This book grew out of lectures on spectral theory which the author gave at the Scuola. Normale Superiore di Pisa in 1985 and at the Universite Laval in 1987. Its aim is to provide a rather quick introduction to the new techniques of subhar­ monic functions and analytic multifunctions in spectral theory. Of course there are many paths which enter the large forest of spectral theory: we chose to follow those of subharmonicity and several complex variables mainly because they have been discovered only recently and are not yet much frequented. In our book Pro­ pri6t6$ $pectrale$ de$ algebre$ de Banach, Berlin, 1979, we made a first incursion, a rather technical one, into these newly discovered areas. Since that time the bushes and the thorns have been cut, so the walk is more agreeable and we can go even further. In order to understand the evolution of spectral theory from its very beginnings, it is advisable to have a look at the following books: Jean Dieudonne, Hutory of Functional AnaIY$u, Amsterdam, 1981; Antonie Frans Monna. , Functional AnaIY$i$ in Hutorical Per$pective, Utrecht, 1973; and Frederic Riesz & Bela SzOkefalvi-Nagy, Le~on$ d'anaIY$e fonctionnelle, Budapest, 1952. However the picture has changed since these three excellent books were written. Readers may convince themselves of this by comparing the classical textbooks of Frans Rellich, Perturbation Theory, New York, 1969, and Tosio Kato, Perturbation Theory for Linear Operator$, Berlin, 1966, with the present work.

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.