Weakly invariant subspaces for multivalued linear operators on Banach spaces
Authors
Gerald Wanjala
 Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, PC 123, Al Khoud, Sultanate of Oman.
Abstract
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 \(\mathcal T\mathcal S(0) = \mathcal T(0)\).
Keywords
 Linear relations
 weakly invariant subspaces
MSC
References

[1]
N. Aronszajn, K. T. Smith, Invariant subspaces of completely continuous operators, Ann. Math., 60 (1954), 345–350.

[2]
C. Constantinescu , \(C^*\)Algebras, Volume 1: Banach spaces, NorthHolland Mathematical Library, Amsterdam (2001)

[3]
R. Cross , Multivalued linear operators, Marcel Dekker Inc., New York (1998)

[4]
C. S. Kubrusly , An introduction to models and decompositions in operator theory, Birkhauser, Boston (1997)

[5]
E. Kreyszig, Introductory functional analysis with applications, John Willy & Sons, New York (1978)

[6]
V. I. Lomonosov, Invariant subspaces for the family of operators which commute with a completely continuous operator, Funct. Anal. Appl., 7 (1973), 213–214.

[7]
R. Meise, D. Vogt, Introduction to functional analysis , Oxford University press, New York (1997)

[8]
P. Saveliev, Lomonosov’s invariant subspace theorem for multivalued linear operators, Proc. Amer. Math. Soc., 131 (2003), 825–834.

[9]
G. Wanjala, The invariant subspace problem for absolutely psumming operators in Krein spaces, J. Inequal. Appl., 2012 (2012), 13 pages.

[10]
G. Wanjala , Operator representation of sectorial linear relations and applications, J. Inequal. Appl., 2015 (2015), 16 pages