On linear operators with an invariant subspace of functions

Abstract

Let us denote $\mathcal{V}$, the finite dimensional vector spaces of functions of the form $\psi (x)=p_n(x)+f(x)p_m(x)$ where $p_n(x)$ and $p_m(x)$ are arbitrary polynomials of degree at most $n$ and $m$ in the variable $x$ while $f(x)$ represents a fixed function of $x$. Conditions on $m,n$ and $f(x)$ are found such that families of linear differential operators exist which preserve $\mathcal{V}$. A special emphasis is accorded to the cases where the set of differential operators represents the envelopping algebra of some abstract algebra.

6 pages, LaTeX, discussion extended, reference added