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

HTML articles powered by AMS MathViewer

- by Alexey I. Popov and Vladimir G. Troitsky
- Proc. Amer. Math. Soc.
**137**(2009), 1793-1800 - DOI: https://doi.org/10.1090/S0002-9939-08-09775-X
- Published electronically: December 29, 2008
- PDF | 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

## Bibliographic Information

**Alexey I. Popov**- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta,Edmonton, Alberta, T6G 2G1, Canada
- MR Author ID: 775644
- Email: apopov@math.ualberta.ca
**Vladimir G. Troitsky**- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta,Edmonton, Alberta, T6G 2G1, Canada
- Email: vtroitsky@math.ualberta.ca
- Received by editor(s): July 22, 2008
- Published electronically: December 29, 2008
- Communicated by: Nigel J. Kalton
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 1793-1800 - MSC (2000): Primary 47B65; Secondary 47A15
- DOI: https://doi.org/10.1090/S0002-9939-08-09775-X
- MathSciNet review: 2470839