Unitary operator

In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space preserving 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.
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.
Definition.
Definition 1. A "unitary operator" is a bounded linear operator on a Hilbert space that 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, or, equivalently, a surjective isometry.
An equivalent definition is the following:
Definition 2. A "unitary operator" is a bounded linear operator on a Hilbert space for which the following hold:
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 on a Hilbert space for which the following hold:
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 or .
Linearity.
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