Toeplitz matrices and determinants with Fisher-Hartwig symbols

If 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 κ_A : $\text{N}$ ∪ {∞} → ($\text{N}$ ∪ {0, ∞}) × {0,1} defined in this paper. If $\text{F}$ denotes the collection of all functions from $\text{N}$ ∪ {∞} into ($\text{N}$ ∪ {0,∞}) × {0,1}, then $\text{F}$ 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 κ_A = κ_B. The paper ends with some comments on the corresponding problem for normal operators.