Boundary values in range spaces of co-analytic truncated Toeplitz operators. (English) Zbl 1253.30087

Given an inner function \(I\) on the unit disk, let \(K_I=H^2\ominus IH^2\) be the model space associated with \(I\) where \(H^2\) is the Hardy space. Let \(P_I\) be the orthogonal projection of \(L^2\), the classical Lebesgue space of the unit circle \(T\) with respect to the normalized arclength measure, onto \(K_I\). Given \(h\in H^2\), let \(A_{\bar h}\) be the (densely-defined) truncated Toeplitz operator on \(K_I\) defined by \(A_{\bar h} f = P_I(\bar h f)\).
In [Acta Math. 124, 191–204 (1970; Zbl 0193.10201); Am. J. Math. 92, 332–342 (1970; Zbl 0197.39202)], P. R. Ahern and D. N. Clark obtained several characterizations for every function in \(K_I\) to have non-tangential limit at a given \(\zeta\in T\).
In this paper the authors restrict \(I\) to Blaschke products and study such boundary behavior even further by generalizing one of the characterizations of Ahern and Clark. The main result states that, for a Blaschke product \(I\) with zeros \(\{\lambda_n\}\) (repeated according to multiplicity) and \(h\in H^2\) for which \(A_{\bar h}\) is bounded on \(K_I\), every function in the range of \(A_{\bar h}\) has a finite non-tangential limit at \(\zeta\in T\) if and only if
\[ \sum_{n\geq 1} |(A_{\bar h}\gamma_n)(\zeta)|^2<\infty,\qquad\text{where}\qquad \gamma_n(z)= {\sqrt{1-|\lambda_n|^2}\over{1-\bar \lambda_n z}} \prod_{k=1}^{n-1} {{z-\lambda_k}\over{1-\bar \lambda_k z}} \]
is the Takenaka-Malquist-Walsh function. A more explicit characterization is given when \(I\) is an interpolating Blaschke product. Several related examples are also provided.


30J10 Blaschke products
30J05 Inner functions of one complex variable
30H10 Hardy spaces
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
Full Text: DOI arXiv