Truncated Toeplitz operators on model spaces have been formally introduced by

D. Sarason in 2007, although some special cases have long ago appeared in the literature, most notably as model operators for completely nonunitary contractions with defect numbers one and for their commutant. This new area of study has been recently very active and many open questions posed by Sarason have now been solved, though some of them remain mysterious. In these lectures, I will make an introduction to this topic which lies at the intersection with model spaces and Toeplitz operators. During all this presentation, I will do a parallel with the theory of Toeplitz operators.

In the first lecture, after an introduction of the main objects (Hardy spaces, model spaces, Toeplitz operators), I will introduce the truncated Toeplitz operators (TTO) and discuss some basics properties. In particular, we will see that, contrary to the case of standard Toeplitz operators, the TTO’s have not a unique symbol, which yields to many difficulties. In that sense, the truncated Toeplitz operators behave more like Hankel operators than Toeplitz operators. Then, I will try to motivate the study of TTO by discussing a bit the model theory of Sz.-Nagy—Foias and the Nevanlinna—Pick interpolation problem through the approach of the commutant lifting theorem.

In the second lecture, I will discuss several characterizations of TTO. A well-known result of Brown-Halmos says that a bounded operator T on the Hardy space H² is a Toeplitz operator if and only if T=STS^*, where S is the shift operator. We give analogues of this result for TTO, and show that the set of bounded TTO is closed in the weak operator topology. We will also discuss the connection between TTO and another interesting class of operators which received recently a lot of attention, the so-called complex symmetric operators. Contrary to the case of Toeplitz operators, we will show that TTO are complex symmetric. Then, I will end this second lecture by explaining the link between TTO and Carleson measures for model spaces.

In the third and last lecture, I will present some results concerning the spectrum of TTO. We will show that for bounded analytic symbols which are continuous up to the boundary, we have a spectral mapping theorem. Then I will discuss the problem of finite rank TTO. It is well known that there are no finite rank Toeplitz operators (in fact even compact), except the trivial one. The situation for TTO is dramatically different and we will exibit all rank-one TTO which is a large class of operators, and as we will see they are connections with Carathéodory points and reproducing kernels on the boundary for model spaces. I will end this mini-course by discussing in details the problem of boundedness of the symbol, which is the following : does any bounded TTO possess a bounded symbol?

List of some references :

- A. Baranov, I. Chalendar, E. Fricain, J. Mashreghi and D. Timotin, Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators, J. Funct. Anal., 259(10):2673--2701, 2010.

- R. Bessonov, A. Baranov and V. Kapustin, Symbols of truncated Toeplitz operators, J. Funct. Anal., 261(12): 3437--3456, 2011.

- R. Bessonov, {Truncated Toeplitz operators of finite rank}, Proc. Amer. Math. Soc., 142(4):1301--1313, 2014.

- A. Brown and P.R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math., 213:89--102, 1963/64.

- S. Garcia, J. Mashreghi and W.T. Ross, Introduction to model spaces and their operators, volume 148 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge 2016.

- S.R. Garcia and W.T. Ross, Recent progress on truncated Toeplitz operators, In Blaschke products and their applications, volume 65 of Filed Insti. Commun., pages 275--319, Springer Verlag,, New York, 2013.

- D. Sarason, Algebraic properties of truncated Toeplitz operators}}, Oper. Matrices, 1(4):491--526, 2007.