Resolventti

Resolventti is the Finnish term for the mathematical concept known in English as the resolvent. It refers to a function or operator that encodes spectral information about a linear operator, typically defined on a complex variable z. For a linear operator A on a Banach space (or a matrix in finite dimensions), the resolvent set ρ(A) consists of all complex numbers z for which A − zI is invertible with bounded inverse. The resolvent operator is then defined by R(z; A) = (A − zI)⁻¹ for z in ρ(A). The complement of ρ(A) in the complex plane is the spectrum σ(A), which consists of the spectral values of A, such as eigenvalues in finite dimensions or more general spectral points in infinite dimensions.

In finite dimensions, the resolvent is well defined wherever det(A − zI) ≠ 0, and the resolvent function R(z; A) is analytic on ρ(A) with poles at eigenvalues. A key identity is the resolvent equation: R(z; A) − R(w; A) = (z − w) R(z; A) R(w; A), which expresses how the resolvent varies with z and w. The resolvent also admits series representations, for example a Neumann series expansion when |z| is large or when perturbations are small in operator norm.

The resolvent is central to spectral theory and functional calculus. It characterizes the spectrum, underpins contour-integral definitions of functions of operators, and appears in applications ranging from differential equations and perturbation theory to quantum mechanics and scattering theory.