Unitary operator
From Exampleproblems
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,
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.
Examples
- The identity function is trivially a unitary operator.
- The bilateral shift on the sequence space
indexed by the integers is unitary. In general, any operator in a Hilbert space which acts by shuffling around an orthonormal basis is unitary.
- A non-obvious example of a unitary operator is the Fourier transform (with proper normalization). That follows from Parseval's theorem.
Properties
- The spectrum of a unitary operator lies on the unit circle. That is, for any complex number λ in the spectrum, one has |λ|=1. This follows by a Neumann series expansion.
