Conjugate transpose of an operator in infinite dimensions
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
![{\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b1ad1b751affab6f762242ec08a6cc976a28860)
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
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
![{\displaystyle \left\langle Ah_{1},h_{2}\right\rangle _{H_{2}}=\left\langle h_{1},A^{*}h_{2}\right\rangle _{H_{1}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b93c417f0e61df17853b4cb082ba36cd29ba0177)
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
![{\displaystyle A^{*}f=f\circ A:u\mapsto f(Au),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/716f005349b11ba3451029e4ef17c8004699d7b0)
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
![{\displaystyle \langle h_{f},h\rangle _{H}=f(Ah).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/798242048ec870bea0f584b58233c68b6b5e96fd)
Definition for unbounded operators between Banach spaces
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
![{\displaystyle D\left(A^{*}\right):=\left\{g\in F^{*}:~\exists c\geq 0:~{\mbox{ for all }}u\in D(A):~|g(Au)|\leq c\cdot \|u\|_{E}\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8153ac85ce1b3c1772d8a0a643eba4e75355447)
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
![{\displaystyle {\begin{aligned}A^{*}:F^{*}\supset D(A^{*})&\to E^{*}\\g&\mapsto A^{*}g={\hat {f}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b44e7414e2a33b9d7339327772db935a99f7c246)
The fundamental defining identity is thus
for ![{\displaystyle u\in D(A).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c23690165f91bb91973395ac316ac13e15e511)
Definition for bounded operators between Hilbert spaces
Suppose H is a complex Hilbert space, with inner product
. Consider a continuous linear operator A : H → H (for linear operators, continuity is equivalent to being a bounded operator). Then the adjoint of A is the continuous linear operator A∗ : H → H satisfying
![{\displaystyle \langle Ax,y\rangle =\left\langle x,A^{*}y\right\rangle \quad {\mbox{for all }}x,y\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7189d3a046917b350fe541cb2303e4bf168cd9d7)
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]
- Involutivity: A∗∗ = A
- If A is invertible, then so is A∗, with
![{\textstyle \left(A^{*}\right)^{-1}=\left(A^{-1}\right)^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce16682f6e6cd40b4eb100b088674c445c484b6c)
- Conjugate linearity:
- "Anti-distributivity": (AB)∗ = B∗A∗
If we define the operator norm of A by
![{\displaystyle \|A\|_{\text{op}}:=\sup \left\{\|Ax\|:\|x\|\leq 1\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/355591cd9a8866f23f0de504eb223bc3a7d9afee)
then
[2]
Moreover,
[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
Definition
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 y ∈ H for which there is a z ∈ H satisfying
![{\displaystyle \langle Ax,y\rangle =\langle x,z\rangle \quad {\mbox{for all }}x\in D(A).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf11deea424783274fa46cf555cd8e1a89971db5)
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 B∗A∗ if A, B and AB are densely defined operators.[5]
ker A*=(im A)⊥
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
If
and
are Hilbert spaces, then
is a Hilbert space with the inner product
![{\displaystyle {\bigl \langle }(a,b),(c,d){\bigr \rangle }_{H_{1}\oplus H_{2}}{\stackrel {\text{def}}{=}}\langle a,c\rangle _{H_{1}}+\langle b,d\rangle _{H_{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/952f2b97d342e4620ece96f243b556bd343cea74)
where
and
Let
be the symplectic mapping, i.e.
Then the graph
![{\displaystyle G(A^{*})=\{(x,y)\mid x\in D(A^{*}),\ y=A^{*}x\}\subseteq H\oplus H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c003a06e2a311f9f744027278cd977dcc9b027)
of
is the orthogonal complement of
![{\displaystyle G(A^{*})=(JG(A))^{\perp }=\{(x,y)\in H\oplus H:{\bigl \langle }(x,y),(-A\xi ,\xi ){\bigr \rangle }_{H\oplus H}=0\;\;\forall \xi \in D(A)\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/192ef2997c921513a8a69dfbac6ea36017e386f6)
The assertion follows from the equivalences
![{\displaystyle {\bigl \langle }(x,y),(-A\xi ,\xi ){\bigr \rangle }=0\quad \Leftrightarrow \quad \langle A\xi ,x\rangle =\langle \xi ,y\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ba504a08c2c5d029bbe8f8751f5512da5c72b2)
and
![{\displaystyle {\Bigl [}\forall \xi \in D(A)\ \ \langle A\xi ,x\rangle =\langle \xi ,y\rangle {\Bigr ]}\quad \Leftrightarrow \quad x\in D(A^{*})\ \&\ y=A^{*}x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef08845a4e6b5c5316f9bdda07e3c38da51f570)
Corollaries
A* is closed
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
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
![{\displaystyle v\in D(A^{*})^{\perp }\ \Leftrightarrow \ (0,v)\in G^{\text{cl}}(A),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22961a00c1250a6e5f26a08091128329b252e637)
which, in turn, is proven through the following chain of equivalencies:
![{\displaystyle {\begin{aligned}v\in D(A^{*})^{\perp }&\Longleftrightarrow (v,0)\in G(A^{*})^{\perp }\Longleftrightarrow (v,0)\in (JG(A))^{\text{cl}}=JG^{\text{cl}}(A)\\&\Longleftrightarrow (0,-v)=J^{-1}(v,0)\in G^{\text{cl}}(A)\\&\Longleftrightarrow (0,v)\in G^{\text{cl}}(A).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59cb99d8bc536e067304cc69e41ae51de79fce85)
A** = Acl
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,
![{\displaystyle {\begin{aligned}\langle J(x_{1},x_{2}),(y_{1},y_{2})\rangle _{H\oplus H}&=\langle (-x_{2},x_{1}),(y_{1},y_{2})\rangle _{H\oplus H}=\langle -x_{2},y_{1}\rangle _{H}+\langle x_{1},y_{2}\rangle _{H}\\&=\langle x_{1},y_{2}\rangle _{H}+\langle x_{2},-y_{1}\rangle _{H}=\langle (x_{1},x_{2}),-J(y_{1},y_{2})\rangle _{H\oplus H}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe914f9b7f4d13b2abc0541f5a1c35c2d36f0bca)
In particular, for every
and every subspace
if and only if
Thus,
and
Substituting
obtain
A* = (Acl)*
For a closable operator
meaning that
Indeed,
![{\displaystyle G\left(\left(A^{\text{cl}}\right)^{*}\right)=\left(JG^{\text{cl}}(A)\right)^{\perp }=\left(\left(JG(A)\right)^{\text{cl}}\right)^{\perp }=(JG(A))^{\perp }=G(A^{*}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e849eb4d07499e10882fea27416e788b3a0e4bd)
Counterexample where the adjoint is not densely defined
Let
where
is the linear measure. Select a measurable, bounded, non-identically zero function
and pick
Define
![{\displaystyle A\varphi =\langle f,\varphi \rangle \varphi _{0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caf4081f5ba558de8952e3f2476e14486980927e)
It follows that
The subspace
contains all the
functions with compact support. Since
is densely defined. For every
and
![{\displaystyle \langle \varphi ,A^{*}\psi \rangle =\langle A\varphi ,\psi \rangle =\langle \langle f,\varphi \rangle \varphi _{0},\psi \rangle =\langle f,\varphi \rangle \cdot \langle \varphi _{0},\psi \rangle =\langle \varphi ,\langle \varphi _{0},\psi \rangle f\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4752444d472d013e323b7b197b643814204b390)
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
A bounded operator A : H → H is called Hermitian or self-adjoint if
![{\displaystyle A=A^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f737c2418b81620e8118c3de35b2392e3eb4498a)
which is equivalent to
[6]
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
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∗ : H → H with the property:
![{\displaystyle \langle Ax,y\rangle ={\overline {\left\langle x,A^{*}y\right\rangle }}\quad {\text{for all }}x,y\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e969441a5946242aea63a3a1741b21590dee1b5d)
Other adjoints
The equation
![{\displaystyle \langle Ax,y\rangle =\left\langle x,A^{*}y\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/03ff59a64abb29817caad5091712f950e9391633)
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
- Mathematical concepts
- Physical applications
References
- 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.
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