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In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding Hermitian adjoint (or adjoint operator). Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a complex Hilbert space as generalized complex numbers, then the adjoint of an operator plays the role of the complex conjugate of a complex number.

In a similar sense, one can define an adjoint operator for linear (and possibly unbounded) operators between Banach spaces.

The adjoint of an operator A may also be called the Hermitian conjugate or Hermitian transpose[1] (after Charles Hermite) of A and is denoted by A or A (the latter especially when used in conjunction with the bra–ket notation). Confusingly, A may also be used to represent the conjugate of A.

Informal definitionEdit

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  . Note the special case where both Hilbert spaces are identical and   is an operator on some Hilbert space.

When one trades the dual pairing for the inner product, one can define the adjoint 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 spacesEdit

Let   be Banach spaces. Suppose   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 spacesEdit

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.

PropertiesEdit

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 spacesEdit

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

 

and A(y) is defined to be the z thus found.[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]

The relationship between the image of A and the kernel of its adjoint is given by:

 

These statements are equivalent. See orthogonal complement for the proof of this and for the definition of  .

Proof of the first equation:[6][clarification needed]

 

The second equation follows from the first by taking the orthogonal complement on both sides. Note that in general, the image need not be closed, but the kernel of a continuous operator[7] always is.[clarification needed]

Hermitian operatorsEdit

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

 

which is equivalent to

 [8]

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 operatorsEdit

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 adjointsEdit

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 alsoEdit

FootnotesEdit

  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. ^ See Rudin 1991, Thm 12.10 for the case of bounded operators
  7. ^ The same as a bounded operator.
  8. ^ Reed & Simon 2003, pp. 187; Rudin 1991, §12.11

ReferencesEdit