Spectral Theory

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.