Quasi Gelfand triples

TL;DR

This paper introduces quasi Gelfand triples, a generalization of Gelfand triples that relaxes the continuous embedding requirement, enabling new applications in boundary space analysis of differential operators.

Contribution

It defines quasi Gelfand triples with closed embeddings, explores their properties, and shows they can be decomposed into two standard Gelfand triples.

Findings

Existence of a minimal space for continuous embedding

Density results for intersections of quasi Gelfand triples

Decomposition of quasi Gelfand triples into two Gelfand triples

Abstract

We generalize the notion of Gelfand triples (also called Banach-Gelfand triples or rigged Hilbert spaces) by dropping the necessity of a continuous embedding. This means in our setting we lack of a chain inclusion. We replace the continuous embedding by a closed embedding of a dense subspace. This new notion will be called quasi Gelfand triple. These triples appear naturally, when we regard the boundary spaces of spatially multidimensional differential operators, e.g., the Maxwell operator. We will show that there is a smallest space where we can continuously embed the entire triple. Moreover, we will show density results for intersections of members of the quasi Gelfand triple. Finally, we show that every quasi Gelfand triple can be decomposed into two "ordinary" Gelfand triples.