Hermitian adjoint

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In mathematics, specifically in operator theory, each linear operator on a Euclidean vector 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 are 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—


Informal definition[edit]

Consider a linear operator 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 antilinear 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 .

Note that 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

Definition for unbounded operators between normed spaces[edit]

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 Hahn–Banach theorem or alternatively through extension by continuity this yields an extension of , called defined on all of . Note that 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

for

Definition for bounded operators between Hilbert spaces[edit]

Suppose H is a complex Hilbert space, with inner product . Consider a continuous linear operator A : HH (for linear operators, continuity is equivalent to being a bounded operator). Then the adjoint of A is the continuous linear operator A : HH 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.

Properties[edit]

The following properties of the Hermitian adjoint of bounded operators are immediate:[2]

  1. Involutivity: A∗∗ = A
  2. If A is invertible, then so is A, with
  3. Anti-linearity:
  4. "Anti-distributivity": (AB) = BA

If we define the operator norm of A by

then

[2]

Moreover,

[2]

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.

Adjoint of densely defined unbounded operators between Hilbert spaces[edit]

Definition[edit]

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 yH for which there is a zH 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 BA if A, B and AB are densely defined operators.[5]

ker A*=(im A)[edit]

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.

Geometric interpretation[edit]

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

Corollaries[edit]

A* is closed[edit]

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.

A* is densely defined ⇔ A is closable[edit]

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:

A** = Acl[edit]

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

A* = (Acl)*[edit]

For a closable operator meaning that Indeed,

Counterexample where the adjoint is not densely defined[edit]

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

Hermitian operators[edit]

A bounded operator A : HH is called Hermitian or self-adjoint if

which is equivalent to

[6]

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 antilinear operators[edit]

For an antilinear operator the definition of adjoint needs to be adjusted in order to compensate for the complex conjugation. An adjoint operator of the antilinear operator A on a complex Hilbert space H is an antilinear operator A : HH with the property:

Other adjoints[edit]

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.

See also[edit]

References[edit]

  1. ^ Miller, David A. B. (2008). Quantum Mechanics for Scientists and Engineers. Cambridge University Press. pp. 262, 280.
  2. ^ a b c d Reed & Simon 2003, pp. 186–187; Rudin 1991, §12.9
  3. ^ See unbounded operator for details.
  4. ^ Reed & Simon 2003, p. 252; Rudin 1991, §13.1
  5. ^ Rudin 1991, Thm 13.2
  6. ^ Reed & Simon 2003, pp. 187; Rudin 1991, §12.11