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Encyclopedia > Unitary operator

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 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.


  Results from FactBites:
 
Unitary operator - Wikipedia, the free encyclopedia (168 words)
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.
A non-obvious example of a unitary operator is the Fourier transform (with proper normalization).
The spectrum of a unitary operator lies on the unit circle.
Unitary operator - encyclopedia article about Unitary operator. (1055 words)
Unitary operators implement isomorphisms isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich.
A non-obvious example of a unitary operator is the Fourier transform The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i.e.
That follows from Parseval's theorem In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.
  More results at FactBites »


 

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