🎭Operator Theory Unit 3 – Compact Operators

Definition and Basic Properties

• Compact operators map bounded sets to relatively compact sets in a Banach space
• Continuous linear operators between Banach spaces with the property that the image of any bounded set is relatively compact
• Relatively compact means the closure of the image is compact
• Compact operators form a closed subspace of the space of bounded linear operators denoted by $\mathcal{K}(X,Y)$
• The composition of a compact operator with a bounded linear operator is compact
• The sum of two compact operators is compact
• The limit of a sequence of compact operators in the operator norm is compact

Types of Compact Operators

• Finite-rank operators are compact and have a finite-dimensional range
  • Examples include projection operators onto finite-dimensional subspaces and operators with a matrix representation
• Integral operators with continuous kernels are compact on spaces of continuous functions
  • Volterra operator $(\mathcal{V}f)(x) = \int_0^x K(x,y)f(y)dy$ is compact on $C[0,1]$
• Hilbert-Schmidt operators on a Hilbert space are compact and characterized by the square summability of their singular values
  • Nuclear operators are a subclass with summable singular values
• Compact perturbations of the identity operator $I+K$ where $K$ is compact

Spectral Theory of Compact Operators

• Compact operators have a discrete spectrum consisting of eigenvalues with finite multiplicities
• The eigenvalues of a compact operator can accumulate only at zero
• Eigenvectors corresponding to distinct eigenvalues are orthogonal in a Hilbert space setting
• The spectral theorem for compact self-adjoint operators provides a decomposition into a sum of rank-one operators
  • $T = \sum_{n=1}^\infty \lambda_n \langle \cdot, e_n \rangle e_n$ where $\lambda_n$ are eigenvalues and $e_n$ are eigenvectors
• The singular value decomposition for compact operators expresses them as a sum of rank-one operators with singular values as coefficients

Fredholm Alternative

• Fundamental result characterizing the solvability of equations involving compact operators
• For a compact operator $K$ and the equation $(I-K)x=y$, exactly one of the following holds:
  1. The equation has a unique solution for every $y$
  2. The homogeneous equation $(I-K)x=0$ has non-trivial solutions
• The dimensions of the kernel and cokernel of $I-K$ are finite and equal (Fredholm index)
• Provides conditions for existence and uniqueness of solutions to integral equations and boundary value problems

Approximation by Finite-Rank Operators

• Compact operators can be approximated in the operator norm by finite-rank operators
• The singular value decomposition allows truncation to obtain finite-rank approximations
  • $T_n = \sum_{k=1}^n \sigma_k \langle \cdot, v_k \rangle u_k$ converges to $T$ in operator norm as $n \to \infty$
• Approximation numbers quantify the rate of approximation by finite-rank operators
  • $a_n(T) = \inf\{\|T-F\| : \text{rank}(F) < n\}$ decay to zero for compact $T$
• Relates to numerical methods for solving integral equations and PDEs

Applications in Integral Equations

• Many integral equations can be formulated as operator equations with compact operators
  • Fredholm integral equations of the second kind: $(I-K)f=g$ with $K$ compact
  • Volterra integral equations: $f(x) - \int_0^x K(x,y)f(y)dy = g(x)$ with compact Volterra operator
• Spectral theory and approximation results for compact operators are used to analyze and solve integral equations
• Numerical methods based on finite-rank approximations (Galerkin, collocation) are effective for compact integral operators
• Applications in physics, engineering, and other fields involving integral equations

Relationship to Other Operator Classes

• Compact operators are a subset of bounded linear operators
• Every compact operator is completely continuous (maps weakly convergent sequences to strongly convergent sequences)
• Compact operators are strictly singular (not an isomorphism on any infinite-dimensional subspace)
• Relationship to Schatten classes of operators defined by summability of singular values
  • Hilbert-Schmidt operators ($p=2$) and trace-class operators ($p=1$) are compact
• Compact operators play a role in the theory of Fredholm operators and index theory

Key Theorems and Proofs

• Spectral theorem for compact self-adjoint operators
  • Proof uses the min-max principle and orthogonality of eigenvectors
• Fredholm alternative theorem
  • Proof relies on the compactness of the operator and properties of the Fredholm index
• Approximation of compact operators by finite-rank operators
  • Proof uses the singular value decomposition and properties of singular values
• Schauder fixed point theorem for compact operators in Banach spaces
  • Proves the existence of fixed points for compact operators on closed convex sets
• Hilbert-Schmidt theorem on integral operators with $L^2$ kernels
  • Characterizes Hilbert-Schmidt operators using the $L^2$ norm of the kernel function