Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T23:13:31.740Z Has data issue: false hasContentIssue false
  • English
  • Français

The Dual of the Compressed Shift

Part of: Function theory on the disc General theory of linear operators

Published online by Cambridge University Press:  17 April 2020

M. C. Câmara  and
W. T. Ross
M. C. Câmara
Affiliation:
Departamento de Matematica, Instituto Superior Tecnico, 1049-001Lisboa, Portugal e-mail: [email protected]
W. T. Ross*
Affiliation:
Department of Mathematics and Computer Science, University of Richmond, Richmond, VA23173, USA
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

For an inner function u, we discuss the dual operator for the compressed shift $P_u S|_{{\mathcal {K}}_u}$, where ${\mathcal {K}}_u$ is the model space for u. We describe the unitary equivalence/similarity classes for these duals as well as their invariant subspaces.

Keywords

Inner functionsHardy spacescompressed shiftunitary equivalencesimilarityinvariant subspaces

MSC classification

Primary: 47A15: Invariant subspaces
Secondary: 47A65: Structure theory 30J05: Inner functions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 98 - 111
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was partially supported by FTC/Portugal through the grant UID/MAT/04459/2019. The second author would like to thank the Center for Mathematical Analysis, Geometry, and Dynamical Systems for their hospitality where the initial research for this paper was done.

References

Benaissa, L. and Guediri, H., Properties of dual Toeplitz operators with applications to Haplitz products on the Hardy space of the polydisk. Taiwanese J. Math. 19(2015), 31–49. https://doi.org/10.11650/tjm.19.2015.4323CrossRefGoogle Scholar
Camara, M. C., Klis-Garlicka, K., Lanucha, B., and Ptak, M., Compressions of the multiplication operator. Preprint.Google Scholar
Camara, M. C., Klis-Garlicka, K., Lanucha, B., and Ptak, M., Fredholmness, invertibility, and kernels of dual truncated Toeplitz operators. arXiv:1912.13266.Google Scholar
Chalendar, I., Fricain, E., and Timotin, D., A survey of some recent results on truncated Toeplitz operators. In: Recent progress on operator theory and approximation in spaces of analytic functions, Contemp. Math., 679, Amer. Math. Soc., Providence, RI, 2016, pp. 59–77. https://doi.org/10.190/conm/679CrossRefGoogle Scholar
Chen, Y., Yu, T., and Zhao, Y. L., Dual Toeplitz operators on orthogonal complement of the harmonic Dirichlet space. Acta Math. Sin. (Engl. Ser.) 33(2017), 383–402. https://doi.org/10.1007/s10114-016-5779-6CrossRefGoogle Scholar
Conway, J. B., The dual of a subnormal operator. J. Operator Theory 5(1981), 195–211.Google Scholar
Ding, X. and Sang, Y., Dual truncated Toeplitz operators. J. Math. Anal. Appl. 461(2018), 929–946. https://doi.org/10.1016/j.jma.2017.12.032CrossRefGoogle Scholar
Ding, X., Sang, Y., and Qin, Y., A theorem of Brown-Halmos type for dual truncated Toeplitz operators. Ann. Funct. Anal. 11(2020), 271–284. https://doi.org/10.1007/s43034-019-00002-7Google Scholar
Ding, X., Sang, Y., and Qin, Y., Dual truncated Toeplitz ${C}^{\ast }$-algebras. Banach J. Math. Anal. 13(2019), 275–292. https://doi.org/10.1215/17358787-2018-0030Google Scholar
Duren, P. L., Theory of ${H}^p$spaces. Pure and Applied Mathematics, 38, Academic Press, New York, 1970.Google Scholar
Garcia, S. R., Mashreghi, J., and Ross, W. T., Introduction to model spaces and their operators. Cambridge Studies in Advanced Mathematics, 148, Cambridge University Press, Cambridge, 2016.CrossRefGoogle Scholar
Garcia, S. R. and Ross, W. T., Recent progress on truncated Toeplitz operators. In: Blaschke products and their applications, Fields Inst. Commun. 65, Springer, New York, 2013, pp. 275–319. https://doi.org/10.1007/978-1-4614-5341-3_15CrossRefGoogle Scholar
Helson, H., Lectures on invariant subspaces. Academic Press, New York-London, 1964.Google Scholar
Hu, Y., Deng, J., Yu, T., Liu, L., and Lu, Y., Reducing subspaces of the dual truncated Toeplitz operator. J. Funct. Spaces (2018), Art. ID 7058401.Google Scholar
Sarason, D., Algebraic properties of truncated Toeplitz operators. Oper. Matrices 1(2007), 491–526. https://doi.org/10.7153/oam-01-29CrossRefGoogle Scholar
Stroethoff, K. and Zheng, D., Algebraic and spectral properties of dual Toeplitz operators. Trans. Amer. Math. Soc. 354(2002), 2495–2520. https://doi.org/10.1090/S0002-9947-02-02954-9CrossRefGoogle Scholar
Li, Y. S. Y. and Ding, X., The commutatant and invariant subspaces for a class of dual truncated Toeplitz operators. Preprint.Google Scholar