Here is the Ringrose Theorem regarding invariant subspace on Peter Lax's Functional Analysis (Maximal invariance nest is a sequence of nested subspace in inclusion which cannot insert subspace into this sequence). I am confused how to generalize the proof to eigenvalues with algebraic multiplicity larger than 1.
For general algebraic multiplicity, I am kind of confused how to generalize the proof. I am planning to construct $M,M_-$ inductively including 1 eigenvector at a time. In the proof, the main step is define $M$ as the smallest subspace in the invariant nest containing $x$ (the eigenvector). But, this $M$ does not guarantee to contain other eigenvectors.
Does anyone have any ideas or comments?
Thanks in advance