The book is intended as a text for a one-semester graduate course in operator theory to be
taught from scratch'' not as a sequel to a functional analysis course with the basics of the
spectral theory of linear operators taking the center stage. The book consists of six chapters
and appendix with the material flowing from the fundamentals of abstract spaces (metric
vector normed vector and inner product) the Banach Fixed-Point Theorem and its applications
such as Picard's Existence and Uniqueness Theorem through the basics of linear operators two
of the three fundamental principles (the Uniform Boundedness Principle and the Open Mapping
Theorem and its equivalents: the Inverse Mapping and Closed Graph Theorems) to the elements of
the spectral theory including Gelfand's Spectral Radius Theorem and the Spectral Theorem for
Compact Self-Adjoint Operators and its applications such as the celebrated Lyapunov Stability
Theorem. Conceived as a text to be used in a classroom the book constantly calls for the
student's actively mastering the knowledge of the subject matter. There are problems at the end
of each chapter starting with Chapter 2 and totaling at 150. Many important statements are
given as problems and frequently referred to in the main body. There are also 432 Exercises
throughout the text including Chapter 1 and the Appendix which require of the student to
prove or verify a statement or an example fill in certain details in a proof or provide an
intermediate step or a counterexample. They are also an inherent part of the material. More
difficult problems are marked with an asterisk many problems and exercises are supplied with
existential'' hints. The book is generous on Examples and contains numerous Remarks
accompanying definitions examples and statements to discuss certain subtleties raise
questions on whether the converse assertions are true whenever appropriate or whether the
conditions are essential. With carefully chosen material proper attention given to
applications and plenty of examples problems and exercises this well-designed text is ideal
for a one-semester Master's level graduate course in operator theory with emphasis on spectral
theory for students majoring in mathematics physics computer science and engineering.
ContentsPreface PreliminariesMetric SpacesVector Spaces Normed Vector Spaces and Banach
SpacesLinear OperatorsElements of Spectral Theory in a Banach Space SettingElements of Spectral
Theory in a Hilbert Space SettingAppendix: The Axiom of Choice and Equivalents
BibliographyIndex
