Шексіз өлшемді Гильберт кеңістігіндегі өнімді сақтайтын сырғақты операторлар

Surjective bounded operator on a Hilbert space preserving the inner product

In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating on a Hilbert space, but the same notion serves to define the concept of isomorphism between Hilbert spaces.

The identity function is trivially a unitary operator. Rotations in 'R'^(2) are the simplest nontrivial example of unitary operators. Rotations do not change the length of a vector or the angle between two vectors. This example can be expanded to 'R'^(3). On the vector space 'C' of complex numbers, multiplication by a number of absolute value 1, that is, a number of the form e^(iθ) 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 2π 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 generalization 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 ℓ^(2) indexed by the integers is unitary. In general, any operator in a Hilbert space that acts by permuting an orthonormal basis is unitary. In the finite dimensional case, such operators are the permutation matrices. The unilateral shift (right shift) is an isometry; its conjugate (left shift) is a coisometry. The Fourier operator is a unitary operator, i. e. the operator that performs the Fourier transform (with proper normalization). This follows from Parseval's theorem. Unitary operators are used in unitary representations. Quantum logic gates are unitary operators. Not all gates are Hermitian. 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 , where I is the multiplicative identity element.

The spectrum of a unitary operator U lies on the unit circle. That is, for any complex number λ in the spectrum, one has This can be seen as a consequence of the spectral theorem for normal operators. By the theorem, U is unitarily equivalent to multiplication by a Borel measurable f on L^(2)(μ), for some finite measure space (X, μ). Now implies , μ a. e. This shows that the essential range of f, therefore the spectrum of U, lies on the unit circle. A linear map is unitary if it is surjective and isometric. (Use Polarization identity to show the only if part.)