[Authors]
Oleg Reinov and Qaisar Latif
[Title]
Grothendieck-Lidskii theorem for subspaces of $L_p$-spaces
[AMS Subj-class]
47B06 Riesz operators; eigenvalue distributions; approximation numbers,
$s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
[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 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 an $S_1$-operator in $L_2(\nu)$ with its spectral trace
$\sum_{k=1}^\infty \mu_k(T).$
We show that {\it 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 a subspace of any $L_p(\nu)$-space
the trace of $T$ is well defined and equals the sum of all eigenvalues of $T.$}
Note that for $p=2$ one has $s=1,$ and for $p=\infty$ one has $s=2/3.$
[Comments]
LaTeX, English, 4 pp.
[Contact e-mail]
orein51@mail.ru