An introduction to operators on the Hardy-Hilbert space, Volume 237
The subject of this book is operator theory on the Hardy space H2, also called the Hardy-Hilbert space. This is a popular area, partially because the Hardy-Hilbert space is the most natural setting for operator theory. A reader who masters the material covered in this book will have acquired a firm foundation for the study of all spaces of analytic functions and of operators on them. The goal is to provide an elementary and engaging introduction to this subject that will be readable by everyone who has understood introductory courses in complex analysis and in functional analysis. The exposition, blending techniques from "soft"and "hard" analysis, is intended to be as clear and instructive as possible. Many of the proofs are very elegant. This book evolved from a graduate course that was taught at the University of Toronto. It should prove suitable as a textbook for beginning graduate students, or even for well-prepared advanced undergraduates, as well as for independent study. There are numerous exercises at the end of each chapter, along with a brief guide for further study which includes references to applications to topics in engineering.
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The Unilateral Shift and Factorization of Functions
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a_2 a_3 a0 a_i a_2 adjoint Algebraic Analysis analytic functions analytic Toeplitz operator Banach space bilateral Blaschke product Borel bounded linear operator bounded operator coanalytic commutes complex numbers composition operator constant function contained continuous function converges Corollary defined Definition denote eigenvalue equation ess ran Exercise exists f(eie factors finite number finite rank fixed point follows Fourier coefficients function g function in H2 G H2 Hankel matrix Hankel operator Hardy-Hilbert space Hence Hilbert space implies inequality inner function invariant subspace invertible lattice Lebesgue measure Lemma log+ modulus multiplicity natural number nonnegative integers nonzero norm Note Number Theory numerical range orthonormal basis outer function polynomial previous theorem Proof Prove rational function real number Recall representation result Riesz self-adjoint sequence singular inner function spectral radius spectrum standard basis Suppose theorem Theorem Toeplitz matrix Toeplitz operator unilateral shift vector