An Introduction to Operators on the Hardy-Hilbert Space

This book offers an elementary and engaging introduction to operator theory on the Hardy-Hilbert space. It provides a firm foundation for the study of all spaces of analytic functions and of the operators on them.

The great mathematician G. H. Hardy told us that Beauty is the ?rst test: there is no permanent place in the world for ugly mathematics (see [24, p. 85]). It is clear why Hardy loved complex analysis: it is a very beautiful partofclassicalmathematics. ThetheoryofHilbertspacesandofoperatorson themisalmostasclassicalandisperhapsasbeautifulascomplexanalysis. The studyoftheHardyHilbertspace(aHilbertspacewhoseelementsareanalytic functions), and of operators on that space, combines these two subjects. The interplay produces a number of extraordinarily elegant results. For example, very elementary concepts from Hilbert space provide simple proofs of the Poisson integral (Theorem 1. 1. 21 below) and Cauchy integral (Theorem 1. 1. 19) formulas. The fundamental theorem about zeros of fu- tions in the HardyHilbert space (Corollary 2. 4. 10) is the central ingredient of a beautiful proof that every continuous function on [0,1] can be uniformly approximated by polynomials with prime exponents (Corollary 2. 5. 3). The HardyHilbert space context is necessary to understand the structure of the invariant subspaces of the unilateral shift (Theorem 2. 2. 12). Conversely, pr- erties of the unilateral shift operator are useful in obtaining results on f- torizations of analytic functions (e. g. , Theorem 2. 3. 4) and on other aspects of analytic functions (e. g. , Theorem 2. 3. 3). The study of Toeplitz operators on the HardyHilbert space is the most natural way of deriving many of the properties of classical Toeplitz mat- ces (e. g. , Theorem 3. 3.

More elementary than the competition Contains clear and beautiful proofs Has been class-tested a number of times Includes supplementary material: sn.pub/extras

Inhalt
The Unilateral Shift and Factorization of Functions.- Toeplitz Operators.- Hankel Operators.- Composition Operators.- Further Reading.