1. Hilbert Space Operators: Adjoint Operators. Self-Adjoint Operators. Normal Operators. Projections and Idempotents. Invariant and Reducing Subspaces. Eigenvalues and Eigenvectors. Compact Operators. The Spectral Theorem for Compact Self-Adjoint Operators. The Spectral Theorem for Compact Normal Operators.
2. Banach Space Operators: Adjoint Operators. Compact Operators. Differentiation Operators. Integral Operators.
3. Banach Algebras: Ideals and Quotients. Invertible Elements and Maximal Ideals. Algebra Homomorphism. Spectrum. The Spectral Mapping Theorem. Spectral Radius. The Spectrum of an Operator. Left and Right Spectra. Approximate Point Spectrum. Numerical Range. Numerical Radius. Abelian Banach Algebra.