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- Unitary_operator abstract "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 U : H → H on a Hilbert space H is called a unitary operator if it 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.An equivalent definition is the following:Definition 2. A bounded linear operator U : H → H on a Hilbert space H is called a unitary operator if:U is surjective, andU preserves the inner product of the Hilbert space, H. In other words, for all vectors x and y in H we have: The following, seemingly weaker, definition is also equivalent:Definition 3. A bounded linear operator U : H → H on a Hilbert space H is called a unitary operator if:the range of U is dense in H, andU 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).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.".
- Unitary_operator wikiPageID "225999".
- Unitary_operator wikiPageRevisionID "596191670".
- Unitary_operator hasPhotoCollection Unitary_operator.
- Unitary_operator subject Category:Operator_theory.
- Unitary_operator subject Category:Unitary_operators.
- Unitary_operator type Abstraction100002137.
- Unitary_operator type Function113783816.
- Unitary_operator type MathematicalRelation113783581.
- Unitary_operator type Operator113786413.
- Unitary_operator type Relation100031921.
- Unitary_operator type UnitaryOperators.
- Unitary_operator comment "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 U : H → H on a Hilbert space H is called a unitary operator if it 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.".
- Unitary_operator label "Operador unitario".
- Unitary_operator label "Operador unitário".
- Unitary_operator label "Operator unitarny".
- Unitary_operator label "Operatore unitario".
- Unitary_operator label "Opérateur unitaire".
- Unitary_operator label "Unitaire operator".
- Unitary_operator label "Unitary operator".
- Unitary_operator label "Unitärer Operator".
- Unitary_operator label "Унитарный оператор".
- Unitary_operator label "ユニタリ作用素".
- Unitary_operator label "幺正算符".
- Unitary_operator sameAs Unitární_operátor.
- Unitary_operator sameAs Unitärer_Operator.
- Unitary_operator sameAs Operador_unitario.
- Unitary_operator sameAs Opérateur_unitaire.
- Unitary_operator sameAs Operatore_unitario.
- Unitary_operator sameAs ユニタリ作用素.
- Unitary_operator sameAs 유니터리_작용소.
- Unitary_operator sameAs Unitaire_operator.
- Unitary_operator sameAs Operator_unitarny.
- Unitary_operator sameAs Operador_unitário.
- Unitary_operator sameAs m.01gyl6.
- Unitary_operator sameAs Q1972470.
- Unitary_operator sameAs Q1972470.
- Unitary_operator sameAs Unitary_operator.
- Unitary_operator wasDerivedFrom Unitary_operator?oldid=596191670.
- Unitary_operator isPrimaryTopicOf Unitary_operator.