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Surjective bounded operator on a Hilbert space preserving the inner product
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 U*U = UU* = I,
where I is the identity element.
Definition 1. A unitary operator is a bounded linear operatorU : H -> H on a Hilbert space H that satisfies U*U = UU* = I, where U* is the adjoint of U, and I : H -> H is the identity operator.
The weaker condition U*U = I defines an isometry. The other condition, UU* = I, defines a coisometry. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry, or, equivalently, a surjective isometry.
An equivalent definition is the following:
Definition 2. A unitary operator is a bounded linear operator U : H -> H on a Hilbert space H for which the following hold:
U preserves the inner product of the Hilbert space, H. In other words, for all vectorsx and y in H we have:
The notion of isomorphism in the category of Hilbert spaces is captured if domain and range are allowed to differ in this definition. Isometries preserve Cauchy sequences, hence the completeness property of Hilbert spaces is preserved
The following, seemingly weaker, definition is also equivalent:
Definition 3. A unitary operator is a bounded linear operator U : H -> H on a Hilbert space H for which the following hold:
U preserves the inner product of the Hilbert space, H. In other words, for all vectors x and y in H we have:
To see that Definitions 1 & 3 are equivalent, notice that U preserving the inner product implies U is an isometry (thus, a bounded linear operator). The fact that U has dense range ensures it has a bounded inverse U-1. It is clear that U-1 = U*.
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 H to itself is sometimes referred to as the Hilbert group of H, denoted Hilb(H) or U(H).
Rotations in R2 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 R3.
On the vector spaceC of complex numbers, multiplication by a number of absolute value1, that is, a number of the form ei? for ? ? R, 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 2? 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 Rn.
The linearity requirement in the definition of a unitary operator can be dropped without changing the meaning because it can be derived from linearity and positive-definiteness of the scalar product:
Analogously you obtain
The spectrum of a unitary operator U lies on the unit circle. That is, for any complex number ? in the spectrum, one has |?| = 1. 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 L2(?), for some finite measure space (X, ?). Now UU* = I implies |f(x)|2 = 1, ?-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.)