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
Wanjala, Gerald 1
@article{10_22436_jnsa_011_07_01,
author = {Wanjala, Gerald },
title = {Weakly invariant subspaces for multivalued linear operators on {Banach} spaces},
journal = {Journal of nonlinear sciences and its applications},
pages = {877-884},
year = {2018},
volume = {11},
number = {7},
doi = {10.22436/jnsa.011.07.01},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.07.01/}
}
TY - JOUR AU - Wanjala, Gerald TI - Weakly invariant subspaces for multivalued linear operators on Banach spaces JO - Journal of nonlinear sciences and its applications PY - 2018 SP - 877 EP - 884 VL - 11 IS - 7 UR - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.07.01/ DO - 10.22436/jnsa.011.07.01 LA - en ID - 10_22436_jnsa_011_07_01 ER -
%0 Journal Article %A Wanjala, Gerald %T Weakly invariant subspaces for multivalued linear operators on Banach spaces %J Journal of nonlinear sciences and its applications %D 2018 %P 877-884 %V 11 %N 7 %U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.07.01/ %R 10.22436/jnsa.011.07.01 %G en %F 10_22436_jnsa_011_07_01
Wanjala, Gerald . Weakly invariant subspaces for multivalued linear operators on Banach spaces. Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 7, p. 877-884. doi: 10.22436/jnsa.011.07.01
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