed anything that seemed attainable at the time, is only part of what Lomonosov proved in an astonishing two-page paper  that introduced nonlinear methods—in particular the Schauder Fixed-Point Theorem—into this supposedly hard-core-linear area of mathematics.
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