# Mini-course on Model Spaces

Model spaces arise as backward-shift invariant subspaces of the Hardy space H2 . A closer inspection reveals that they are of independent interest for their function-theoretic behavior and operator-theoretic connections. For example, functions in model spaces are characterized by a form of generalized analytic continuation and compressions of the unilateral shift to model spaces provide concrete functional models for a certain class of Hilbert-space contractions.

This chapter is a brief and selective overview of the theory of model spaces, an invitation to the novice rather than a refresher for the experienced. These notes parallel the series of three introductory lectures on model spaces given by the author as part of the “Focus Program on Analytic Function Spaces and Their Applications” which was held virtually at the Fields Institute in 2021. The author gave a similar series of lectures in 2013 at the Centre de Recherches Mathematiques. Those presentations later evolved into a survey article [5] ´ (coauthored with W. T. Ross) and a book devoted to model spaces [4] (coauthored with J. Mashreghi and W. T. Ross). Both sources are good starting points for readers looking for more information than this brief sketch provides.

This chapter, which owes much to [4, 5], is part of a volume that covers many related topics, both introductory and advanced. Consequently, we do not reinvent the wheel or step on the toes of the other lecturers. Overall, we present the material in an economical fashion and in roughly the same order and context as they appeared in the lectures. We keep things short and sup press many of the proofs since they can be found elsewhere. In many cases, we present finite-dimensional examples, for we find them the friendliest point of entry for most topics. Although experts may be unimpressed by this approach, we hope that the novice will find it instructive and informative.

In a brief survey such as this, many topics get overlooked. We focus here on model spaces themselves, rather than operators that interact with them, although we consider shift operators and a few other examples in so far as they illuminate the structure of model spaces and their elements. On the other hand, commutant lifting, interpolation problems, Hankel operators, and truncated Toeplitz operators do not appear. Fortunately, these topics and much more are covered in other chapters in this volume. More thorough treatments of model spaces can be found in [1–4, 6–10] and the references therein.

References:

[1] J. Agler and J. E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259 (2003b:47001)

[2] H. Bercovici, Operator theory and arithmetic in H∞, Mathematical Surveys and Monographs, vol. 26, American Mathematical Society, Providence, RI, 1988. MR 954383 (90e:47001)

[3] J. A. Cima and W. T. Ross, The backward shift on the Hardy space, Mathematical Surveys and Monographs, vol. 79, American Mathematical Society, Providence, RI, 2000. MR 1761913 (2002f:47068)

[4] Stephan Ramon Garcia, Javad Mashreghi, and William T. Ross, Introduction to model spaces and their operators, Cambridge Studies in Advanced Mathematics, vol. 148, Cambridge University Press, Cambridge, 2016. MR 3526203

[5] Stephan Ramon Garcia and William T. Ross, Model spaces: a survey, Invariant subspaces of the shift operator, Contemp. Math., vol. 638, Amer. Math. Soc., Providence, RI, 2015, pp. 197–245. MR 3309355

[6] N. Nikolski, Treatise on the shift operator, Springer-Verlag, Berlin, 1986.

[7] , Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002, Hardy, Hankel, and Toeplitz, Translated from the French by Andreas Hartmann. MR 1864396 (2003i:47001a)

[8] , Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002, Model operators and systems, Translated from the French by Andreas Hartmann and revised by the author. MR 1892647(2003i:47001b)

[9] W. T. Ross and H. S. Shapiro, Generalized analytic continuation, University Lecture Series, vol. 25, American Mathematical Society, Providence, RI, 2002. MR 1895624 (2003h:30003)

[10] Bela Sz.-Nagy, Ciprian Foias, Hari Bercovici, and Laszlo Kerchy, Harmonic analysis of operators on Hilbert space, second ed., Universitext, Springer, New York, 2010. MR 2760647 (2012b:47001)