Hermitian adjoint


In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint) operator on that space according to the rule

where is the inner product on the vector space.

The adjoint may also be called the Hermitian conjugate or simply the Hermitian[1] after Charles Hermite. It is often denoted by A in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators can be represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose).

The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces . The definition has been further extended to include unbounded densely defined operators, whose domain is topologically dense in, but not necessarily equal to,

Informal definition edit

Consider a linear map   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  , which is linear in the first coordinate and conjugate linear in the second coordinate. Note the special case where both Hilbert spaces are identical and   is an operator on that Hilbert space.

When one trades the inner product for the dual pairing, one can define the adjoint, also called the transpose, 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  .

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 Banach spaces edit

Let   be Banach spaces. Suppose   and  , and suppose that   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 the Hahn–Banach theorem, or alternatively through extension by continuity, this yields an extension of  , called  , defined on all of  . 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


Definition for bounded operators between Hilbert spaces edit

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.

Properties edit

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. Conjugate linearity:
  4. "Anti-distributivity": (AB) = BA

If we define the operator norm of A by






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

Definition edit

Let the inner product   be linear in the first argument. 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


Owing to the density of   and Riesz representation theorem,   is uniquely defined, and, by definition,  [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]

ker A*=(im A) edit

For every   the linear functional   is identically zero, and hence  

Conversely, the assumption that   causes the functional   to be identically zero. Since the functional is obviously bounded, the definition of   assures that   The fact that, for every     shows that   given that   is dense.

This property shows that   is a topologically closed subspace even when   is not.

Geometric interpretation edit

If   and   are Hilbert spaces, then   is a Hilbert space with the inner product


where   and  

Let   be the symplectic mapping, i.e.   Then the graph


of   is the orthogonal complement of  


The assertion follows from the equivalences




Corollaries edit

A* is closed edit

An operator   is closed if the graph   is topologically closed in   The graph   of the adjoint operator   is the orthogonal complement of a subspace, and therefore is closed.

A* is densely defined ⇔ A is closable edit

An operator   is closable if the topological closure   of the graph   is the graph of a function. Since   is a (closed) linear subspace, the word "function" may be replaced with "linear operator". For the same reason,   is closable if and only if   unless  

The adjoint   is densely defined if and only if   is closable. This follows from the fact that, for every  


which, in turn, is proven through the following chain of equivalencies:

A** = Acl edit

The closure   of an operator   is the operator whose graph is   if this graph represents a function. As above, the word "function" may be replaced with "operator". Furthermore,   meaning that  

To prove this, observe that   i.e.   for every   Indeed,


In particular, for every   and every subspace     if and only if   Thus,   and   Substituting   obtain  

A* = (Acl)* edit

For a closable operator     meaning that   Indeed,


Counterexample where the adjoint is not densely defined edit

Let   where   is the linear measure. Select a measurable, bounded, non-identically zero function   and pick   Define


It follows that   The subspace   contains all the   functions with compact support. Since     is densely defined. For every   and  


Thus,   The definition of adjoint operator requires that   Since   this is only possible if   For this reason,   Hence,   is not densely defined and is identically zero on   As a result,   is not closable and has no second adjoint  

Hermitian operators edit

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


which is equivalent to


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 conjugate-linear operators edit

For a conjugate-linear operator the definition of adjoint needs to be adjusted in order to compensate for the complex conjugation. An adjoint operator of the conjugate-linear operator A on a complex Hilbert space H is an conjugate-linear operator A : HH with the property:


Other adjoints edit

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 also edit

References edit

  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. ^ Reed & Simon 2003, pp. 187; Rudin 1991, §12.11
  • Brezis, Haim (2011), Functional Analysis, Sobolev Spaces and Partial Differential Equations (first ed.), Springer, ISBN 978-0-387-70913-0.
  • Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8.
  • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.