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 Title: Hermitian adjoint Author: World Heritage Encyclopedia Language: English Subject: Collection: Operator Theory Publisher: World Heritage Encyclopedia Publication Date:

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.

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

## Contents

• Definition for bounded operators 1
• Properties 2
• Adjoint of densely defined operators 3
• Hermitian operators 4
• Adjoints of antilinear operators 5
• Footnotes 8
• References 9

## Definition for bounded operators

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.

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:

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}. 

Moreover,

\| A^* A \| _{op} = \| A \| _{op}^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 operators

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. 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.

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.

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:

\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 always is.

## Hermitian operators

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. 

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.

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.

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.

• Mathematical concepts
• Physical applications

## Footnotes

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