Grothendieck-Lidskii theorem for subspaces and factor spaces of L_p-spaces

TL;DR

This paper extends classical trace theorems to subspaces and factor spaces of Lp-spaces, showing that for certain nuclear operators, the trace equals the sum of eigenvalues.

Contribution

It generalizes Grothendieck-Lidskii theorems to a broader class of subspaces and factor spaces of Lp-spaces, establishing trace-eigenvalue equality.

Findings

Trace of s-nuclear operators is well-defined in these spaces.

Trace equals the sum of all eigenvalues for these operators.

Extension of classical theorems to new Banach space classes.

Abstract

In 1955, A. Grothendieck has shown that if the linear operator $T$ in a Banach subspace of an $L_\infty$-space is 2/3-nuclear then the trace of $T$ is well defined and is equal to the sum of all eigenvalues $\{\mu_k(T)\}$ of $T.$ V.B. Lidski\v{\i}, in 1959, proved his famous theorem on the coincidence of the trace of the $S_1$-operator in $L_2(\nu)$ with its spectral trace $\sum_{k=1}^\infty \mu_k(T).$ We show that for $p\in[1,\infty]$ and $s\in (0,1]$ with $1/s=1+|1/2-1/p|,$ and for every $s$-nuclear operator $T$ in every subspace of any $L_p(\nu)$-space the trace of $T$ is well defined and equals the sum of all eigenvalues of $T.$