Abstract
In an attempt to solve the invariant subspace problem, we introduce a certain orthonormal basis of Hilbert spaces, and prove that a bounded linear operator on a Hilbert space must have an invariant subspace once this basis fulfills certain conditions. Ultimately, this basis is used to show that every bounded linear operator on a Hilbert space is the sum of a shift and an upper triangular operators, each of which having an invariant subspace.
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Aronszajn, N., Smith, K.T.: Invariant subspaces of completely continuous operators. Ann. Math. 60, 345–350 (1954)
Bernstein, A.R., Robinson, A.: Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pac. J. Math. 16, 421–431 (1966)
Chalendar, I., Partington, J.R.: An overview of some recent developments on the invariant subspace problem. Concr. Oper. 1, 1–10 (2012)
Enflo, P.: On the invariant subspace problem for Banach spaces. Acta Math. 158, 213–313 (1987)
Lomonosov, V.I.: Invariant subspaces of the family of operators that commute with a completely continuous operator. Funkc. Anal. i Pril. 7(3), 55–56 (1973)
Read, C.J.: A solution to the invariant subspace problem on the space \(\ell ^1\). Bull. Lond. Math. Soc. 17, 305–317 (1985)
Yadav, B.S.: The present state and heritages of the invariant subspace problem. Milan J. Math. 73, 289–316 (2005)
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Communicated by Kumam Poom.
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Sababheh, M., Yousef, A. & Khalil, R. On the Invariant Subspace Problem. Bull. Malays. Math. Sci. Soc. 39, 699–705 (2016). https://doi.org/10.1007/s40840-015-0135-z
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DOI: https://doi.org/10.1007/s40840-015-0135-z