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In functional analysis, a branch of mathematics, a unitary operator (not to be confused with a unity operator) is defined as follows: Definition 1. A bounded linear operator on a Hilbert space is called a ''unitary operator'' if it satisfies , where is the adjoint of , and is the identity operator. The weaker condition defines an ''isometry''. The other condition, , defines a ''coisometry''. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry. An equivalent definition is the following: Definition 2. A bounded linear operator on a Hilbert space is called a unitary operator if: * is surjective, and * preserves the inner product of the Hilbert space, . In other words, for all vectors and in we have: :: The following, seemingly weaker, definition is also equivalent: Definition 3. A bounded linear operator on a Hilbert space is called a unitary operator if: *the range of is dense in , and * preserves the inner product of the Hilbert space, . In other words, for all vectors and in we have: :: To see that Definitions 1 & 3 are equivalent, notice that preserving the inner product implies is an isometry (thus, a bounded linear operator). The fact that has dense range ensures it has a bounded inverse . It is clear that . Thus, unitary operators are just automorphisms of Hilbert spaces, 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 to itself is sometimes referred to as the Hilbert group of , denoted . A unitary element is a generalization of a unitary operator. In a unital *-algebra, an element of the algebra is called a unitary element if , where is the identity element.〔 〕 ==Examples== * The identity function is trivially a unitary operator. * Rotations in 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 . * On the vector space of complex numbers, multiplication by a number of absolute value , that is, a number of the form for , is a unitary operator. is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of modulo does not affect the result of the multiplication, and so the independent unitary operators on are parametrized by a circle. The corresponding group, which, as a set, is the circle, is called . * 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 . * The bilateral shift on the sequence space indexed by the integers 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 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 which performs the Fourier transform (with proper normalization). This follows from Parseval's theorem. * Unitary operators are used in unitary representations. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Unitary operator」の詳細全文を読む スポンサード リンク
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