adjoint

Adjoint

In mathematics, the term adjoint describes a companion operation to another object that reverses a procedure while preserving a pairing. It appears in several contexts, notably linear algebra, functional analysis, differential equations, physics, and category theory.

In linear algebra and functional analysis, the adjoint of a linear operator T between inner product spaces is the operator T* that satisfies ⟨Tx, y⟩ = ⟨x, T*y⟩ for all x and y in appropriate domains. If T is a bounded operator, T* is defined on the entire target space and is unique. In finite-dimensional complex spaces, the matrix of T* is the conjugate transpose of the matrix of T; in real spaces, it is the transpose. In Hilbert spaces, the adjoint has a precise domain: y lies in Dom(T*) exactly when x ↦ ⟨Tx, y⟩ is a continuous linear functional, and then ⟨Tx, y⟩ = ⟨x, T*y⟩ for all x in Dom(T). Properties include T** = T if T is densely defined, and ||T|| = ||T*||. Notions such as self-adjoint (T* = T), skew-adjoint (T* = -T), normal (T*T = TT*), and unitary (T* T = I) operators arise from adjoints.

Adjoints also feature in differential equations, where adjoint operators arise via integration by parts and variational methods, and in quantum mechanics as Hermitian conjugates.

In category theory, adjoint functors generalize the idea beyond linear operators. Given functors F: C → D and G: D → C, F is left adjoint to G (and G right adjoint to F) if there is a natural isomorphism Hom_D(FX, Y) ≅ Hom_C(X, GY) for all objects X, Y. This adjunction is equivalent to the existence of unit and counit satisfying triangular identities and underpins many constructions, such as free-forgetful adjunctions and limits.