×

The existence of hyperinvariant subspaces for weighted shift operators. (English) Zbl 06902461

Summary: We introduce some classes of Banach spaces for which the hyperinvariant subspace problem for the shift operator has positive answer. Moreover, we provide sufficient conditions on weights which ensure that certain subspaces of \(\ell^2_{\beta}(\mathbb{Z})\) are closed under convolution. Finally we consider some cases of weighted spaces for which the problem remains open.

MSC:

47A15 Invariant subspaces of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
PDF BibTeX XML Cite
Full Text: DOI Euclid Link

References:

[1] A. Beurling, Mittag-Leffler lectures in complex analysis, Collected works of A. Beurling, (1977-78), pp. 361-443.
[2] A. Beurling, On two problems concerning linear transformations on Hilbert spaces, Acta. Math. 81 (1948), 17 pp. · Zbl 0033.37701
[3] I. Chalendar and J. Partington, Modern approaches to the invariant-subspace problem, Cambridge Tracts in Mathematics, 188. Cambridge University Press, Cambridge, 2011. · Zbl 1231.47005
[4] J. Esterle, Singular inner functions and invariant subspaces for dissymmetric weighted shifts, J. Funct. Anal. 144 (1997), 64-104. · Zbl 0938.47003
[5] J. Esterle and A. Volberg, Asymptotically holomorphic functions and translation invariant subspaces of weighted Hilbert spaces of sequences, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 2, 185-230. · Zbl 1059.47034
[6] J. Esterle and A. Volberg, Analytic left-invariant subspace of weighted Hilbert spaces of sequence, J. Operator Theory 45 (2001), 265-301. · Zbl 1002.47001
[7] A. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, pp. 49-128. Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974.
[8] J. Wermer, The existence of invariant subspaces, Duke Math. J. 19 (1952), 615-622. · Zbl 0047.35806
[9] N. Wiener, Tauberian theorems, Ann. Math. (2) 33 (1932), no. 1, 1-100. · JFM 58.0226.02
[10] B. S. Yadav, The present state and heritages of the invariant subspace problem, Milan J. Math. 73 (2005), 289-316. · Zbl 1227.47004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.