integraloperatorer

An integral operator is a linear operator defined by integrating against a kernel. Formally, on a domain Ω with measure μ, an operator T acts on a suitable function f by (Tf)(x) = ∫Ω K(x,y) f(y) dμ(y), where K: Ω×Ω → C (or R) is the kernel. The operator typically acts on spaces such as Lp(Ω) or C(Ω), and the domain of T depends on the integrability of K and f.

Two principal families are recognized. Fredholm integral operators use a fixed domain of integration: (Tf)(x) = ∫Ω K(x,y) f(y) dμ(y). Volterra operators have a triangular structure, for example (Tf)(x) = ∫a^x K(x,y) f(y) dy, reflecting a causal or cumulative integration.

Key properties include boundedness and compactness. If K ∈ L2(Ω×Ω), then T is a Hilbert–Schmidt operator on L2(Ω) and is compact. More generally, kernels with suitable integrability conditions yield bounded operators on Lp spaces. Convolution operators are a notable special case: (Tf)(x) = ∫ f(y) k(x−y) dy, which are translation-invariant and are diagonalized by the Fourier transform on L2.

Spectral theory for integral operators often centers on compact operators, whose spectrum consists of 0 together with eigenvalues that accumulate only at 0. This underpins the solvability of Fredholm equations of the second kind: f − λTf = g.

Applications are widespread, including potential theory, boundary-value problems, quantum mechanics, and signal processing. Numerically, integral operators are approximated by matrices via quadrature, enabling discrete eigenvalue problems and solution of integral equations. In Swedish contexts, the term integraloperatorer is used for plural references to integral operators.