Weakly invariant subspaces for multivalued linear operators on Banach spaces

Volume 11, Issue 7, pp 877--884 http://dx.doi.org/10.22436/jnsa.011.07.01 Publication Date: May 16, 2018       Article History


Gerald Wanjala - Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, PC 123, Al Khoud, Sultanate of Oman


Peter Saveliev generalized Lomonosov's invariant subspace theorem to the case of linear relations. In particular, he proved that if \(\mathcal S\) and \(\mathcal T\) are linear relations defined on a Banach space \(X\) and having finite dimensional multivalued parts and if \(\mathcal T\) right commutes with \(\mathcal S\), that is, \(\mathcal S \mathcal T \subset \mathcal T\mathcal S\), and if \(\mathcal S\) is compact then \(\mathcal T\) has a nontrivial weakly invariant subspace. However, the case of left commutativity remained open. In this paper, we develop some operator representation techniques for linear relations and use them to solve the left commutativity case mentioned above under the assumption that \(\mathcal S\mathcal T(0) = \mathcal S(0)\) and \(\mathcal T\mathcal S(0) = \mathcal T(0)\).