Compact embeddings and Pitt's property for weighted sequence spaces of Sobolev type

TL;DR

This paper introduces a new class of weighted Sobolev-type sequence spaces, proves their compact embedding properties, and establishes a Pitt's theorem for bounded linear operators between these spaces, with applications in mathematical physics.

Contribution

It generalizes existing compact embedding results and proves a Pitt's type theorem for these new weighted sequence spaces.

Findings

Established several compact embedding theorems for the new class

Proved a Pitt's type theorem for bounded linear transformations

Applied results to spectral analysis in non-equilibrium statistical mechanics

Abstract

In this article we introduce a new class of weighted sequence spaces of Sobolev type and prove several compact embedding theorems for them. It is our contention that the chosen class is general enough so as to allow applications in various areas of mathematics and mathematical physics. In particular, our results constitute a generalization of those compact embeddings recently obtained in relation to the spectral analysis of a class of master equations with non-constant coefficients arising in non-equilibrium statistical mechanics. As a byproduct of our considerations, we also prove a theorem of Pitt's type asserting that under some conditions every linear bounded transformation from one weighted sequence space of the class into another is compact.