We consider three problems connected with coinvariant subspaces of the backward shift operator in Hardy spaces H p :-properties of truncated Toeplitz operators; -Carleson-type embedding theorems for the coinvariant subspaces; -factorizations of pseudocontinuable functions from H 1 . These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by Donald Sarason.

We characterize even measures µ = w dx + µs on the real line R with finite entropy integral R log w(t) 1+t 2 dt > −∞ in terms of 2 × 2 Hamiltonians generated by µ in the sense of the inverse spectral theory. As a corollary, we obtain criterion for spectral measure of Krein string to have converging logarithmic integral.2010 Mathematics Subject Classification. 42C05, 34L40, 34A55.The aim of this paper is to complement assertions (a), (b) in Krein-Wiener theorem with a necessary and sufficient condition similar to condition (c) in Szegő theorem. Instead of recurrence relation Φ n+1 (z) = zΦ n (z) −ᾱ n Φ * n (z), we will consider canonical Hamiltonian system JM ′ = zHM which naturally appears from µ via Krein -de Branges spectral theory.Consider the Cauchy problem for a canonical Hamiltonian system on the half-axis R + = [0, +∞),( 1.2) and real-valued on R, so v(x) = 0 again and the logarithmic integral diverges. More details on Theorem 2 can be found in Section 6.Historical remarks. Except for Krein-Wiener theorem, all previously known results on Szegő theorem in the continuous setting were proved for the so-called Krein systems, i.e., differential systems that appear as a result of "orthogonalization process with continuous parameter" invented by Krein in [19]. Krein systems with locally summable coefficients can be reduced to the canonical Hamiltonian systems with absolutely continuous Hamiltonians H (see, e.g, [1] for this reduction in the diagonal case). The class of Hamiltonians considered in Theorem 1 is considerably wider. Krein himself formulated a restricted version of Szegő theorem for Krein systems in [19]. In [5], the second author of this paper characterized Krein systems with coefficients from a Stummel class whose spectral measures belong to Sz(R). In [25], Teplyaev fixed an error in the original formulation of Szegő theorem in [19]. The reader interested in Szegő theory for Krein systems can find further information in monograph [6]. In [15] and [16], Killip and Simon proved analogs of Szegő theorem for Jacobi matrices and Schrödinger operators. See also the work [20] by Nazarov, Peherstorfer, Volberg, and Yuditskii for a closely related subject of sum rules for Jacobi matrices.9

We prove existence of modified wave operators for one-dimensional Dirac operators whose spectral measures belong to the Szegő class on the real line.2010 Mathematics Subject Classification. 34L40, 42C05.