Closability of differential operators and subjordan operators

A (bounded linear) operator J on a Hilbert space is said to be jordan if J = S + N where S = S* and N² = 0. The operator T is subjordan if T is the restriction of a jordan operator to an invariant subspace, and pure subjordan if no nonzero restriction of T to an invariant subspace is jordan. The main operator theoretic result of the paper is that a compact subset of the real line is the spectrum of some pure subjordan operator if and only if it is the closure of its interior. The result depends on understanding when the operator D = θ + d/dx : L²(μ) —> L²(v) is closable. Here θ is an L²(μ) function, μ and v are two finite regular Borel measures with compact support on the real line, and the domain of D is taken to be the polynomials. Approximation questions more general than what is needed for the operator theory result are also discussed. Specifically, an explicit characterization of the closure of the graph of D for a large class of (θ, μ, v) is obtained, and the closure of the graph of D in other topologies is analyzed. More general results concerning spectral synthesis in a certain class of Banach algebras and extensions to the complex domain are also indicated.