Unitarity and time evolution

In functional analysis, a '''unitary operator''' is a bounded linear operator $U$ on a [[Hilbert space]] satisfying :$U^*U=UU^*=I$ where $I$ is the identity operator. This property is equivalent to any of the following: * $U$ is a surjective isometry * $U$ is surjective and preserves the inner product on the Hilbert space, so that for all vectors $x$ and $y$ in the Hilbert space, :$\backslash langle\; Ux,\; Uy\; \backslash rangle\; =\; \backslash langle\; x,\; y\; \backslash rangle.$ Unitary matrices are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalisation of the notion of a unitary matrix. Unitary operators implement isomorphisms between operator algebras. [[Category:Evolutions and Operations]] [[Category:Handbook of Quantum Information]]