## Banach Journal of Mathematical Analysis

### Characterizations of asymmetric truncated Toeplitz operators

#### Abstract

The aim of this paper is to investigate asymmetric truncated Toeplitz operators with $L^{2}$-symbols between two different model spaces given by inner functions such that one divides the other. The class of symbols corresponding to the zero operator is described. Asymmetric truncated Toeplitz operators are characterized in terms of operators of rank at most $2$, and the relations with the corresponding symbols are studied.

#### Article information

Source
Banach J. Math. Anal. (2017), 24 pages.

Dates
Accepted: 15 December 2016
First available in Project Euclid: 30 August 2017

https://projecteuclid.org/euclid.bjma/1504080092

Digital Object Identifier
doi:10.1215/17358787-2017-0029

#### Citation

Câmara, Crisina; Jurasik, Joanna; Kliś-Garlicka, Kamila; Ptak, Marek. Characterizations of asymmetric truncated Toeplitz operators. Banach J. Math. Anal., advance publication, 30 August 2017. doi: 10.1215/17358787-2017-0029. https://projecteuclid.org/euclid.bjma/1504080092

#### References

• [1] R. Adamczak, “On the operator norm of random rectangular Toeplitz matrices” in High Dimensional Probability VI, Progr. Probab. 66, Birkhäuser/Springer, Basel, 2013, 247–260.
• [2] F. Andersson and M. Carlsson, On general domain truncated correlation and convolution operators with finite rank, Integral Equations Operator Theory 82 (2015), no. 3, 339–370.
• [3] T. Bäckström, Vandermonde factorization of Toeplitz matrices and applications in filtering and warping, IEEE Trans. Signal Process. 61 (2013), no. 24, 6257–6263.
• [4] A. Baranov, I. Chalendar, E. Fricain, J. Mashreghi, and D. Timotin, Bounded symbols and reproducing kernels thesis for truncated Toeplitz operators, J. Funct. Anal. 259 (2010), no. 10, 2673–2701.
• [5] H. Bercovici, Operator Theory and Arithmetic in $H^{\infty}$, Math. Surveys Monogr. 26, Amer. Math. Soc., Providence, 1988.
• [6] M. C. Câmara, M. T. Malheiro, and J. R. Partington, Model spaces and Toeplitz kernels in reflexive Hardy spaces, Oper. Matrices 10 (2016), no. 1, 127–148.
• [7] M. C. Câmara and J. R. Partington, Spectral properties of truncated Toeplitz operators by equivalence after extension, J. Math. Anal. Appl. 433 (2016), no. 2, 762–784.
• [8] M. C. Câmara and J. R. Partington, Asymmetric truncated Toeplitz operators and Toeplitz operators with matrix symbol, J. Operator Theory 77 (2017), no. 2, 455–479.
• [9] I. Chalendar, P. Gorkin, and J. R. Partington, Inner functions and operator theory, North-West. Eur. J. Math. 1 (2015), 9–28.
• [10] I. Chalendar and D. Timotin, Commutation relations for truncated Toeplitz operators, Oper. Matrices 8 (2014), no. 3, 877–888.
• [11] J. A. Cima, W. T. Ross, and W. R. Wogen, Truncated Toeplitz operators on finite dimensional spaces, Oper. Matrices 2 (2008), no. 3, 357–369.
• [12] S. R. Garcia, J. Mashreghi, and W. T. Ross, Introduction to model spaces and their operators, Cambridge Univ. Press, Cambridge, MA, 2016.
• [13] S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1285–1315.
• [14] S. R. Garcia and M. Putinar, Complex symmetric operators and applications, II, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3913–3931.
• [15] S. R. Garcia and W. T. Ross, Recent Progress on Truncated Toeplitz Operators, Blaschke Products and Their Applications, Fields Inst. Commun. 65, Springer, New York, 2013.
• [16] M. H. Gutknecht, Stable row recurrences for the Pad table and generically superfast lookahead solvers for non-Hermitian Toeplitz systems, Linear Algebra Appl. 188–189 (1993), 351–421.
• [17] G. Heinig and K. Rost, Algebraic Methods for Toeplitz-Like Matrices and Operators, Oper. Theory Adv. Appl. 13, Birkhäuser, Basel, 1984.
• [18] K. Kliś-Garlicka and M. Ptak, C-symmetric operators and reflexivity, Oper. Matrices 9 (2015), no. 1, 225–232.
• [19] D. Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1 (2007), no. 4, 491–526.
• [20] F.-O. Speck, General Wiener–Hopf factorization methods, Res. Notes in Math. 119, Pitman, Boston, 1985.