Hermitian adjoint

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In mathematics, specifically in functional analysis, each bounded linear operator on a complex 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 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 there can be defined an adjoint operator for linear (and possibly unbounded) operators between Banach spaces.

The adjoint of an operator A may also be called the Hermitian adjoint, 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).

Informal definition[edit]

Consider a linear operator A: H_1\to H_2 between Hilbert spaces. Without taking care of any details, the adjoint operator is the (in most cases uniquely defined) linear operator A^* : H_2 \to H_1 fulfilling

\langle Ah_1, h_2 \rangle_{H_2} = \langle h_1, A^* h_2 \rangle_{H_1},

where \langle\cdot, \cdot \rangle_{H_i} is the inner product in the Hilbert space H_i. Note the special case where the both Hilbert spaces are identical and A 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 A: E\to F, where E, F are Banach spaces with corresponding norms \|\cdot\|_E, \|\cdot\|_F. Here (again not considering any technicalities), its adjoint operator is defined as A^*: F^*\to E^* with

A^*f = (u\mapsto f(Au)),

i.e. (A^*f)(u) = f(Au) for f\in F^*, u\in E.

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 A: H \to E, where H is a Hilbert space and E is a Banach space. The dual is then defined as A^*: E^* \to H with A^*f = h_f such that

 \langle h_f, h\rangle_H = f(Au).

Definition for unbounded operators between normed spaces[edit]

Let (E, \|\cdot\|_E), (F,\|\cdot\|_F) be Banach spaces. Suppose  A: E \supset D(A) \to F is a (possibly unbounded) linear operator which is densely defined (i.e. D(A) is dense in E). Then its adjoint operator A^* is defined as follows. The domain is

D(A^*):=\{g\in F^*:~ \exists c \geq 0:~ \forall u\in D(A):~ |g(Au)| \leq c\cdot \|u\|_E \}.

Now for arbitrary but fixed g\in D(A^*) we set f: D(A) \to \R with f(u) = g(Au). By choice of g and definition of D(A^*), f is (uniformly) continuous on D(A) as |f(u)|=|g(Au)| \leq c\cdot \|u\|_E. Then by Hahn–Banach_theorem or alternatively through extension by continuity this yields an extension of f, called \hat f defined on all of E^*. Note that this technicality is necessary to later obtain A^* as an operator D(A^*)\to E^* instead of D(A^*) \to (D(A))^*. Remark also that this does not mean that A can be extended on all of E but the extension only worked for specific elements g\in D(A^*).

Now we can define the adjoint of A as

\begin{align}A^*: F^*&\to E^*\\ g &\mapsto A^*g = \hat f\end{align}

The fundamental defining identity is thus

g(Au) = (A^*g)(u) for u\in D(A),~ v\in D(A^*).

Definition for bounded operators between Hilbert spaces[edit]

Suppose H is a complex 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.[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. A** = Ainvolutiveness
  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 λantilinearity (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}. [2]

Moreover,

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

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 lies 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

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

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 B*A* 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:

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

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

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


\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[7] always is.[clarification needed]

Hermitian operators[edit]

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

  A  = A^{*}

which is equivalent to

 \langle Ax , y \rangle = \langle x , A y \rangle \mbox{ for all } x,y\in H. [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 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:

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

Other adjoints[edit]

The equation

 \langle Ax , y \rangle = \langle x , A^* y \rangle

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. ^ David A. B. Miller (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, pp. 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

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 .
  • Brezis, Haim (2011), Functional Analysis, Sobolev Spaces and Partial Differential Equations (first ed.), Springer, ISBN 978-0-387-70913-0 .