6th Summer Workshop on Operator Theory

9th - 13th July 2018 Kraków

Joanna Jurasik

Asymmetric truncated Toeplitz operators on finite-dimensional spaces

Let H2 be the usual Hardy space, the subspace of L2 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 Kα, defined by Kα=H2⊖αH2 .
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∈ L2. An asymmetric truncated Toeplitz operator Aφα,β is the operator from Kα into Kβ given by

Aφα,βf=Pβ(φf),  f ∈ Kα,

where Pβ is the orthogonal projection of L2 onto Kβ.

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.

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