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The invariant subspaces of the shift plus integer multiple of the Volterra operator on Hardy spaces. (English) Zbl 1471.47005

Arch. Math. 111, No. 5, 513-522 (2018); correction ibid. 112, No. 5, 559-560 (2019).
For the operator \(T\) acting on the Hardy space \(H_p(\mathbb D)\) (\(1<p<\infty\)) and defined by the formula \((Tf)(z)=zf(z)+n\int_0^zf(\omega)\,d\omega\), \(z\in\mathbb D\), where \(n\) is a positive integer, there is given a complete characterization of the lattice of invariant subspaces. The paper extends results obtained in [Ž. Čučković and B. Paudyal, Arch. Math. 110, No. 5, 477–486 (2018; Zbl 1471.47003)].

MSC:

47A15 Invariant subspaces of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
30H10 Hardy spaces
45P05 Integral operators
47A46 Chains (nests) of projections or of invariant subspaces, integrals along chains, etc.

Citations:

Zbl 1471.47003
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References:

[1] Aleman, A.; Korenblum, B., Volterra invariant subspaces of \(H^p\), Bull. Sci. Math., 132, 510-528, (2008) · Zbl 1168.46011 · doi:10.1016/j.bulsci.2007.08.001
[2] Aleman, A.; Richter, S.; Sundberg, S., Beurling’s theorem for the Bergman space, Acta Math., 177, 275-310, (1996) · Zbl 0886.30026 · doi:10.1007/BF02392623
[3] Aronszajn, N.; Smith, K., Invariant subspaces of completely continuous operators, Ann. Math., 60, 345-390, (1954) · Zbl 0056.11302 · doi:10.2307/1969637
[4] Bernstein, A.; Robinson, A., Solution of an invariant subspace problem of K.T. Smith and P.R. Halmos, Pac. J. Math., 16, 421-431, (1966) · Zbl 0141.12903 · doi:10.2140/pjm.1966.16.421
[5] Beurling, A., On two problems concerning linear transformations in Hilbert space, Acta Math., 81, 239-255, (1949) · Zbl 0033.37701 · doi:10.1007/BF02395019
[6] Chalendar, I., Partington, J.: Modern Approaches to the Invariant-Subspace Problem, Cambridge Tracts in Mathematics, vol. 188. Cambridge University Press, Cambridge (2011) · Zbl 1231.47005 · doi:10.1017/CBO9780511862434
[7] Cowen, C.; Wahl, R., Shift-invariant subspaces invariant for composition operators on Hardy-Hilbert space, Proc. Am. Math. Soc., 142, 4143-4154, (2014) · Zbl 1309.47020 · doi:10.1090/S0002-9939-2014-12132-0
[8] Čučković, Ž; Paudyal, B., Invariant subspaces of the shift plus complex Volterra operator, J. Math. Anal. Appl., 426, 1174-1181, (2015) · Zbl 1308.47006 · doi:10.1016/j.jmaa.2015.01.056
[9] Čučković, Ž; Paudyal, B., The lattices of invariant subspaces of a class of operators on the Hardy space, Arch. Math., 110, 477-486, (2018) · Zbl 1471.47003 · doi:10.1007/s00013-017-1142-0
[10] Duren, P.: Theory of \(H^p\) Spaces. Academic Press, New York (1970) · Zbl 0215.20203
[11] Enflo, P., On the invariant subspace problem for Banach spaces, Acta Math., 158, 213-313, (1987) · Zbl 0663.47003 · doi:10.1007/BF02392260
[12] Halmos, P., Invariant subspaces of polynomially compact operators, Pac. J. Math., 16, 433-437, (1966) · Zbl 0141.12904 · doi:10.2140/pjm.1966.16.433
[13] Korenblum, B., Invariant subspaces of the shift operator in certain weighted Hilbert spaces of sequences, (Russian), Dokl. Akad. Nauk. SSSR, 202, 1258-1260, (1972)
[14] Lin, Q.; Liu, J.; Wu, Y., Volterra type operators on \(S^p({\mathbb{D}})\) spaces, J. Math. Anal. Appl., 461, 1100-1114, (2018) · Zbl 06852149 · doi:10.1016/j.jmaa.2018.01.038
[15] Matache, V., Invariant subspaces of composition operators, J. Oper. Theory, 73, 243-264, (2015) · Zbl 1399.47087 · doi:10.7900/jot.2013nov14.2041
[16] Montes-Rodriguez, A.; Ponce-Escudero, M.; Shkarin, S., Invariant subspaces of parabolic self-maps in the Hardy space, Math. Res. Lett., 17, 1174-1181, (2010) · Zbl 1244.47006 · doi:10.4310/MRL.2010.v17.n1.a8
[17] Ong, B., Invariant subspace lattices for a class of operators, Pac. J. Math., 94, 385-405, (1981) · Zbl 0431.47002 · doi:10.2140/pjm.1981.94.385
[18] Read, C., A solution to the invariant subspace problem on the space \(l_1\), Bull. Lond. Math. Soc., 17, 305-317, (1985) · Zbl 0574.47006 · doi:10.1112/blms/17.4.305
[19] Read, C., A short proof concerning the invariant subspace problem, J. Lond. Math. Soc., 34, 335-348, (1986) · Zbl 0664.47006 · doi:10.1112/jlms/s2-34.2.335
[20] Šamojan, F., The structure of closed ideals in certain algebras of functions analytic in the disc and smooth up to its boundary, (Russian) Akad, Nauk Armjan. SSR Dokl., 60, 133-136, (1975)
[21] Sarason, D., A remark on the Volterra operator, J. Math. Anal. Appl., 12, 244-246, (1965) · Zbl 0138.38801 · doi:10.1016/0022-247X(65)90035-1
[22] Sarason, Donald, Invariant subspaces, 1-47, (1974), Providence, Rhode Island · Zbl 0302.47003
[23] Yadav, B., The present state and heritages of the invariant subspace problem, Milan J. Math., 73, 289-316, (2005) · Zbl 1227.47004 · doi:10.1007/s00032-005-0048-7
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