antilinear

Antilinear, also called conjugate-linear, describes a map T between complex vector spaces that reverses scalar multiplication: T(α v) = ᾱ T(v) for all scalars α in C and vectors v. Equivalently, T(α v + β w) = ᾱ T(v) + β̄ T(w). This contrasts with linear maps, which satisfy T(α v) = α T(v).

Examples include the complex conjugation map c: C → C, c(z) = z̄, which is antilinear. On C^n, the componentwise conjugation z ↦ z̄ is antilinear, and more generally, any map T(z) = A z̄ with a fixed matrix A is an antilinear map from C^n to C^m.

Properties of antilinear maps include that the sum of two antilinear maps is antilinear. The composition with a linear map (on either side) is also antilinear: if L is linear and C is antilinear, then L ∘ C and C ∘ L are antilinear. The composition of two antilinear maps is linear, since ᾱ ᾱ = α.

In inner product spaces over the complex numbers, antilinear maps frequently interact with sesquilinear forms. A prominent class is antiunitary operators, which are bijective and antilinear and preserve inner products: ⟨Ux, Uy⟩ = ⟨x, y⟩. Antiunitary operators, such as the quantum-mechanical time-reversal operator, play a key role in physics.

Terminology varies: some authors use conjugate-linear to mean the same concept as antilinear. In all cases, antilinearity is distinct from linearity and is characterized by conjugation of scalars under scalar multiplication.