# Unitary operator

In mathematics, a unitary operator is a bijective linear operator between two Hilbert spaces that receives the scalar product . Unitary operators are special orthogonal or unitary mappings and always norm-preserving , distance- preserving , bounded and, if both Hilbert spaces are equal, normal . The inverse operator of a unitary operator is equal to its adjoint operator . The eigenvalues ​​of a unitary operator in a Hilbert space all have the absolute value one. Unitary operators between finite-dimensional vector spaces of the same dimension can be represented by unitary matrices , depending on the choice of an orthonormal basis . Important examples of unitary operators between infinite-dimensional function spaces are the Fourier transform and the time evolution operators of quantum mechanics .

## definition

A unitary operator is a bijective linear operator between two Hilbert spaces and such that ${\ displaystyle T \ colon V \ to W}$ ${\ displaystyle (V, \ langle \ cdot, \ cdot \ rangle _ {V})}$${\ displaystyle (W, \ langle \ cdot, \ cdot \ rangle _ {W})}$

${\ displaystyle \ langle Tu, Tv \ rangle _ {W} = \ langle u, v \ rangle _ {V}}$

holds for all vectors . A unitary operator is therefore an isomorphism between two Hilbert spaces that receives the scalar product . A unitary operator between two real Hilbert spaces is sometimes called an orthogonal operator . ${\ displaystyle u, v \ in V}$

## properties

In the following, the additions to the scalar products are omitted, since the argument makes it clear which space is involved. ${\ displaystyle V, W}$

### Basic characteristics

Every unitary operator represents a unitary mapping (in the real case orthogonal mapping ). The linearity therefore already follows from the conservation of the scalar product and therefore does not have to be required separately. A unitary operator still receives the scalar product norm of a vector, that is, it holds

${\ displaystyle \ | Tv \ | = {\ sqrt {\ langle Tv, Tv \ rangle}} = {\ sqrt {\ langle v, v \ rangle}} = \ | v \ |}$,

and thus also the distance between two vectors. The figure thus represents an isometry and the two spaces and are therefore isometrically isomorphic . The eigenvalues ​​of a unitary operator all have the amount one. More generally, the spectrum of a unitary operator lies in the edge of the unit circle . ${\ displaystyle T}$${\ displaystyle V}$${\ displaystyle W}$${\ displaystyle T \ colon V \ to V}$

### Operator norm

The following applies to the operator norm of a unitary operator due to the maintenance of the norm ${\ displaystyle T}$

${\ displaystyle \ | T \ | = \ sup _ {\ | v \ | = 1} \ | Tv \ | = \ sup _ {\ | v \ | = 1} \ | v \ | = 1}$.

A unitary operator is therefore always bounded and therefore continuous .

### Inverse

The inverse operator of a unitary operator is equal to its adjoint operator , so ${\ displaystyle T ^ {- 1}}$${\ displaystyle T}$ ${\ displaystyle T ^ {\ ast}}$

${\ displaystyle T ^ {- 1} = T ^ {\ ast}}$,

because it applies

${\ displaystyle \ langle u, T ^ {\ ast} v \ rangle = \ langle Tu, v \ rangle = \ langle Tu, TT ^ {- 1} v \ rangle = \ langle u, T ^ {- 1} v \ rangle}$.

Conversely, if the inverse and adjoint of a linear operator coincide, then this is unitary because it holds

${\ displaystyle \ langle Tu, Tv \ rangle = \ langle u, T ^ {\ ast} Tv \ rangle = \ langle u, T ^ {- 1} Tv \ rangle = \ langle u, v \ rangle}$.

### normality

Due to the agreement of the inverse and adjoint, a unitary operator in the case is always normal , that is ${\ displaystyle V = W}$

${\ displaystyle T ^ {\ ast} T = TT ^ {\ ast} = I}$.

The spectral theorem applies to unitary operators on complex Hilbert spaces and self-adjoint unitary operators on real Hilbert spaces .

### Base transformation

Is a unitary operator and is a Hilbert basis (a complete orthonormal system) of , then is a Hilbert basis of , because it holds ${\ displaystyle T}$${\ displaystyle (v_ {i}) _ {i \ in I}}$${\ displaystyle V}$${\ displaystyle (Tv_ {i}) _ {i \ in I}}$${\ displaystyle W}$

${\ displaystyle \ langle Tv_ {i}, Tv_ {j} \ rangle = \ langle v_ {i}, v_ {j} \ rangle = \ delta _ {ij}}$.

If vice versa and Hilbert bases of and and is linear, then the unitarity of follows , because one obtains ${\ displaystyle (v_ {i}) _ {i \ in I}}$${\ displaystyle (Tv_ {i}) _ {i \ in I}}$${\ displaystyle V}$${\ displaystyle W}$${\ displaystyle T}$${\ displaystyle T}$

{\ displaystyle {\ begin {aligned} \ langle Tu, Tv \ rangle & = {\ big \ langle} T {\ big (} {\ textstyle \ sum _ {i}} \ lambda _ {i} v_ {i} {\ big)}, T {\ big (} {\ textstyle \ sum _ {j}} \ mu _ {j} v_ {j} {\ big)} {\ big \ rangle} = {\ big \ langle} {\ textstyle \ sum _ {i}} \ lambda _ {i} Tv_ {i}, {\ textstyle \ sum _ {j}} \ mu _ {j} Tv_ {j} {\ big \ rangle} = {\ textstyle \ sum _ {i}} {\ textstyle \ sum _ {j}} \ lambda _ {i} {\ bar {\ mu}} _ {j} {\ big \ langle} Tv_ {i}, Tv_ {j } {\ big \ rangle} = \\ & = {\ textstyle \ sum _ {i}} {\ textstyle \ sum _ {j}} \ lambda _ {i} {\ bar {\ mu}} _ {j} \ delta _ {ij} = {\ textstyle \ sum _ {i}} {\ textstyle \ sum _ {j}} \ lambda _ {i} {\ bar {\ mu}} _ {j} \ langle v_ {i }, v_ {j} \ rangle = {\ big \ langle} {\ textstyle \ sum _ {i}} \ lambda _ {i} v_ {i}, {\ textstyle \ sum _ {j}} \ mu _ { j} v_ {j} {\ big \ rangle} = \ langle u, v \ rangle. \ end {aligned}}}