%0 Journal Article %T Weakly invariant subspaces for multivalued linear operators on Banach spaces %A Wanjala, Gerald %J Journal of Nonlinear Sciences and Applications %D 2018 %V 11 %N 7 %@ ISSN 2008-1901 %F Wanjala2018 %X 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 \(\mathcal T\mathcal S(0) = \mathcal T(0)\). %9 journal article %R 10.22436/jnsa.011.07.01 %U http://dx.doi.org/10.22436/jnsa.011.07.01 %P 877--884 %0 Journal Article %T Invariant subspaces of completely continuous operators %A N. Aronszajn %A K. T. Smith %J Ann. Math. %D 1954 %V 60 %F Aronszajn1954 %0 Book %T \(C^*\)-Algebras, Volume 1: Banach spaces %A C. Constantinescu %D 2001 %I North-Holland Mathematical Library %C Amsterdam %F Constantinescu 2001 %0 Book %T Multivalued linear operators %A R. Cross %D 1998 %I Marcel Dekker Inc. %C New York %F Cross 1998 %0 Book %T An introduction to models and decompositions in operator theory %A C. S. Kubrusly %D 1997 %I Birkhauser %C Boston %F Kubrusly 1997 %0 Book %T Introductory functional analysis with applications %A E. Kreyszig %D 1978 %I John Willy & Sons %C New York %F Kreyszig1978 %0 Journal Article %T Invariant subspaces for the family of operators which commute with a completely continuous operator %A V. I. Lomonosov %J Funct. Anal. Appl. %D 1973 %V 7 %F Lomonosov1973 %0 Book %T Introduction to functional analysis %A R. Meise %A D. Vogt %D 1997 %I Oxford University press %C New York %F Meise1997 %0 Journal Article %T Lomonosov’s invariant subspace theorem for multivalued linear operators %A P. Saveliev %J Proc. Amer. Math. Soc. %D 2003 %V 131 %F Saveliev2003 %0 Journal Article %T The invariant subspace problem for absolutely p-summing operators in Krein spaces %A G. Wanjala %J J. Inequal. Appl. %D 2012 %V 2012 %F Wanjala2012 %0 Journal Article %T Operator representation of sectorial linear relations and applications %A G. Wanjala %J J. Inequal. Appl. %D 2015 %V 2015 %F Wanjala 2015