We prove that if T is an operator on an infinite-dimensional Hilbert space
whose spectrum and essential spectrum are both connected and whose Fredholm
index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces
of T are the finite-dimensional ones. We also characterize norm-stable
invariant subspaces of any weighted unilateral shift operator. Along the way,
using the results of Yngve Domar, we prove that the nontrivial invariant
subspaces of a quasianalytic unilateral weighted shift all have
finite-codimension, thereby answering a question posed by Allen Shields. We
show that quasianalytic shift operators are points of norm continuity of the
lattice of the invariant subspaces. We also provide a necessary condition for
strongly stable invariant subspaces for certain operators.