Abstract
This paper is concerned with asymmetric truncated Toeplitz operators, which are compressions of multiplication operators acting between two model spaces. We give necessary and sufficient conditions so that the product of two asymmetric truncated Toeplitz operators is also an asymmetric truncated Toeplitz operator. Then we define analogue classes to the Sedlock classes, which consist on the algebras of truncated Toeplitz operators, and deal with the case of two inner functions such that one divides the other. Some examples of the product of rank-one asymmetric truncated Toeplitz operators involving inner functions are also given.
Similar content being viewed by others
References
Ahern, P.R., Clark, D.N.: Radial limits and invariant subspaces. Amer. J. Math. 92, 332–342 (1970)
Baranov, A., Bessonov, R., Kapustin, V.: Symbols of truncated Toeplitz operators. J. Funct. Anal. 261, 3437–3456 (2011)
Câmara, C., Jurasik, J., K-Garlicka, K., Ptak, M.: Characterizations of asymmetric truncated Toeplitz operators. Banach J. Math. Anal 11(4), 899–922 (2017)
Câmara, C., Partington, J.R.: Spectral properties of truncated Toeplitz operators by equivalence after extension. J. Math. Anal. Appl. 433(2), 762–784 (2016)
Câmara, M.C., Partington, J.R.: Asymmetric truncated Toeplitz operators and Toeplitz operators with matrix symbol. J. Oper. Theory 77(2), 455–479 (2017)
Chalendar, I., Timotin, D.: Commutation relations for truncated Toeplitz operators. Oper. Matrices 8(3), 877–888 (2014)
Cima, J.A., Garcia, S.R., Ross, W.T., Wogen, W.R.: Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity. Indiana Univ. Math. J 59(2), 595–620 (2010)
Garcia, S.R., Mashreghi, J.E., Ross, W.: Introduction to Model Spaces and their Operators. Cambridge Studies in Advanced mathematics, University Press, Cambridge (2016)
Jurasik, J., Lanucha, B.: Asymmetric truncated Toeplitz operators equal to the zero operator. Ann. Univ. Mariae Curie Sklodowska Sect. A 70(2), 51–62 (2016)
Jurasik, J., Lanucha, B.: Asymmetric truncated Toeplitz operators on finite-dimensional spaces. Oper. Matrices 11(1), 245–262 (2017)
Lanucha, B.: Asymmetric truncated Toeplitz operators on finite-dimensional spaces II, in press, arXiv:1611.01231
Lanucha, B.: On rank-one asymmetric truncated Toeplitz operators on finite-dimensional model spaces. J. Math. Anal. Appl. 454, 961–980 (2017)
Sarason, D.: Algebraic properties of truncated Toeplitz operators. Oper. Matrices 1(4), 491–526 (2007)
Sedlock, N.A.: Algebras of truncated Toeplitz operators. Oper. Matrices 5, 309–326 (2011)
Yagoub, A., Zarrabi, M.: Semigroups of truncated Toeplitz operators. Oper. Matrices 12(3), 603–618 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yuri Karlovich.
Rights and permissions
About this article
Cite this article
Yagoub, A. Products of asymmetric truncated Toeplitz operators. Adv. Oper. Theory 5, 233–247 (2020). https://doi.org/10.1007/s43036-019-00014-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s43036-019-00014-z
Keywords
- Model spaces
- Truncated Toeplitz operator
- Asymmetric truncated Toeplitz operator
- Product of asymmetric truncated Toeplitz operators