Eigenvalues of weighted generalized shifts over direct products of vector spaces

TL;DR

This paper computes all eigenvalues of weighted generalized shift operators on vector spaces over arbitrary sets, extending understanding of their spectral properties in a broad algebraic context.

Contribution

It provides a complete characterization of eigenvalues for weighted generalized shift operators on vector spaces over arbitrary index sets, generalizing previous results.

Findings

Eigenvalues are explicitly characterized for weighted generalized shifts.

Results apply to shifts over arbitrary nonempty index sets.

The spectral analysis extends classical shift operator theory.

Abstract

In the following text for vector space $V$ over field $F$ we compute all eigenvalues of weighted generalized shift $\sigma_{\varphi,\mathfrak{w}}:V^\Gamma\to V^\Gamma$ (and generalized shift $\sigma_\varphi:V^\Gamma\to V^\Gamma$) for nonempty set $\Gamma$, weight vector $\mathfrak{w}\in F^\Gamma$ and self--map $\varphi:\Gamma\to \Gamma$.