Definition and Examples of Compact Operators

• Definition: Operator K is compact if image of unit ball {Ku : ||u|| ≤ 1} closure is compact in H
• Example 1: Ka = (a₁/1, a₂/2, a₃/3, ...) for a ∈ ℓ² is compact operator
• Example 2: Integral operator Tf(x) = ∫₀¹ K(x,y)f(y)dy on L²[0,1] with continuous kernel K(x,y)
  • Assignment proves T is compact using Arzelà-Ascoli theorem
  • Solution operators for differential equations take this form
  • Example: u'' = f, u(0) = u(1) = 0 has solution operator as compact operator
• Non-example: Identity operator I on ℓ² is not compact
  • Standard basis {eₙ} has ||Ieₙ - Ieₘ|| = √2 for n ≠ m
  • No convergent subsequence exists
  • Identity not compact on any infinite dimensional Hilbert space

Main Theorem: Characterization of Compact Operators

• Theorem: For separable Hilbert space H, bounded linear operator T is compact iff there exists sequence {Tₙ} of finite rank operators converging to T in operator norm
• Proof (⟹): If T compact
  • H separable ⟹ has orthonormal basis {eₖ}
  • Compact set has equismall tails: ∀ε > 0, ∃N such that Σₖ>ₙ |⟨Tu, eₖ⟩|² < ε² for ||u|| ≤ 1
  • Define Tₙu = Σₖ₌₁ⁿ ⟨Tu, eₖ⟩eₖ (finite rank operator)
  • ||Tₙ - T|| ≤ ε for n ≥ N
• Proof (⟸): If T = lim Tₙ in operator norm with Tₙ finite rank
  • Show T(unit ball closure) can be approximated by finite dimensional subspaces
  • Use characterization: compact iff closed, bounded, and approximable by finite dimensional subspaces

Algebraic Properties of Compact Operators

• K(H) is closed subspace of bounded linear operators
• If T compact, then T* compact
  • Proof: T = lim Tₙ with Tₙ finite rank ⟹ T* = lim Tₙ* in operator norm
• Multiplication property: If A, B bounded linear and T compact, then ATB compact
  • Proof: ||ATₙB - ATB|| ≤ ||A|| ||Tₙ - T|| ||B|| → 0
• K(H) is *-closed two-sided ideal in algebra of bounded linear operators

Spectrum Theory Introduction

• Resolvent set: res(A) = {λ ∈ ℂ : A - λI invertible}
• Spectrum: spec(A) = ℂ \ res(A)
• Example: For diagonal matrix diag(λ₁, λ₂), spec(A) = {λ₁, λ₂}
• Eigenvalue: λ ∈ spec(A) such that A - λI not injective
  • ∃u ≠ 0: Au = λu
• Non-eigenvalue spectrum elements: Can occur when A - λI injective but not surjective (range dense but not closed)

Examples of Spectrum

• Compact operator example: T(a₁, a₂, a₃, ...) = (a₁/1, a₂/2, a₃/3, ...)
  • Eigenvalues: 1/n for n = 1, 2, 3, ...
  • 0 ∈ spec(T) (T injective but not surjective)
• Multiplication operator: Tf(x) = xf(x) on L²[0,1]
  • No eigenvalues
  • spec(T) = [0,1]

General Properties of Spectrum

• Theorem: For bounded linear operator A:
  • spec(A) is closed subset of ℂ
  • spec(A) ⊆ {λ ∈ ℂ : |λ| ≤ ||A||}
  • Therefore spec(A) is compact
• Proof outline:
  • res(A) open (follows from GL(H) being open set)
  • If |λ| > ||A||, then ||A/λ|| < 1, so I - A/λ invertible by Neumann series
  • Therefore A - λ = -λ(I - A/λ) invertible
• Spectrum non-empty (requires complex analysis - Liouville's theorem)

Self-Adjoint Operators Preparation

• Properties for self-adjoint A = A*:
  1. ⟨Au, u⟩ ∈ ℝ for all u ∈ H
  2. ||A|| = sup{|⟨Au, u⟩| : ||u|| = 1}
• Proof of (1): ⟨Au, u⟩* = ⟨u, Au⟩ = ⟨A*u, u⟩ = ⟨Au, u⟩
• Proof of (2): Uses polarization identity and parallelogram law
  • Key step: ⟨Au, v⟩ = ¼[⟨A(u+v), u+v⟩ - ⟨A(u-v), u-v⟩] for real part