Operators commuting with translation by one Part I. Representation theorems

Theory of Operators Part I. Single Operators

The work of von Neumann on Operator Theory as distinct from the later work on Rings of operators, extends from 1928 to 1932 with certain later additions. I t is a remarkable development in which the inadequacy of formula mathematics for quantum mechanics was established and a new abstract approach introduced. This approach required the abstract definition of Hubert space and reformulation of th...

Computing semi-commuting differential operators in one and multiple variables

We discuss the concept of what we refer to as semi-commuting linear differential operators. Such operators hold commuting operators as a special case. In particular, we discuss a heuristic by which one may construct such operators. Restricting to the case in which one such operator is of degree two, we construct families of linear differential operators semi-commuting with some named operators ...

Polynomial Extension Operators. Part I

In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to i...

PART I . One –

On Commuting Differential Operators

The theory of commuting linear differential expressions has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs (P,L). Gesztesy and the present author have established a relationship of this circle of ideas with the property that all solutions of the differential equations Ly = zy, z ∈ C, are meromorphic. In this paper this relationship is explored ...