In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint) operator on that space according to the rule
where is the inner product on the vector space.
The adjoint may also be called the Hermitian conjugate or simply the Hermitian^{[1]} after Charles Hermite. It is often denoted by A^{†} 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 operators on Hilbert spaces . The definition has been further extended to include unbounded densely defined operators, whose domain is topologically dense in, but not necessarily equal to,
Consider a linear map between Hilbert spaces. Without taking care of any details, the adjoint operator is the (in most cases uniquely defined) linear operator fulfilling
where is the inner product in the Hilbert space , which is linear in the first coordinate and conjugate linear in the second coordinate. Note the special case where both Hilbert spaces are identical and 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, of an operator , where are Banach spaces with corresponding norms . Here (again not considering any technicalities), its adjoint operator is defined as with
I.e., for .
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. Then it is only natural that we can also obtain the adjoint of an operator , where is a Hilbert space and is a Banach space. The dual is then defined as with such that
Let be Banach spaces. Suppose and , and suppose that is a (possibly unbounded) linear operator which is densely defined (i.e., is dense in ). Then its adjoint operator is defined as follows. The domain is
Now for arbitrary but fixed we set with . By choice of and definition of , f is (uniformly) continuous on as . Then by the Hahn–Banach theorem, or alternatively through extension by continuity, this yields an extension of , called , defined on all of . This technicality is necessary to later obtain as an operator instead of Remark also that this does not mean that can be extended on all of but the extension only worked for specific elements .
Now, we can define the adjoint of as
The fundamental defining identity is thus
Suppose H is a complex Hilbert space, with inner product . Consider a continuous linear operator A : H → H (for linear operators, continuity is equivalent to being a bounded operator). Then the adjoint of A is the continuous linear operator A^{∗} : H → H satisfying
Existence and uniqueness of this operator follows from the Riesz representation theorem.^{[2]}
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.
The following properties of the Hermitian adjoint of bounded operators are immediate:^{[2]}
If we define the operator norm of A 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 H together with the adjoint operation and the operator norm form the prototype of a C*-algebra.
Let the inner product be linear in the first argument. A densely defined operator A from a complex Hilbert space H to itself is a linear operator whose domain D(A) is a dense linear subspace of H and whose values lie in H.^{[3]} By definition, the domain D(A^{∗}) of its adjoint A^{∗} is the set of all y ∈ H for which there is a z ∈ H satisfying
Owing to the density of and Riesz representation theorem, is uniquely defined, and, by definition, ^{[4]}
Properties 1.–5. hold with appropriate clauses about domains and codomains.^{[clarification needed]} For instance, the last property now states that (AB)^{∗} is an extension of B^{∗}A^{∗} if A, B and AB are densely defined operators.^{[5]}
For every the linear functional is identically zero, and hence
Conversely, the assumption that causes the functional to be identically zero. Since the functional is obviously bounded, the definition of assures that The fact that, for every shows that given that is dense.
This property shows that is a topologically closed subspace even when is not.
If and are Hilbert spaces, then is a Hilbert space with the inner product
where and
Let be the symplectic mapping, i.e. Then the graph
of is the orthogonal complement of
The assertion follows from the equivalences
and
An operator is closed if the graph is topologically closed in The graph of the adjoint operator is the orthogonal complement of a subspace, and therefore is closed.
An operator is closable if the topological closure of the graph is the graph of a function. Since is a (closed) linear subspace, the word "function" may be replaced with "linear operator". For the same reason, is closable if and only if unless
The adjoint is densely defined if and only if is closable. This follows from the fact that, for every
which, in turn, is proven through the following chain of equivalencies:
The closure of an operator is the operator whose graph is if this graph represents a function. As above, the word "function" may be replaced with "operator". Furthermore, meaning that
To prove this, observe that i.e. for every Indeed,
In particular, for every and every subspace if and only if Thus, and Substituting obtain
For a closable operator meaning that Indeed,
Let where is the linear measure. Select a measurable, bounded, non-identically zero function and pick Define
It follows that The subspace contains all the functions with compact support. Since is densely defined. For every and
Thus, The definition of adjoint operator requires that Since this is only possible if For this reason, Hence, is not densely defined and is identically zero on As a result, is not closable and has no second adjoint
A bounded operator A : H → H 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.
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 A on a complex Hilbert space H is an conjugate-linear operator A^{∗} : H → H with the property:
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 from.