Introduction to model spaces and their operators (Q2814156)

The theory of model spaces evolved from the seminal 1949 paper of A.~Beurling, wherein the non-trivial invariant subspaces of the unilateral shift \(Sf=zf\) on \(H^2\) were completely characterized as \(uH^2\), \(u\) being an inner function on the open unit disk. The elements of \(uH^2\) are simple to understand (all \(H^2\) multiples of the inner function \(u\)). By taking orthogonal complements, we come to the invariant subspaces of the backward shift operator \(S^*f=(f-f(0))/z\) on \(H^2\). Such subspaces \(K_u=(uH^2)^{\bot}\) are known as the \textit{model spaces}. In contrast to the subspaces \(uH^2\), the model spaces are much more intricate and troublesome. For instance, it is unclear which functions actually belong to \(K_u\) or what properties these functions have.NEWLINENEWLINEA major breakthrough in the study of model spaces occurred in 1970, with the publication of the paper of Douglas-Shapiro-Shields. Based on a new type of continuation, called a pseudocontinuation, discovered and formalized in earlier papers of Shapiro, they showed that such continuation turned out to be the determining characterization of functions in \(K_u\). From this starting point, the function theoretic aspects of model spaces have truly grown. For instance, the subtle relationship between the boundary behavior of functions in \(K_u\) and the angular derivative of \(u\) became clear through the papers of Ahern and Clark.NEWLINENEWLINEOn the operator theory level, work of Sz.-Nagy and Foias provided the models for certain types of contractions as the compression of the unilateral shift to a model space, which gave reason to broaden the study of model spaces from the scalar case to the vector one. The climax of their research, now regarded as one of the gems of operator theory, became known as the commutant lifting theorem. In 1972 Clark examined unitary rank-one perturbations of the compressed shift and came up with an exact spectral realization of such unitary operator using a family of measures known now as the Aleksandrov-Clark measures. They arise in various areas of analysis and mathematical physics, and enjoy some truly fascinating properties.NEWLINENEWLINESome other areas of mathematics such as approximation theory (classical inequalities for rational functions), harmonic analysis (minimal sequences, Riesz bases), engineering (the Darlington synthesis problem, control theory), and mathematical physics (completeness problems for differential operators) should be mentioned with regard to the model spaces.NEWLINENEWLINEIn preparing a series of lectures within the last decade, the authors came to realize the need for a friendly introduction to model spaces and their operators. This largely self-contained book sets out the basic ideas and quickly takes a newcomer to the frontiers of this area of mathematical analysis. Open questions point to potential areas of future research, offering plenty of inspiration to graduate students wishing to advance further.