Unitary Operator

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Unitary Operator

 

an extension of the notation of a rotation of Euclidean space to the infinite-dimensional case. More precisely, a unitary operator is a rotation of a Hilbert space about the origin. In analytic terms, an operator U that maps the Hilbert space H onto itself is said to be unitary if (f, g) = (Uf,Ug) for any two vectors f, g in H. A unitary operator preserves lengths of vectors and angles between vectors and is a linear operator. A unitary operator U has a unitary inverse U–1 such that U–1 = U*, where U* is the adjoint of U.

An example of a unitary operator is the Fourier-Plancherel operator, which associates to each function f(x), – ∞ < x < ∞, with square integrable absolute value the function