🧐Functional Analysis Unit 7 – Compact Operators: Properties and Analysis

Definition and Basic Concepts

• Compact operators map bounded sets to relatively compact sets in a Banach space
• Let $X$ and $Y$ be Banach spaces, an operator $T: X \to Y$ is compact if for every bounded sequence $(x_n)$ in $X$, the sequence $(Tx_n)$ has a convergent subsequence in $Y$
• Compactness is a generalization of the notion of a continuous function on a compact metric space
• Every compact operator is bounded and continuous, but the converse is not always true
• The set of compact operators forms a closed subspace of the space of bounded operators with the operator norm
  • This subspace is denoted as $\mathcal{K}(X,Y)$
• Compact operators can be approximated by finite-rank operators in the operator norm
• The composition of a compact operator with a bounded operator is compact

Types of Compact Operators

• Finite-rank operators are compact, they map $X$ into a finite-dimensional subspace of $Y$
  • Example: Let $\{e_1, \ldots, e_n\}$ be a basis for a subspace $M \subset Y$, and define $T: X \to Y$ by $Tx = \sum_{i=1}^n f_i(x)e_i$, where $f_i \in X^*$
• Integral operators with continuous kernels are compact
  • For example, the Fredholm integral operator $T: L^2[a,b] \to L^2[a,b]$ defined by $(Tf)(x) = \int_a^b K(x,y)f(y)dy$, where $K$ is continuous on $[a,b] \times [a,b]$
• Hilbert-Schmidt operators are compact, they have a finite Hilbert-Schmidt norm
  • The Hilbert-Schmidt norm is defined as $\|T\|_{HS} = (\sum_{i,j} |\langle Te_i, f_j \rangle|^2)^{1/2}$, where $\{e_i\}$ and $\{f_j\}$ are orthonormal bases for $X$ and $Y$
• Compact operators on Hilbert spaces can be characterized using the singular value decomposition
• Compact perturbations of the identity, i.e., operators of the form $I + K$, where $K$ is compact

Properties of Compact Operators

• The set of compact operators $\mathcal{K}(X,Y)$ is a closed subspace of the space of bounded operators $\mathcal{B}(X,Y)$
• $\mathcal{K}(X,Y)$ is a two-sided ideal in $\mathcal{B}(X,Y)$, i.e., if $T \in \mathcal{K}(X,Y)$ and $S \in \mathcal{B}(X,Y)$, then $ST$ and $TS$ are in $\mathcal{K}(X,Y)$
• If $X$ is a Banach space, then $\mathcal{K}(X)$ (the space of compact operators from $X$ to itself) is a closed subalgebra of $\mathcal{B}(X)$
• The limit of a sequence of compact operators (in the operator norm) is compact
• Compact operators map weakly convergent sequences to strongly convergent sequences
  • If $x_n \rightharpoonup x$ in $X$ and $T$ is compact, then $Tx_n \to Tx$ in $Y$
• The adjoint of a compact operator is compact
• If $T$ is compact and injective, then $T^{-1}$ is bounded on the range of $T$

Spectral Theory for Compact Operators

• The spectrum of a compact operator consists of eigenvalues and possibly 0
  • Eigenvalues are elements $\lambda \in \mathbb{C}$ such that $Tx = \lambda x$ for some nonzero $x \in X$
• Each nonzero eigenvalue has finite multiplicity (the dimension of the corresponding eigenspace)
• If $X$ is infinite-dimensional, then 0 is always in the spectrum of a compact operator
• Eigenvalues can accumulate only at 0, i.e., for every $\varepsilon > 0$, there are at most finitely many eigenvalues with absolute value greater than $\varepsilon$
• Compact self-adjoint operators on a Hilbert space have a spectral decomposition
  • $T = \sum_{n=1}^\infty \lambda_n \langle \cdot, e_n \rangle e_n$, where $\{e_n\}$ is an orthonormal basis of eigenvectors and $\lambda_n$ are the corresponding eigenvalues
• The spectral theorem for compact operators is a powerful tool for analyzing their properties and behavior

Examples and Applications

• Integral equations, such as Fredholm and Volterra equations, often involve compact integral operators
  • For example, the Fredholm equation of the second kind: $u(x) - \lambda \int_a^b K(x,y)u(y)dy = f(x)$
• Compact operators arise in the study of differential equations, particularly in the theory of elliptic PDEs
  • For instance, the solution operator for the Dirichlet problem on a bounded domain is compact
• Compact operators are used in the approximation theory, as they can be approximated by finite-rank operators
  • This is the basis for numerical methods like the Galerkin method and the finite element method
• In quantum mechanics, compact operators are used to model observables with discrete spectra
• Compact operators play a role in the theory of inverse problems, as they often appear as the forward operators in such problems
  • For example, in computerized tomography, the Radon transform is a compact operator

Relationship to Other Operator Classes

• Every compact operator is bounded, but not every bounded operator is compact
  • For example, the identity operator on an infinite-dimensional space is bounded but not compact
• Compact operators form a strict subset of the class of completely continuous operators (operators that map weakly convergent sequences to strongly convergent sequences)
• Compact operators are a generalization of finite-rank operators
  • Every finite-rank operator is compact, but not every compact operator is finite-rank
• In Hilbert spaces, the class of Hilbert-Schmidt operators contains the class of compact operators
  • Every compact operator is Hilbert-Schmidt, but not every Hilbert-Schmidt operator is compact
• Trace-class operators, which have a finite trace norm, form a subset of the compact operators in Hilbert spaces
• The Fredholm alternative theorem relates compact operators to Fredholm operators (operators with a finite-dimensional kernel and cokernel)

Theorems and Proofs

• Theorem: Let $X$ and $Y$ be Banach spaces. An operator $T: X \to Y$ is compact if and only if for every bounded sequence $(x_n)$ in $X$, the sequence $(Tx_n)$ has a convergent subsequence in $Y$.
  • Proof: (Sketch) The forward direction follows from the definition of compactness. For the converse, consider the unit ball in $X$ and use the fact that $(Tx_n)$ has a convergent subsequence to construct a finite cover of the image of the unit ball.
• Theorem: (Spectral Theorem for Compact Self-Adjoint Operators) Let $H$ be a Hilbert space and $T: H \to H$ a compact self-adjoint operator. Then there exists an orthonormal basis $\{e_n\}$ of $H$ consisting of eigenvectors of $T$, and the corresponding eigenvalues $\{\lambda_n\}$ satisfy $\lim_{n \to \infty} \lambda_n = 0$.
  • Proof: (Sketch) The proof involves constructing the eigenvectors and eigenvalues iteratively using the Hilbert-Schmidt theorem and the Gram-Schmidt process.
• Theorem: (Fredholm Alternative) Let $X$ be a Banach space and $T: X \to X$ a compact operator. Then either (i) the equation $x - Tx = y$ has a unique solution for every $y \in X$, or (ii) the homogeneous equation $x - Tx = 0$ has a nonzero solution.
  • Proof: (Sketch) The proof relies on the properties of the Fredholm operator $I - T$ and the fact that compact operators map weakly convergent sequences to strongly convergent sequences.

Exercises and Problem-Solving Techniques

• To show that an operator is compact, try to prove that it maps bounded sets to relatively compact sets or that it maps bounded sequences to sequences with convergent subsequences
  • Example: Show that the operator $T: C[0,1] \to C[0,1]$ defined by $(Tf)(x) = \int_0^x f(t)dt$ is compact
• When dealing with integral operators, check if the kernel is continuous or square-integrable to determine compactness
  • Example: Prove that the operator $T: L^2[0,1] \to L^2[0,1]$ defined by $(Tf)(x) = \int_0^1 e^{-|x-y|} f(y)dy$ is compact
• To find the spectrum of a compact operator, look for eigenvalues and check if 0 is in the spectrum
  • Example: Find the spectrum of the operator $T: \ell^2 \to \ell^2$ defined by $T(x_1, x_2, \ldots) = (0, \frac{x_1}{2}, \frac{x_2}{3}, \ldots)$
• Use the spectral theorem for compact self-adjoint operators to diagonalize them and analyze their properties
  • Example: Let $T: L^2[0,1] \to L^2[0,1]$ be defined by $(Tf)(x) = \int_0^1 \min\{x,y\}f(y)dy$. Find the eigenvalues and eigenfunctions of $T$
• Apply the Fredholm alternative to solve equations involving compact operators or to show the existence of solutions
  • Example: Use the Fredholm alternative to prove that the equation $x(t) - \lambda \int_0^1 t^2 x(t)dt = t$ has a unique solution for all $\lambda \neq \frac{1}{3}$