A version of Lomonosov’s theorem for collections of positive operators

Popov, Alexey; Troitsky, Vladimir

doi:10.1090/S0002-9939-08-09775-X

A version of Lomonosov’s theorem for collections of positive operators
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by Alexey I. Popov and Vladimir G. Troitsky PDF

Proc. Amer. Math. Soc. 137 (2009), 1793-1800 Request permission

Abstract:

It is known that for every Banach space $X$ and every proper $WOT$-closed subalgebra $\mathcal A$ of $L(X)$, if $\mathcal A$ contains a compact operator, then it is not transitive; that is, there exist non-zero $x\in X$ and $f\in X^*$ such that $\langle f,Tx\rangle =0$ for all $T\in \mathcal A$. In the case of algebras of adjoint operators on a dual Banach space, V. Lomonosov extended this result as follows: without having a compact operator in the algebra, one has $\bigl \lvert \langle f,Tx\rangle \bigr \rvert \le \lVert T_*\rVert _e$ for all $T\in \mathcal A$. In this paper, we prove a similar extension of a result of R. Drnovšek. Specifically, we prove that if $\mathcal C$ is a collection of positive adjoint operators on a Banach lattice $X$ satisfying certain conditions, then there exist non-zero $x\in X_+$ and $f\in X^*_+$ such that $\langle f,Tx\rangle \le \lVert T_*\rVert _e$ for all $T\in \mathcal C$.

References

Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002. MR 1921782, DOI 10.1090/gsm/050
Sheldon Axler, Nicholas Jewell, and Allen Shields, The essential norm of an operator and its adjoint, Trans. Amer. Math. Soc. 261 (1980), no. 1, 159–167. MR 576869, DOI 10.1090/S0002-9947-1980-0576869-9
Roman Drnovšek, Common invariant subspaces for collections of operators, Integral Equations Operator Theory 39 (2001), no. 3, 253–266. MR 1818060, DOI 10.1007/BF01332655
R. Drnovšek, D. Kokol-Bukovšek, L. Livshits, G. MacDonald, M. Omladič, and H. Radjavi, An irreducible semigroup of non-negative square-zero operators, Integral Equations Operator Theory 42 (2002), no. 4, 449–460. MR 1885443, DOI 10.1007/BF01270922
Donald Hadwin, Eric Nordgren, Mehdi Radjabalipour, Heydar Radjavi, and Peter Rosenthal, A nil algebra of bounded operators on Hilbert space with semisimple norm closure, Integral Equations Operator Theory 9 (1986), no. 5, 739–743. MR 860869, DOI 10.1007/BF01195810
Mikael Lindström and Georg Schlüchtermann, Lomonosov’s techniques and Burnside’s theorem, Canad. Math. Bull. 43 (2000), no. 1, 87–89. MR 1749953, DOI 10.4153/CMB-2000-013-8
V. I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator, Funkcional. Anal. i Priložen. 7 (1973), no. 3, 55–56 (Russian). MR 0420305
V. Lomonosov, An extension of Burnside’s theorem to infinite-dimensional spaces, Israel J. Math. 75 (1991), no. 2-3, 329–339. MR 1164597, DOI 10.1007/BF02776031
A. J. Michaels, Hilden’s simple proof of Lomonosov’s invariant subspace theorem, Adv. Math. 25 (1977), no. 1, 56–58. MR 500214, DOI 10.1016/0001-8708(77)90089-5
Ben de Pagter, Irreducible compact operators, Math. Z. 192 (1986), no. 1, 149–153. MR 835399, DOI 10.1007/BF01162028
Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682, DOI 10.1007/978-3-642-65574-6