arXiv.org e-Print archive: math.FA/9805124

**Formal abstract:**
Let T:*l _{1}*->

**Brief review:**
Several examples of
operators without invariant subspaces have been constructed by P.Enflo and
C.J.Read. During the last several years there has been a noticeable
increase of interest in the invariant subspace problem for positive
operators on Banach lattices. In particular, it is still an open problem
if every positive operator has an invariant subspace. It has recently been
proved by Yu.Abramovich, C.Aliprantis, and O.Burkinshaw that every
positive quasinilpotent operator on *l _{p}* has an invariant
subspace.
Keeping in mind that each operator on

**A follow-up comment:**
In the paper we mention the following fact:
(**Theorem 2**) * If the essential spectral radius of a
positive operator on a Banach lattice is strictly less than
the usual spectral radius, then the spectral radius is an
eigenvalue of the
operator, corresponding to a positive eigenvector.* We attribute
this result to R.Nussbaum, 1981. (Actually, Nussbaum proved a
somewhat more general result.)
I have recently learned from
A.Schep that the same result was also published in the following
paper:

Original paper in Russian:
П.П.Забрейко, С.В.Смицких, **"Об одной теореме М.Г.Крейна-М.А.Рутмана",**
*Функциональный анализ и его приложения*
**13**, no3, (1979), 81-82.

English translation:
P.P. Zabreiko and S.V. Smitskikh, **"A theorem of M.G.Krein and M.A.Rutman",**
*Functional Analysis and its
Applications,* (1980), 222-223.