Unitary Operator

In ''H'' satisfying

U=UU^---=I

where ''U''^∗ is the operator. This property is equivalent to the following:

#The range of ''U'' is dense, and
#''U'' preserves the Inner Product ⟨ , ⟩ on the Hilbert space, i.e. for all Vector s ''x'' and ''y'' in the Hilbert space,
::

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

To see this, notice that ''U'' preserves 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 Automorphism s of Hilbert Space s, 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'').

EXAMPLES

The Identity Function is trivially a unitary operator.

On the Vector Space C of Complex Number s, multiplication by a number of Absolute Value 1, that is, a number of the form ''e''^{''i θ''} for ''θ'' ∈ '''R''', is a unitary operator. ''θ'' is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of ''θ'' modulo 2''π'' does not affect the result of the multiplication, and so the independent unitary operators on C are parametrized by a circle. The corresponding group, which, as a set, is the circle, is called U(1).

More generally, 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. Orthogonal Matrices are the special case of unitary matrices in which all entries are real. They are the unitary operators on R^''n''.

The Bilateral Shift on the Sequence Space $\ell^2$ indexed by the Integer s is unitary. In general, any operator in a Hilbert space which acts by shuffling around an Orthonormal Basis is unitary. In the finite dimensional case, such operators are the Permutation Matrices .

The Fourier Operator is a unitary operator, i.e. the operator which performs the Fourier Transform (with proper normalization). This follows from Parseval's Theorem .

Unitary operators are used in Unitary Representation s.

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Information About

Unitary Operator