7.4 Fredholm alternative and its applications

The Fredholm Alternative is a key theorem in functional analysis that helps determine when equations involving compact operators have solutions. It states that for an equation (I - K)x = y, where K is compact, either a unique solution exists for all y, or the homogeneous equation has non-zero solutions.

This theorem has wide-ranging applications, from integral equations to boundary value problems. It provides a powerful tool for analyzing the existence and uniqueness of solutions in various mathematical contexts, especially when dealing with compact operators in Banach spaces.

Fredholm Alternative

Fredholm alternative for compact operators

• Fundamental theorem in functional analysis characterizes solvability of equations with compact operators
• Let $X$ be a Banach space and $K: X \to X$ a compact operator
• For equation $(I - K)x = y$, where $I$ is identity operator and $y \in X$, exactly one holds:
  • For every $y \in X$, equation $(I - K)x = y$ has unique solution $x \in X$
  • Homogeneous equation $(I - K)x = 0$ has non-zero solution $x \in X$
• Proof outline:
  • Show kernel (null space) of $(I - K)$ is finite-dimensional
  • Prove range of $(I - K)$ is closed
  • Use Riesz lemma to show either range of $(I - K)$ is all of $X$ or codimension equals dimension of kernel

Applications to integral equations

• Fredholm integral equations of second kind: $u(x) - \lambda \int_a^b K(x, t)u(t)dt = f(x)$
  • Integral operator $K$ often compact, allows application of Fredholm alternative
  • If $\lambda$ not eigenvalue of $K$, equation has unique solution for every $f$
• Volterra integral equations: $u(x) - \lambda \int_a^x K(x, t)u(t)dt = f(x)$
  • Fredholm alternative does not directly apply, but can transform into Fredholm equation
• Boundary value problems:
  • Consider second-order linear differential equation with homogeneous boundary conditions
  • Green's function $G(x, t)$ converts problem into Fredholm integral equation
  • Integral operator associated with Green's function often compact
  • Apply Fredholm alternative to determine existence and uniqueness of solutions

Solutions with compact operators

• Use Fredholm alternative to determine solvability of equations $(I - K)x = y$, where $K$ compact operator
  • If $y$ in range of $(I - K)$, solution exists
  • If kernel of $(I - K)$ trivial (only zero element), solution unique
• Eigenvalues and eigenfunctions of compact operators:
  • If $\lambda$ eigenvalue of compact operator $K$, equation $(I - \lambda K)x = 0$ has non-zero solution (eigenfunction)
  • Fredholm alternative implies $(I - \lambda K)x = y$ has solution iff $y$ orthogonal to all eigenfunctions corresponding to eigenvalue $\lambda$

Solvability in Banach spaces

• Consider linear equation $Ax = y$ in Banach space $X$, where $A: X \to X$ bounded linear operator
  • If $A$ compact operator, Fredholm alternative directly applies
  • If $A$ not compact, can still use Fredholm alternative by decomposing $A$ into sum of compact operator and bounded invertible operator
    • Write $A = I - K$, where $K$ compact
    • Solvability of $Ax = y$ equivalent to solvability of $(I - K)x = y$
• Riesz-Schauder theory:
  • Extends Fredholm alternative to closed operators (not necessarily bounded) in Banach spaces
  • Provides conditions for existence and uniqueness of solutions to linear equations with closed operators