Is a self-adjoint operator determined by its invariant subspace lattice?

AbstractIf A and B are self-adjoint operators, this paper shows that A and B have order isomorphic invariant subspace lattices if and only if there are Borel subsets E and F of σ(A) and σ(B), respectively, whose complements have spectral measure zero, and there is a bijective function φ: E → F such that (i) Δ is a Borel subset of E if and only if φ(Δ) is a Borel subset of F; (ii) a Borel subset Δ of E has A-spectral measure zero if and only if φ(Δ) has B-spectral measure zero; (iii) B is unitarily equivalent to φ(A). If A is any self-adjoint operator, there is an associated function κA : N ∪ {∞} → (N ∪ {0, ∞}) × {0,1} defined in this paper. If F denotes the collection of all functions from N ∪ {∞} into (N ∪ {0,∞}) × {0,1}, then F is a parameter space for the isomorphism classes of the invariant subspace lattices of self-adjoint operators. That is, two self-adjoint operators A and B have isomorphic invariant subspace lattices if and only if κA = κB. The paper ends with some comments on the corresponding problem for normal operators