 Unitary operator

A unitary operator $U:H\to K$ is a surjective linear isometry between complex inner product spaces. It preserves every inner product, not just the norm of a selected vector:Its adjoint operator is its inverse function, soFor an operator on one space, both identities have the same identity operator on the right. In finite-dimensional orthonormal bases, its matrix is a unitary matrix; the real counterpart is an orthogonal transformation.

Surjectivity matters in infinite dimension. An isometric embedding can satisfy $U^{\ast }U=I$ without satisfying $U{U}^{\ast }=I$: a proper image leaves some vectors outside its range. A unitary operator instead gives a reversible change of orthonormal coordinates, preserving all lengths and orthogonality.

In quantum mechanics and functional analysis, a **unitary operator** is a type of linear operator that preserves the inner product in a Hilbert space. This means that it is a transformation that maintains the length of vectors and angles between them, which is crucial for ensuring the conservation of probability in quantum systems.