eksponentioperatiori

Eksponentioperatiori refers to the operator obtained by applying the exponential function to a linear operator, typically denoted exp(A). In mathematics, it generalizes the scalar exponential to operators and plays a key role in solving linear evolution equations, studying symmetry groups, and in quantum mechanics. For matrices, it is the matrix exponential.

Definition for bounded operators: If A is a bounded linear operator on a Banach space X, the exponential is defined by the convergent power series exp(A) = sum_{n=0}^∞ A^n / n!. The operator is invertible with inverse exp(-A). If A and B commute, exp(A)exp(B) = exp(A+B). In finite dimensions, this specializes to the matrix exponential.

Computation: In finite dimensions, if A is diagonalizable A = P D P^{-1}, then exp(A) = P exp(D) P^{-1}, with exp(D) formed by exponentiating the diagonal entries. More generally, use Jordan form, or numerical methods like Padé approximants with scaling-and-squaring.

Unbounded operators and semigroups: For unbounded operators, the series definition is not generally applicable. Exponentials are defined via functional calculus or as a strongly continuous one-parameter semigroup {T(t)} with T(0)=I and T'(t)=A T(t). The existence requires conditions such as A generating a C0-semigroup (Hille–Yosida theory).

Applications and related: Used to describe time evolution in differential equations, quantum dynamics, control theory, and Lie group theory through the exponential map. Related notions include the matrix exponential, spectral theorem, and the Baker–Campbell–Hausdorff formula.