In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint on that space according to the rule

is the inner product on the vector space.

The adjoint may also be called the Hermitian conjugate or simply the Hermitian after Charles Hermite. It is often denoted by in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators can be represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose).

The above definition of an adjoint operator extends verbatim to bounded linear operatorss on Hilbert spaces. The definition has been further extended to include unbounded densely defined operators, whose domain is topologically dense

Informal definition

Consider a linear map between Hilbert spaces fulfilling

is the inner product, which is linear in the first coordinate and conjugate linear is an operator on that Hilbert space.

When one trades the inner product for the dual pairing, one can define the adjoint, also called the transpose are Banach spaces with corresponding norms with

.

The above definition in the Hilbert space setting is really just an application of the Banach space case when one identifies a Hilbert space with its dual (via the Riesz representation theorem such that

Definition for unbounded operators between Banach spaces

be Banach spaces is a (possibly unbounded) linear operator which is densely defined is defined as follows. The domain is

. Then by the Hahn–Banach theorem.

as

The fundamental defining identity is thus

== Definition for bounded operators between Hilbert spaces == Suppose is a complex Hilbert space, with inner product. Consider a continuous linear operator (for linear operators, continuity is equivalent to being a bounded operator). Then the adjoint of is the continuous linear operator satisfying

Existence and uniqueness of this operator follows from the Riesz representation theorem.

This can be seen as a generalization of the adjoint matrix of a square matrix which has a similar property involving the standard complex inner product.

Properties

The following properties of the Hermitian adjoint of bounded operators are immediate:

1. Involutivity:

2. If is invertible, then so is

Conjugate linearity:

• , where denotes the complex conjugate of the complex number

• "Anti-distributivity":

If we define the operator norm of by

then

Moreover,

One says that a norm that satisfies this condition behaves like a "largest value", extrapolating from the case of self-adjoint operators.

The set of bounded linear operators on a complex Hilbert space together with the adjoint operation and the operator norm form the prototype of a C*-algebra.

Adjoint of densely defined unbounded operators between Hilbert spaces

Definition

be linear in the first argument. A densely defined operator from a complex Hilbert space to itself is a linear operator whose domain is a dense linear subspace of and whose values lie in . By definition, the domain of its adjoint is the set of all for which there is a satisfying

and Riesz representation theorem

Properties 1.–5. hold with appropriate clauses about domains and codomains. For instance, the last property now states that is an extension of if , and are densely defined operators.

ker A* = (im A)⊥

is dense.

is not.

Geometric interpretation

is a Hilbert space with the inner product

be the symplectic mapping Then the graph

is the orthogonal complement

The assertion follows from the equivalences

and

Corollaries

A* is closed

is closed is the orthogonal complement of a subspace, and therefore is closed.

A* is densely defined ⇔ A is closable

is closable

which, in turn, is proven through the following chain of equivalencies:

A** = Acl

The closure

Indeed,

A* = (Acl)*

Indeed,

Counterexample where the adjoint is not densely defined

Define

Hermitian operators

A bounded operator is called Hermitian or self-adjoint if

which is equivalent to

In some sense, these operators play the role of the real numbers (being equal to their own "complex conjugate") and form a real vector space. They serve as the model of real-valued observables in quantum mechanics. See the article on self-adjoint operators for a full treatment.

Adjoints of conjugate-linear operators

For a conjugate-linear operator the definition of adjoint needs to be adjusted in order to compensate for the complex conjugation. An adjoint operator of the conjugate-linear operator on a complex Hilbert space is an conjugate-linear operator with the property:

Other adjoints

The equation

is formally similar to the defining properties of pairs of adjoint functors in category theory, and this is where adjoint functors got their name.

The notion of adjoint operators on Hilbert spaces is also abstracted in the definition of a dagger category or dagger symmetric monoidal category.

See also

Mathematical concepts

• Conjugate transpose

• Hermitian operator

• Transpose of linear maps

Physical applications

• Operator (physics)

• †-algebra

References

• .

• .