Classification of operator extensions, monad liftings and distributive laws for differential algebras and Rota–Baxter algebras

Generalizing the algebraic formulation of the First Fundamental Theorem of Calculus (FFTC), a class of constraints involving a pair of operators was considered in [Extensions of operators, liftings of monads, and mixed distributive laws, Appl. Categ. Struct. 26 (2018) 747–765]. For a given constraint, the existences of extensions of differential and Rota–Baxter operators, of liftings of monads and comonads, and of mixed distributive laws are shown to be equivalent. In this paper, we give a classification of the constraints satisfying these equivalent conditions.

Positive Self-adjoint Operator Extensions with Applications to Differential Operators

In this paper we consider extensions of positive operators. We study the connections between the von Neumann theory of extensions and characterisations of positive extensions via decompositions of the domain of the associated form. We apply the results to elliptic second order differential operators and look in particular at examples of the Laplacian on a disc and the Aharonov–Bohm operator.