arXiv.org e-Print archive: math.FA/9805124
Formal abstract: Let T:l1->l1 be the quasinilpotent operator without an invariant subspace constructed by C.J.Read. We prove that the modulus of this operator has an invariant subspace (and even an eigenvector). This answers a question posed by Y.Abramovich, C.Aliprantis and O.Burkinshaw.
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 lp has an invariant subspace. Keeping in mind that each operator on l1 has a modulus and that some of C.J.Read's counterexamples of operators without invariant subspaces were constructed on l1, it was suggested by Yu.Abramovich, C.Aliprantis, and O.Burkinshaw that the modulus of some of these operators might be a natural candidate for a counterexample to the above problem. Following this suggestion, we deal with the modulus of the quasinilpotent operator T constructed by C.J.Read in 1996. It turns out, quite surprisingly, that not only does |T| have an invariant subspace but it even has a positive eigenvector. This result increases the chances for an affirmative answer to the hypothesis that every positive operator has an invariant subspace.
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
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.