Includes sections on the spectral resolution and spectral representation of self adjoint operators, invariant subspaces, strongly continuous one-parameter semigroups, the index of operators, the trace formula of Lidskii, the Fredholm determinant, and more.
Assumes prior knowledge of Naive set theory, linear algebra, point set topology, basic complex variable, and real variables.
Includes an appendix on the Riesz representation theorem.
Foreword. Linear Spaces. Linear Maps. The Hahn-Banach Theorem. Applications of the Hahn-Banach Theorem. Normed Linear Spaces. Hilbert Space. Applications of Hilbert Space Results. Duals of Normed Linear Space. Applications of Duality. Weak Convergence. Applications of Weak Convergence. The Weak and Weak Topologies. Locally Convex Topologies and the Krein-Milman Theorem. Examples of Convex Sets and their Extreme Points. Bounded Linear Maps. Examples of Bounded Linear Maps. Banach Algebras and their Elementary Spectral Theory. Gelfand's Theory of Commutative Banach Algebras. Applications of Gelfand's Theory of Commutative Banach Algebras. Examples of Operators and their Spectra. Compact Maps. Examples of Compact Operators. Positive Compact Operators. Fredholm's Theory of Integral Equations. Invariant Subspaces. Harmonic Analysis on a Halfline. Index Theory. Compact Symmetric Operators in Hilbert Space. Examples of Compact Symmetric Operators. Trace Class and Trace Formula. Spectral Theory of Symmetric, Normal and Unitary Operators. Spectral Theory of Self-Adjoint Operators. Examples of Self-Adjoint Operators. Semigroups of Operators. Groups of Unitary Operators. Examples of Strongly Continuous Semigroups. Scattering Theory. A Theorem of Beurling. Appendix A: The Riesz-Kakutani Representation Theorem. Appendix B: Theory of Distributions. Appendix C: Zorn's Lemma. Author Index. Subject Index.