🎵Spectral Theory Unit 4 – Compact Operators: Spectra and Properties

Definition and Basic Concepts

• Compact operators map bounded sets to relatively compact sets in a Banach space
• Characterized by the property that the image of any bounded sequence has a convergent subsequence
• Can be approximated by finite-rank operators in the operator norm topology
• Play a crucial role in the study of integral equations and differential equations
• Compact operators form a closed subspace of the space of bounded linear operators
  • This subspace is an ideal in the algebra of bounded linear operators
• The composition of a compact operator with a bounded operator is compact
• The adjoint of a compact operator is also compact

Types of Compact Operators

• Finite-rank operators are the simplest examples of compact operators
  • An operator has finite rank if its range is finite-dimensional
• Integral operators with continuous kernels are compact on appropriate function spaces
  • For example, the Fredholm integral operator $(Kf)(x) = \int_a^b k(x,y)f(y)dy$ is compact on $C[a,b]$ if $k$ is continuous
• Hilbert-Schmidt operators are compact on Hilbert spaces
  • An operator $T$ is Hilbert-Schmidt if $\sum_{n=1}^\infty \|Te_n\|^2 < \infty$ for some orthonormal basis $\{e_n\}$
• Trace class operators are compact and have well-defined traces
  • An operator $T$ is trace class if $\sum_{n=1}^\infty \langle |T|e_n, e_n \rangle < \infty$ for some orthonormal basis $\{e_n\}$
• Compact perturbations of the identity operator $I + K$, where $K$ is compact

Spectral Properties

• The spectrum of a compact operator consists of eigenvalues and possibly 0
  • Eigenvalues are isolated points in the spectrum
• The set of eigenvalues is at most countable and can only accumulate at 0
• If $\lambda \neq 0$ is an eigenvalue, then its algebraic multiplicity is finite
  • Algebraic multiplicity is the dimension of the generalized eigenspace associated with $\lambda$
• The eigenspaces corresponding to distinct eigenvalues are linearly independent
• For self-adjoint compact operators, the eigenvalues are real, and the eigenvectors form an orthonormal basis
  • This leads to the spectral decomposition of the operator
• The spectral radius of a compact operator equals its operator norm

Eigenvalues and Eigenvectors

• Eigenvalues are complex numbers $\lambda$ for which the operator equation $Tx = \lambda x$ has non-trivial solutions
  • The non-trivial solutions $x$ are called eigenvectors
• Eigenvectors corresponding to distinct eigenvalues are linearly independent
• For self-adjoint compact operators, eigenvectors corresponding to different eigenvalues are orthogonal
• The eigenspace associated with an eigenvalue $\lambda$ is the nullspace of $T - \lambda I$
  • Its dimension is called the geometric multiplicity of $\lambda$
• The algebraic multiplicity of an eigenvalue is the order of the pole of the resolvent operator at that eigenvalue
• For compact operators, the geometric and algebraic multiplicities of non-zero eigenvalues are finite and equal

Fredholm Alternative

• Provides a characterization of the solvability of the equation $Tx = y$ for compact operators $T$
• States that either the homogeneous equation $Tx = 0$ has only the trivial solution, or the non-homogeneous equation $Tx = y$ is solvable if and only if $y$ is orthogonal to the solutions of the adjoint homogeneous equation $T^*x = 0$
• Implies that the dimensions of the nullspaces of $T$ and $T^*$ are equal (Fredholm index)
• Useful in the study of integral equations and boundary value problems
  • Helps determine the existence and uniqueness of solutions
• Generalizes to Fredholm operators, which are compact perturbations of the identity

Applications in Integral Equations

• Integral equations often lead to compact operators on function spaces
  • For example, Fredholm integral equations of the second kind: $u(x) - \lambda \int_a^b k(x,y)u(y)dy = f(x)$
• The compactness of the integral operator allows the application of the Fredholm alternative
  • Determines the existence and uniqueness of solutions
• Eigenvalue problems for integral equations are related to the spectral properties of compact operators
  • Eigenvalues and eigenfunctions of the integral operator correspond to those of the compact operator
• Numerical methods for integral equations often rely on the approximation of compact operators by finite-rank operators
  • Galerkin methods and collocation methods are examples
• Singular integral equations, such as the Cauchy integral equation, also lead to compact operators on appropriate spaces

Relation to Other Operator Classes

• Compact operators are a subset of bounded linear operators
  • Every compact operator is bounded, but not every bounded operator is compact
• Finite-rank operators are a subset of compact operators
  • Every finite-rank operator is compact, but not every compact operator has finite rank
• Hilbert-Schmidt operators and trace class operators are subsets of compact operators
  • Every Hilbert-Schmidt operator is compact, and every trace class operator is Hilbert-Schmidt
• Fredholm operators are compact perturbations of the identity
  • They have similar spectral properties to compact operators
• Compact operators are closely related to completely continuous operators
  • An operator is completely continuous if it maps weakly convergent sequences to strongly convergent sequences
• Compact operators play a role in the study of Banach algebras and C*-algebras
  • The set of compact operators forms a closed ideal in these algebras

Key Theorems and Proofs

• Spectral Theorem for Compact Self-Adjoint Operators
  • States that a compact self-adjoint operator has a countable orthonormal basis of eigenvectors with real eigenvalues
  • Proof relies on the existence of a maximal eigenvalue and the orthogonality of eigenvectors
• Fredholm Alternative Theorem
  • Characterizes the solvability of the equation $Tx = y$ for compact operators $T$
  • Proof uses the closed range theorem and the orthogonality of the nullspaces of $T$ and $T^*$
• Hilbert-Schmidt Theorem
  • Shows that a Hilbert-Schmidt operator is compact and has a square-summable sequence of singular values
  • Proof uses the spectral theorem for compact self-adjoint operators applied to $T^*T$
• Schauder Fixed Point Theorem
  • States that a continuous mapping of a convex, compact subset of a Banach space into itself has a fixed point
  • Proof relies on the Leray-Schauder degree theory and the compactness of the mapping
• Atkinson's Theorem
  • Characterizes Fredholm operators as operators that are invertible modulo compact operators
  • Proof uses the stability of the Fredholm index under compact perturbations