unitary operator
A unitary operator is a rigid rotation of a Hilbert space — it moves vectors around but never changes any length or angle. It is the infinite-dimensional version of an orthogonal/rotation matrix. Because it preserves all geometry, it is invertible, and its inverse is just its adjoint, which makes it extraordinarily easy to handle.
Formally, a bounded operator U on a Hilbert space is unitary if it is surjective and preserves the inner product: <U x, U y> = <x, y> for all x, y. Equivalently U is invertible with U* = U^{-1}, equivalently U U* = U* U = I. Preserving the inner product forces ||U x|| = ||x|| (it is an isometry), and surjectivity upgrades that isometry to a full unitary.
Surjectivity is not automatic and must not be dropped: the right shift on l^2 preserves the inner product (it is an isometry, U* U = I) but is NOT surjective, so it is NOT unitary (U U* is not the identity). The spectrum of a unitary operator always lies on the unit circle |lambda| = 1, mirroring that orthogonal matrices have eigenvalues of modulus one.
On L^2(R), the Fourier transform is a unitary operator (Plancherel's theorem says it preserves the L^2 inner product). Multiplication by e^{i theta(x)} for a real function theta is also unitary.
Fourier transform as a rotation of L^2.