adjoint of a bounded operator

The adjoint generalizes the conjugate transpose of a matrix, but it is defined without ever choosing coordinates — purely through the inner product. The adjoint T^* is the operator that lets you move T from one side of an inner product to the other. It is the single most important construction for classifying operators, because the familiar finite-dimensional families (self-adjoint, unitary, normal) carry over verbatim.

Precisely: for a bounded operator T on a Hilbert space H, the adjoint T^* is the unique bounded operator satisfying <T x, y> = <x, T^* y> for all x, y in H. Its existence and uniqueness are guaranteed by the Riesz representation theorem. Key properties: (S T)^* = T^* S^*, (T^*)^* = T, ||T^*|| = ||T||, and the C*-identity ||T^* T|| = ||T||^2.

Why the finite-dimensional vocabulary survives: T is self-adjoint when T^* = T (the analog of a symmetric or Hermitian matrix), unitary when T^* T = T T^* = I (the analog of an orthogonal or unitary matrix, an inner-product-preserving bijection), and normal when T^* T = T T^* (commuting with its adjoint). These classes have exactly the spectral properties you would hope: self-adjoint operators have real spectrum, unitary operators have spectrum on the unit circle.

A subtlety for later: for bounded operators everything is clean. For unbounded operators the adjoint is far more delicate — its domain must be determined, and self-adjointness becomes a strictly stronger condition than mere symmetry. But within the bounded world, the adjoint behaves exactly like the conjugate transpose you already trust.