Fredholm Alternative

A dichotomy for linear equations involving compact (or Fredholm) operators: for an operator A of Fredholm type and a scalar λ, either (I − λA) is invertible and the inhomogeneous equation has a unique solution for every right-hand side, or the homogeneous equation has nontrivial solutions and solvability of the inhomogeneous problem requires compatibility (orthogonality) conditions against the adj

A linear algebraic/operator-theoretic dichotomy for equations of the form (I − K)x = y where K is a compact (or Fredholm) operator: either the homogeneous equation has only the trivial solution and the inhomogeneous equation is solvable for every y, or the homogeneous equation has a nontrivial finite-dimensional solution space and the inhomogeneous equation is solvable only for y orthogonal (or an

A solvability principle for linear compact or Fredholm operators stating that for a linear equation Lx = f either the homogeneous problem Lx = 0 has only the trivial solution and L is onto the codomain (so Lx=f solvable for every f), or the homogeneous problem has nontrivial solutions and Lx=f is solvable precisely when f is orthogonal to every element of the adjoint homogeneous solution space.