Some remarks on spectra of nuclear operators

TL;DR

This paper explores the spectral symmetry of nuclear operators across various Banach spaces, extending previous criteria and providing sharp generalizations to broader classes of operators and spaces.

Contribution

It extends Zelikin's spectral symmetry criterion for nuclear operators to subspaces of quotients of Lp-spaces and generalizes Mityagin's results to non-compact operators.

Findings

Spectral symmetry criterion does not extend to all Banach spaces.

Constructs a nuclear operator in l1 with trace 1 and square zero.

Provides sharp generalizations of previous theorems to broader spaces.

Abstract

It was shown by M. I. Zelikin (2007) that the spectrum of a nuclear operator in a separable Hilbert space is central-symmetric iff the spectral traces of all odd powers of the operator equal zero. The criterium can not be extended to the case of general Banach spaces: It follows from Grothendieck-Enflo results that there exists a nuclear operator $U$ in the space $l_1$ with the property that $\operatorname{trace}\, U=1$ and $U^2=0.$ B. Mityagin (2016) has generalized Zelikin's criterium to the case of compact operators (in Banach spaces) some of which powers are nuclear. We give sharp generalizations of Zelikin's theorem (to the cases of subspaces of quotients of $L_p$-spaces) and of Mityagin's result (for the case where the operators are not necessarily compact).