  We exhibit a general class of unbounded operators in Banach spaces which can
be shown to have the single-valued extension property, and for which the local
spectrum at suitable points can be determined. We show that a local spectral
radius formula holds, analogous to that for a globally defined bounded operator
on a Banach space with the single-valued extension property. An operator of the
class under consideration can occur in practice as (an extension of) a
differential operator which, roughly speaking, can be diagonalised on its
domain of smooth test functions via a discrete transform, such that the
diagonalising transform establishes an isomorphism of topological vector spaces
between the domain of the differential operator, in its own topology, and a
sequence space. We give concrete examples of (extensions of) such operators
(constant coefficient differential operators on the d-torus, Jacobi operators,
the Hermite operator, Laguerre operators) and indicate further perspectives.
