Truncated Toeplitz operators (Q6613350)

The following is based on the author's introduction.\N\NThis is a set of lecture notes to accompany a series of talks given as part of the Fields Institute session on \textit{Truncated Toeplitz Operators} from July--December 2021\N\NThe truncated Toeplitz operators live on the model spaces \(K_\Theta\), which are the closed invariant subspaces for the backward shift operator \(S^*\) acting on the Hardy space \(H^2\). Given a model space \(K_\Theta\) and a function \(\phi\in L_2(\mathbb T)\), the truncated Toeplitz operator \(A_\phi^\Theta\) is defined on a dense subspace of \(K_\Theta\) as the compression to \(K_\Theta\) of multiplication by \(\phi\). The function \(\phi\) is then called a symbol of this operator.\N\NAfter the preliminary Sect.~6.2 with generalities about Hardy spaces, model spaces, one-component inner functions, Toeplitz and Hankel operators, and Carleson measures, truncated Toeplitz operators are introduced in Sect. 6.3. Then, in Sect. 6.4, it is explained how truncated Toeplitz operators appear in several natural problems in operator theory and complex function theory. In Sect. 6.5, the non-uniqueness of symbols of truncated Toeplitz operators is discussed in detail. In Sects. 6.6 and 6.7, several useful characterization of truncated Toeplitz operators are given and connections with the so-called complex symmetric operators are discussed. In Sect. 6.8, norm estimates are provided, while in Sect. 6.9, a complete description of the spectrum of truncated Toeplitz operators with \(H^\infty\) symbols is obtained. Finite rank and compact truncated Toeplitz operators are treated in Sects. 6.10 and 6.11, respectively. Finally, the last Sect. 6.12 is devoted to the open problem of the existence of bounded symbols for a given truncated Toeplitz operator.\N\NFor the entire collection see [Zbl 1527.30001].