adjoint operator

The adjoint is the operator's 'reflection across the inner product'. In matrix language it is the conjugate transpose: the operator you get by flipping T to the other side of an inner-product pairing. It is the abstract device that lets you move T from acting on the first slot to acting on the second slot, and it underlies the entire classification (self-adjoint, normal, unitary).

On a Hilbert space H, the adjoint T* of a bounded operator T is the unique bounded operator satisfying <T x, y> = <x, T* y> for all x, y in H. Existence and uniqueness follow from the Riesz representation theorem. The map T -> T* is conjugate-linear, reverses products ((ST)* = T* S*), satisfies (T*)* = T, and is isometric: ||T*|| = ||T|| and in fact ||T* T|| = ||T||^2 (the C*-identity).

For a bounded operator on a Hilbert space the adjoint always exists and is bounded. For an unbounded densely defined operator the adjoint is defined more delicately — its domain consists of those y for which x -> <T x, y> is bounded — and matching domains is the heart of distinguishing symmetric from self-adjoint. On a general Banach space one instead uses the dual (transpose) operator acting between dual spaces, which agrees with the Hilbert adjoint up to the Riesz identification.