Elements of Operator Theory (PDF)

{\it Elements of Operatory Theory} is aimed at graduate students
as well as a new generation of mathematicians and scientists who need
to apply operator theory to their field. Written in a user-friendly,
motivating style, fundamental topics are presented in a systematic
fashion, i.e., set theory, algebraic structures, topological
structures, Banach spaces, Hilbert spaces, culminating with the
Spectral Theorem, one of the landmarks in the theory of operators on
Hilbert spaces. The exposition is concept-driven and as much as
possible avoids the formula-computational approach.
Key features of this largely self-contained work include:
* required background material to each chapter
* fully rigorous proofs, over 300 of them, are specially tailored to
the presentation and some are new
* more than 100 examples and, in several cases, interesting
counterexamples that demonstrate the frontiers of an important theorem
* over 300 problems, many with hints
* both problems and examples underscore further auxiliary results and
extensions of the main theory; in this non-traditional framework, the
reader is challenged and has a chance to prove the principal theorems
anew
This work is an excellent text for the classroom as well as a
self-study resource for researchers. Prerequisites include an
introduction to analysis and to functions of a complex variable, which
most first-year graduate students in mathematics, engineering, or
another formal science have already acquired. Measure theory and
integration theory are required only for the last section of the final
chapter.