Is a self-adjoint operator determined by its invariant subspace lattice? (Q1822022)

Let \({\mathcal H}\) be a separable complex Hilbert space and let Lat T denote the lattice of invariant subspaces of a bounded linear operator T on \({\mathcal H}\). The aim of this paper is to address the following question: if A and B are two self-adjoint operators on \({\mathcal H}\) with Lat A and Lat B order isomorphic, how are A and B related? This question is answered by characterizing the property that Lat A and Lat B are order isomorphic both in terms of the spectral structure of A and B and in terms of their functional calculi. Each self-adjoint operator A has a canonical representation as \[ A=A_{\mu_{\infty}}^{(\infty)}\oplus A_{\mu_ 1}^{(1)}\oplus A_{\mu_ 2}^{(2)}\oplus..., \] where \(\mu_{\infty},\mu_ 1,\mu_ 2,..\). are mutually singular compactly supported measures on \({\mathbb{R}}\) and \(A_{\mu_ j}^{(j)}\) is the direct sum of j copies of multiplication by the independent variable on \(L^ 2(\mu_ j)\). Define a function \(\kappa_ A:\) \({\mathbb{N}}\cup \{\infty \}\to ({\mathbb{N}}\cup \{0,\infty \})\times \{0,1\}\) by setting \(\kappa_ A(j)=(n,\epsilon)\), where n is the number of atoms of \(\mu_ j\) and \(\epsilon =0\) or 1 according as the nonatomic part of \(\mu_ j\) is zero or non-zero. The main result of the paper asserts that, for self-adjoint operators A and B on \({\mathcal H}\), the following statements are equivalent. (i) Lat A and Lat B are order isomorphic. (ii) \(\kappa_ A=\kappa_ B.\) (iii) There is a unitary operator U such that Lat B\(=\{U{\mathcal N}:\) \({\mathcal N}\in Lat A\}.\) (iv) B is unitarily equivalent to a generator of the von Neumann algebra generated by A. (v) There are a Borel subset E of the spectrum of A with full A-spectral measure, a Borel subset F of the spectrum of B with full B-spectral measure, and a bijection \(\phi\) : \(E\to F\) such that: (a) \(\Delta\) is a Borel subset of E if and only if \(\phi\) (\(\Delta)\) is a Borel subset of F; (b) a Borel subset \(\Delta\) of E has A-spectral measure zero if and only if \(\phi\) (\(\Delta)\) has B-spectral measure zero; (c) B is unitarily equivalent to \(\phi\) (A). This result is deduced from a similar one characterizing those pairs of normal operators with order isomorphic lattices of reducing subspaces. A key idea in the proof is that of the central elements in such reducing subspace lattices. The paper ends with some remarks concerning the problem of characterizing pairs N and M of normal operators for which Lat N is order isomorphic to Lat M.