Unitary operator

For unitarity in physics, see Unitarity (physics).

In functional analysis, a branch of mathematics, a unitary operator (not to be confused with a unity operator) is defined as follows:

Definition 1. A unitary operator is a bounded linear operator U : H  H on a Hilbert space H that satisfies U*U = UU* = I, where U* is the adjoint of U, and I : H  H is the identity operator.

The weaker condition U*U = I defines an isometry. The other condition, UU* = I, defines a coisometry. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry.[1]

An equivalent definition is the following:

Definition 2. A unitary operator is a bounded linear operator U : H  H on a Hilbert space H for which the following hold:

\langle Ux, Uy \rangle_H = \langle x, y \rangle_H.

The following, seemingly weaker, definition is also equivalent:

Definition 3. A unitary operator is a bounded linear operator U : H  H on a Hilbert space H for which the following hold:

\langle Ux, Uy \rangle_H = \langle x, y \rangle_H.

To see that Definitions 1 & 3 are equivalent, notice that U preserving the inner product implies U is an isometry (thus, a bounded linear operator). The fact that U has dense range ensures it has a bounded inverse U−1. It is clear that U−1 = U*.

Thus, unitary operators are just automorphisms of Hilbert spaces, i.e., they preserve the structure (in this case, the linear space structure, the inner product, and hence the topology) of the space on which they act. The group of all unitary operators from a given Hilbert space H to itself is sometimes referred to as the Hilbert group of H, denoted Hilb(H).

A unitary element is a generalization of a unitary operator. In a unital *-algebra, an element U of the algebra is called a unitary element if U*U = UU* = I, where I is the identity element.[2]:55

Examples

Linearity

The linearity requirement in the definition of a unitary operator can be dropped without changing the meaning because it can be derived from linearity and positive-definiteness of the scalar product:

\begin{align}
\| \lambda U(x) -U(\lambda x) \|^2 &= \langle \lambda U(x) -U(\lambda x), \lambda U(x)-U(\lambda x) \rangle \\
&= \| \lambda U(x) \|^2 + \| U(\lambda x) \|^2 - \langle U(\lambda x), \lambda  U(x) \rangle - \langle \lambda U(x), U(\lambda  x) \rangle \\
&= |\lambda|^2 \| U(x)\|^2 + \| U(\lambda x) \|^2 - \overline{\lambda} \langle U(\lambda x), U(x) \rangle - \lambda  \langle U(x), U(\lambda x) \rangle \\
&= |\lambda|^2 \| x \|^2 + \| \lambda x \|^2 - \overline{\lambda} \langle \lambda x, x \rangle - \lambda \langle x, \lambda x \rangle \\
&= 0
\end{align}

Analogously you obtain

\| U(x+y)-(Ux+Uy)\| = 0.

Properties

See also

Footnotes

  1. (Halmos 1982, Sect. 127, page 69)
  2. Doran, Robert S.; Victor A. Belfi (1986). Characterizations of C*-Algebras: The Gelfand-Naimark Theorems. New York: Marcel Dekker. ISBN 0-8247-7569-4.

References

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