Trace, determinant and eigenvalues of nuclear operators

TL;DR

This paper explores the spectral properties of nuclear operators in Banach spaces, applying advanced determinant and trace theories to establish new theorems on eigenvalue distribution and trace coincidence, including novel classes of generalized nuclear operators.

Contribution

It introduces new theorems on eigenvalue distribution and trace coincidence for nuclear operators, utilizing determinant and trace theories, and defines new classes of generalized nuclear operators.

Findings

New theorems on eigenvalue distribution

Conditions for spectral and nuclear trace coincidence

Introduction of generalized nuclear operators of Lorentz–Lapresté type

Abstract

It is shown how the results in the theory of determinants and traces as well as in the theory of quasi-normed tensor products can be applied for getting new theorems on distribution of eigenvalues of nuclear operators in Banach spaces and on coincidence of spectral and nuclear traces such operators. As examples, it is considered the new classes of operators -- generalized nuclear operators of Lorentz--Laprest\'e $N_{(r,s),p}.$