Abstract
It is known that every bounded operator on an infinite dimensional separable Hilbert space \({\mathcal{H}}\) has an invariant subspace if and only if each pair of idempotents on \({\mathcal{H}}\) has a common invariant subspace. We show that the same equivalence holds for operators and pairs of idempotents that are essentially selfadjoint. We also show that each pair of idempotents on \({\mathcal{H}}\) has a common almost-invariant half-space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory, Graduate Studies in Mathematics, vol. 50. American Mathematical Society, Providence (2002)
Allan G.R., Zemánek J.: Invariant subspaces for pairs of projections. J. Lond. Math. Soc. 57, 449–468 (1998)
Androulakis G., Popov A.I., Tcaciuc A., Troitsky V.G.: Almost invariant half-spaces of operators on Banach spaces. Integr. Equ. Oper. Theory 65, 473–484 (2009)
Davis C.: Generators of the ring of bounded operators. Proc. Am. Math. Soc. 6, 970–972 (1955)
Foias C., Hamid S., Onica C., Pearcy C.: Operators with compact imaginary part. Indiana Univ. Math. J. 58, 2297–2304 (2009)
Lomonosov V.: On real invariant subspaces of bounded operators with compact imaginary part. Proc. Am. Math. Soc. 115, 775–777 (1992)
Marcoux L.W., Popov A.I., Radjavi H.: On almost-invariant subspaces and approximate commutation. J. Funct. Anal. 264, 1088–1111 (2013)
Nordgren E.A., Radjabalipour M., Radjavi H., Rosenthal P.: Quadratic operators and invariant subspaces. Studia Math. 88, 263–268 (1988)
Nordgren E.A., Radjavi H., Rosenthal P.: A geometric equivalent of the invariant subspace problem. Proc. Am. Math. Soc. 61, 66–68 (1976)
Popov A.I., Tcaciuc A.: Every operator has almost-invariant subspaces. J. Funct. Anal. 265, 257–265 (2013)
Radjavi H., Rosenthal P.: Invariant Subspaces. Springer, New York (1973)
Read C.J.: A solution to the invariant subspace problem on the sppace \({\ell_1}\). Bull. Lond. Math. Soc. 17, 305–317 (1985)
Simonič A.: An extension of Lomonosov’s techniques to non-compact operators. Trans. Am. Math. Soc. 348, 975–995 (1996)
Sirotkin G., Wallis B.: The structure of almost-invariant half-spaces for some operators. J. Funct. Anal. 267, 2298–2312 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
J. Bernik was supported in part by Research and Development Agency of Slovenia. H Radjavi was supported by NSERC of Canada.
Rights and permissions
About this article
Cite this article
Bernik, J., Radjavi, H. Invariant and Almost-Invariant Subspaces for Pairs of Idempotents. Integr. Equ. Oper. Theory 84, 283–288 (2016). https://doi.org/10.1007/s00020-015-2260-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-015-2260-3