OAI.Hyperinvariant.main_theorem

The theorem states that, for every infinite-dimensional (that is, not finite-dimensional) separable complex Hilbert space H, which is complete, the defined proposition FullClaim holds for H (the proof is admitted in the source). FullClaim asserts that there exists a bounded linear operator T on H such that T is nonzero; the spectral-radius-type quantity ‖T^n‖^(1/n) tends to 0 as n tends to infinity (n ranging over natural numbers); T has a transitive commutant, meaning that every closed linear subspace K of H invariant under every bounded operator A commuting with T (A T = T A) is either {0} or all of H; the commutant of T, the subalgebra of bounded operators commuting with T, is not the whole algebra of bounded operators; and this commutant is closed in the strong operator topology, defined as the topology induced by the maps A ↦ (x ↦ A x), that is, the topology of pointwise convergence on vectors of H.