Anzeige
Erschienen in:
2014 | OriginalPaper | Buchkapitel
14. Spectral Theory
verfasst von : Joseph Muscat
Erschienen in: Functional Analysis
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
The spectrum of an element of a Banach algebra is defined and shown to be a compact set with a specific formula for its spectral radius. Examples are given of calculating the spectrum of an operator, such as the shift operators, including cases with approximate eigenvalues, continuous and residual spectrum, and adjoint operators. The theory is complete for compact operators, where the main Riesz-Schauder theorem is proved via finite ascents and descents. All this allows for the powerful analytic functional calculus to be defined, culminating with the spectral mapping theorem. Quasinilpotents and the Jacobson radical, as well as the State Space of Banach algebras, lead naturally to the Gelfand transform, taking a particularly pleasant form for commutative algebras.