Chapter II. Compact Self-Adjoint Operators

<!-- *** Custom HTML *** --> This chapter proves a first version of the Spectral Theorem and shows how it applies to complete the analysis in Sturm’s Theorem of Section I.3. Section 1 introduces compact linear operators from a Hilbert space into itself and characterizes them as the limits in the operator norm topology of the linear operators of finite rank. The adjoint of a compact operator is compact. Section 2 proves the Spectral Theorem for compact self-adjoint operators on a Hilbert space, showing that such operators have orthonormal bases of eigenvectors with eigenvalues tending to 0. Section 3 establishes two versions of the Hilbert–Schmidt Theorem concerning self-adjoint integral operators with a square-integrable kernel. The abstract version gives an $L^{2}$ expansion of the members of the image of the operator in terms of eigenfunctions, and the concrete version, valid when the kernel is continuous and the space is compact metric, proves that the eigenfunctions are continuous and the expansion in terms of eigenfunctions is uniformly convergent. Section 4 introduces unitary operators on a Hilbert space, establishing the equivalence of three conditions that may be used to define them. Section 5 studies compact linear operators on an abstract Hilbert space, with special attention to two kinds—the Hilbert–Schmidt operators and the operators of trace class. All three sets of operators—compact, Hilbert–Schmidt, and trace-class—are ideals in the algebra of all bounded linear operators and are closed under the operation of adjoint. Trace-class implies Hilbert–Schmidt, which implies compact. The product of two Hilbert–Schmidt operators is of trace class.