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ON THE DECOMPOSITION OF OPERATORS WITH SEVERAL ALMOST-INVARIANT SUBSPACES

Part of: General theory of linear operators

Published online by Cambridge University Press:  04 January 2019

AMANOLLAH ASSADI ,
MOHAMAD ALI FARZANEH  and
HAJI MOHAMMAD MOHAMMADINEJAD
AMANOLLAH ASSADI
Affiliation:
Department of Mathematical and Statistical Sciences, University of Birjand, PO Box 97175/615, Birjand, Iran email assadi-aman@birjand.ac.ir
MOHAMAD ALI FARZANEH*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Birjand, PO Box 97175/615, Birjand, Iran email farzaneh@birjand.ac.ir
HAJI MOHAMMAD MOHAMMADINEJAD
Affiliation:
Department of Mathematical and Statistical Sciences, University of Birjand, PO Box 97175/615, Birjand, Iran email hmohammadin@birjand.ac.ir
*
✉farzaneh@birjand.ac.ir
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Abstract

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We seek a sufficient condition which preserves almost-invariant subspaces under the weak limit of bounded operators. We study the bounded linear operators which have a collection of almost-invariant subspaces and prove that a bounded linear operator on a Banach space, admitting each closed subspace as an almost-invariant subspace, can be decomposed into the sum of a multiple of the identity and a finite-rank operator.

Keywords

subspacealmost-invariant subspacehalf-spacefinite-rank operatordefect

MSC classification

Primary: 47A15: Invariant subspaces
Secondary: 47A55: Perturbation theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 2 , April 2019 , pp. 274 - 283
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

References

Androulakis, G., Popov, A. I., Tcaciuc, A. and Troitsky, V. G., ‘Almost invariant half-spaces of operators on Banach spaces’, Integral Equations Operator Theory 65 (2009), 473–484.Google Scholar
Assadi, A., Farzaneh, M. A. and Mohammadinejad, H. M., ‘Invariant subspaces close to almost invariant subspaces for bounded linear operators’, Aust. J. Math. Anal. Appl. 15(2) (2018), Article ID 4, 9 pages.Google Scholar
Cross, R. W., Ostrovskii, M. I. and Shechik, V. V., ‘Operator ranges in Banach spaces I’, Math. Nachr. 173 (1995), 91–114.Google Scholar
Enflo, P., ‘On the invariant subspace problem for Banach spaces’, Acta Math. 158(34) (1987), 213–313.Google Scholar
Marcoux, L. W., Popov, A. I. and Radjavi, H., ‘On almost-invariant subspaces and approximate commutation’, J. Funct. Anal. 264(4) (2013), 1088–1111.Google Scholar
Popov, A. I., ‘Almost invariant half-spaces of algebras of operators’, Integral Equations Operator Theory 67(2) (2010), 247–256.Google Scholar
Popov, A. I. and Tcaciuc, A., ‘Every operator has almost-invariant subspaces’, J. Funct. Anal. 265(2) (2013), 257–265.Google Scholar
Read, C. J., ‘A solution to the invariant subspace problem on the space l 1 ’, Bull. Lond. Math. Soc. 17(4) (1985), 305–317.Google Scholar
Read, C. J., ‘Quasinilpotent operators and the invariant subspace problem’, J. Lond. Math. Soc. (2) 56(3) (1997), 595–606.Google Scholar
Sirotkin, G. and Wallis, B., ‘The structure of almost-invariant half-spaces for some operators’, J. Funct. Anal. 267 (2014), 2298–2312.Google Scholar
Sirotkin, G. and Wallis, B., ‘Almost-invariant and essentially-invariant halfspaces’, Linear Algebra Appl. 507 (2016), 399–413.Google Scholar
Tcaciuc, A., ‘The almost-invariant subspace problem for rank one perturbations’, Duke Math. J., to appear, arXiv:1707.07836 [math.FA].Google Scholar