This book presents the basic tools of modern analysis within the context
of what might be called the fundamental problem of operator theory: to
c- culate spectra of speci?c operators on in?nite-dimensional spaces,
especially operators on Hilbert spaces. The tools are diverse, and they
provide the basis for more re?ned methods that allow one to approach
problems that go well beyond the computation of spectra; the
mathematical foundations of quantum physics, noncommutative K-theory,
and the classi?cation of sim- ? ple C -algebras being three areas of
current research activity that require mastery of the material presented
here. The notion of spectrum of an operator is based on the more
abstract notion of the spectrum of an element of a complex Banach
algebra. - ter working out these fundamentals we turn to more concrete
problems of computing spectra of operators of various types. For normal
operators, this amounts to a treatment of the spectral theorem. Integral
operators require 2 the development of the Riesz theory of compact
operators and the ideal L of Hilbert-Schmidt operators. Toeplitz
operators require several important tools; in order to calculate the
spectra of Toeplitz operators with continuous symbol one needs to know
the theory of Fredholm operators and index, the ? structure of the
Toeplitz C -algebra and its connection with the topology of curves, and
the index theorem for continuous symbols.