**Asymmetric truncated Toeplitz operators on finite-dimensional spaces**

Let H^{2} be the usual Hardy space, the subspace of L^{2} of normalized Lebesgue measure on T whose negative indexed Fourier coefficients are all zero.

With any nonconstant inner function α we associate the model space K_{α}, defined by K_{α}=H^{2}⊖αH^{2} .

Truncated Toepltz operators are compressions of classical Toeplitz operators to model spaces. We consider their generalizations, the so-called asymmetric truncated Topelitz operators.

Let α, β be two inner functions and let φ∈ L^{2}. An asymmetric truncated Toeplitz operator A_{φ}^{α,β} is the operator from K_{α} into K_{β} given by

A_{φ}^{α,β}f=P_{β}(φf),_{ } f ∈ K_{α},

where P_{β} is the orthogonal projection of L^{2} onto K_{β}.

We present some known properties of asymmetric truncated Toeplitz operators. In particular, we present their characterizations in terms of matrix representations and compare with the characterizations of truncated Toeplitz operators.

The talk is based on joint work with Bartosz Łanucha.

- C. C\^{a}mara, K. Kliś-Garlicka, J. Jurasik, M. Ptak,
*Characterizations of asymmetric truncated Toeplitz operators*, Banach J. Math. Anal. 11 (2017), no. 4, 899–922. - J. A. Cima, W. T. Ross, W. R. Wogen,
*Truncated Toeplitz operators on finite dimensional spaces*, Oper. Matrices 2 (2008), no. 3, 357-369. - B. Łanucha,
*Matrix representations of truncated Toeplitz operators*, J. Math. Anal. Appl. 413 (2014), 430–437. - J. Jurasik, B. Łanucha,
*Asymmetric truncated Toeplitz operators equal to the zero operator*, Ann. Univ. Mariae Curie-Skłodowska Sect. A 70 (2016), no. 2, 51-62. - J. Jurasik, B. Łanucha,
*Asymmetric truncated Toeplitz operators on finite-dimensional spaces*, Oper. Matrices 11 (2017), no. 1, 245-262. - D. Sarason,
*Algebraic properties of truncated Toeplitz operators*, Operators and Matrices 1 (2007), no. 4, 491-526.