A note on the surjectivity of operators on vector bundles over discrete spaces

Abstract

In this note we give a short and self-contained proof for a criterion of Eidelheit on the solvability of linear equations in infinitely many variables. We use this criterion to study the surjectivity of magnetic Schrödinger operators on bundles over graphs.

From Hardy to Rellich inequalities on graphs

“…In general, solutions of the equation do not need to exist. However, if is a nonnegative Schrödinger operator on a connected, locally finite graph, then is surjective as an operator from to [19, Theorem 2.2] (see also [5, 14]), where we have for locally finite graphs. Using the results of the previous section, we show the existence of solutions of the above equation on not necessarily locally finite graphs under the assumption of the validity of a Rellich inequality.…”