A Primer on Spectral Theory: Universitext

I. Some Reminders of Functional Analysis.- II. Some Classes of Operators.- §1. Finite-Dimensional Operators.- §2. Bounded Linear Operators on a Banach Space.- §3. Bounded Linear Operator on a Hilbert Space.- III. Banach Algebras.- §1. Definition and Examples.- §2. Invertible Elements and Spectrum.- §3. Holomorphic Functional Calculus.- §4. Analytic Properties of the Spectrum.- IV. Representation Theory.- §1. Gelfand Theory for Commutative Banach Algebras.- §2. Representation Theory for Non-Commutative Banach Algebras.- V. Some Applications of Subharmonicity.- §1. Some Elementary Applications.- §2. Spectral Characterizations of Commutative Banach Algebras.- §3. Spectral Characterizations of the Radical.- §4. Spectral Characterizations of Finite-Dimensional Banach Algebras.- §5. Automatic Continuity for Banach Algebra Morphisms.- §6. Elements with Finite Spectrum.- §7. Inessential Elements.- VI. Representation of C?-algebras and the Spectral Theorem.- §1. Banach Algebras with Involution.- §2. C?-algebras.- §3. The Spectral Theorem.- §4. Applications.- VII. An Introduction to Analytic Multifunctions.- §1. Definitions and General Properties.- §2. The Oka-Nishino Theorem and Its Applications.- §3. Distribution of Values of Analytic Multifunctions.- §4. Conclusion.- §1. Subharmonic Functions and Capacity.- §2. Functions of Several Complex Variables.- References.- Author and Subject Index.