Hermitian adjoint

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

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

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

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

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

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

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

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 $y \in H$ for which there is a $z \in H$ 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. 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]

\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.

Hermitian operators

A bounded operator $A : H \to H$ 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

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 $A * : H \to H$ with the property:

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

Other adjoints

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.

Footnotes

↑ David A. B. Miller (2008). Quantum Mechanics for Scientists and Engineers. Cambridge University Press. pp. 262, 280.
1 2 3 4 Reed & Simon 2003, pp. 186–187; Rudin 1991, §12.9
↑ See unbounded operator for details.
↑ Reed & Simon 2003, pp. 252; Rudin 1991, §13.1
↑ Rudin 1991, Thm 13.2
↑ See Rudin 1991, Thm 12.10 for the case of bounded operators
↑ The same as a bounded operator.
↑ Reed & Simon 2003, pp. 187; Rudin 1991, §12.11

References

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 .

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