The central topic of this book is the spectral theory of bounded
and unbounded self-adjoint operators on Hilbert spaces. After
introducing the necessary prerequisites in measure theory and
functional analysis, the exposition focuses on operator theory and
especially the structure of self-adjoint operators. These can be
viewed as infinite-dimensional analogues of Hermitian matrices; the
infinite-dimensional setting leads to a richer theory which goes
beyond eigenvalues and eigenvectors and studies self-adjoint
operators in the language of spectral measures and the Borel
functional calculus. The main approach to spectral theory adopted
in the book is to present it as the interplay between three main
classes of objects: self-adjoint operators, their spectral measures
and Herglotz functions, which are complex analytic functions
mapping the upper half-plane to itself. Self-adjoint operators
include many important classes of recurrence and differential
operators; the later part of this book is dedicated to two of the
most studied classes, Jacobi operators and one-dimensional
Schrodinger operators. This text is intended as a course textbook
or for independent reading for graduate students and advanced
undergraduates. Prerequisites are linear algebra, a first course in
analysis including metric spaces, and for parts of the book, basic
complex analysis. Necessary results from measure theory and from
the theory of Banach and Hilbert spaces are presented in the first
three chapters of the book. Each chapter concludes with a number of
helpful exercises.
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