Fredholm

Fredholm refers primarily to Erik Ivar Fredholm (1866–1927), a Swedish mathematician whose work on integral equations gave rise to Fredholm theory. His studies of linear integral equations with kernels led to foundational concepts and techniques in functional analysis and the theory of integral equations, and many related notions bear his name.

In functional analysis, a Fredholm operator is a bounded linear operator T between Banach spaces with finite-dimensional kernel and cokernel and with closed range. The index of T is defined as ind(T) = dim ker(T) − dim coker(T). Fredholm operators are stable under compact perturbations, and their index remains invariant under such perturbations. This stability and the finite-dimensional nature of the kernel and cokernel make Fredholm operators central to global analysis and spectral theory.

Fredholm equations and the Fredholm alternative are key components of the theory. A common example is the Fredholm integral equation of the second kind, φ(x) − λ ∫ K(x, t) φ(t) dt = f(x), where K is a kernel. For compact kernels, the Fredholm alternative provides solvability criteria: either the homogeneous equation has nontrivial solutions, or the inhomogeneous equation has solutions for all right-hand sides f that satisfy an orthogonality condition with respect to the adjoint homogeneous problem.

The Fredholm determinant, det(I − K), is defined for trace-class (and related) operators and serves as a tool to study spectral properties. Zeros of the Fredholm determinant correspond to eigenvalues of the underlying operator and aid in analyzing solvability of integral equations.

Fredholm theory extends to elliptic differential operators on compact manifolds and is a foundational element of index theory, including the Atiyah–Singer index theorem, which connects analytical indices of Fredholm operators to topological invariants.