Hermitian adjoint

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

The adjoint of an operator A is also sometimes called the Hermitian conjugate (after Charles Hermite) of A and is denoted by A* or A (the latter especially when used in conjunction with the bra–ket notation).

Definition for bounded operators[edit]

Suppose H is a Hilbert space, with inner product \langle\cdot,\cdot\rangle. 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

 \langle Ax , y \rangle = \langle x , A^* y \rangle \quad \mbox{for all } x,y\in H.

Existence and uniqueness of this operator follows from the Riesz representation theorem.[1]

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:[1]

  1. A** = A – involutiveness
  2. If A is invertible, then so is A*, with (A*)−1 = (A−1)*
  3. (A + B)* = A* + B*
  4. A)* = λA*, where λ denotes the complex conjugate of the complex number λ – antilinearity (together with 3.)
  5. (AB)* = B* A*

If we define the operator norm of A by

 \| A \| _{op} := \sup \{ \|Ax \| : \| x \| \le 1 \}

then

 \| A^* \| _{op} = \| A \| _{op}. [1]

Moreover,

 \| A^* A \| _{op} = \| A \| _{op}^2. [1]

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 Hilbert space H together with the adjoint operation and the operator norm form the prototype of a C*-algebra.

Adjoint of densely defined operators[edit]

A densely defined operator A on a Hilbert space H is a linear operator whose domain D(A) is a dense linear subspace of H and whose co-domain is H.[2] Its adjoint A* has as domain D(A*) the set of all yH for which there is a zH satisfying

 \langle Ax , y \rangle = \langle x , z \rangle \quad \mbox{for all } x \in D(A),

and A*(y) equals the z defined thus.[3]

Properties 1.–5. hold with appropriate clauses about domains and codomains. For instance, the last property now states that (AB)* is an extension of B*A* if A, B and AB are densely defined operators.[4]

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

 \ker A^* = \left( \operatorname{im}\ A \right)^\bot (see orthogonal complement)
 \left( \ker A^* \right)^\bot = \overline{\operatorname{im}\ A}

Proof of the first equation:[5]


\begin{align}
A^* x = 0 &\iff
\langle A^*x,y \rangle = 0 \quad \forall y \in H \\ &\iff
\langle x,Ay \rangle = 0 \quad \forall y \in H \\ &\iff
x\ \bot \ \operatorname{im}\ A
\end{align}

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[6] always is.

Hermitian operators[edit]

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

  A  = A^{*}

which is equivalent to

 \lang Ax , y \rang = \lang x , A y \rang \mbox{ for all } x,y\in H. [7]

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 Hilbert space H is an antilinear operator A* : HH with the property:

 \lang Ax , y \rang = \overline{\lang x , A^* y \rang} \quad \text{for all } x,y\in H.

Other adjoints[edit]

The equation

 \lang Ax , y \rang = \lang x , A^* y \rang

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]

Footnotes[edit]

  1. ^ a b c d Reed & Simon 2003, pp. 186­­–187; Rudin 1991, §12.9
  2. ^ See unbounded operator for details.
  3. ^ Reed & Simon 2003, pp. 252; Rudin 1991, §13.1
  4. ^ Rudin 1991, Thm 13.2
  5. ^ See Rudin 1991, Thm 12.10 for the case of bounded operators
  6. ^ The same as a bounded operator.
  7. ^ Reed & Simon 2003, pp. 187; Rudin 1991, §12.11

References[edit]

  • Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8 .
  • Rudin, Walter (1991), Functional Analysis (second ed.), McGraw-Hill, ISBN 0-07-054236-8 .