Adjoint operator

In functional analysis , an adjoint operator (sometimes also a dual operator ) can be defined for every tightly defined linear operator . ${\displaystyle T}$ ${\displaystyle T^{*}}$

Linear operators can be defined between two vector spaces with a common basic body . Adjoint operators are often only considered on Hilbert spaces , for example (finite-dimensional) Euclidean spaces. On finite-dimensional spaces the adjoint operator corresponds to the adjoint matrix . In the matrix calculation with real entries, the formation of the adjoint operator corresponds to the transposition , in the case of complex entries to the (complex) conjugation and transposition of the output matrix . In physics and engineering, in analogy to matrix theory, the adjoint operator is usually not referred to with but with . ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}$ ${\displaystyle T^{\ast }\,,}$ ${\displaystyle T^{\dagger }}$

definition

In this section the adjoint of an operator between Hilbert spaces is defined. The first subsection is restricted to restricted operators . In the second section, the concept is expanded to include unrestricted operators .

Limited operators

Let and Hilbert spaces and be a linear bounded operator. The adjoint operator is given by the equation ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle T\colon H_{1}\to H_{2}}$ ${\displaystyle T^{*}\colon H_{2}\to H_{1}}$

${\displaystyle \langle Tx,y\rangle _{H_{2}}=\langle x,T^{*}y\rangle _{H_{1}}}$

Are defined.

Alternatively, the figure can be viewed for each . This is a linear continuous functional defined on the whole Hilbert space. The representation theorem of Fréchet-Riesz says that for every continuous linear functional there is a uniquely determined element , so that applies to all . So in total there is exactly one element with for each . Now is set. This construction is equivalent to the above definition. ${\displaystyle y\in H_{2}}$ ${\displaystyle x\mapsto \langle Tx,y\rangle _{H_{2}}}$ ${\displaystyle z\in H_{1}}$ ${\displaystyle \langle Tx,y\rangle _{H_{2}}=\langle x,z\rangle _{H_{1}}}$ ${\displaystyle x\in H_{1}}$ ${\displaystyle y\in H_{2}}$ ${\displaystyle z\in H_{1}}$ ${\displaystyle \langle Tx,y\rangle _{H_{2}}=\langle x,z\rangle _{H_{1}}}$ ${\displaystyle T^{*}y:=z}$

Unlimited operators

Be and Hilbert dreams. With which is domain of the linear unbounded operator called. The operators and are formally adjoint to one another , if ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle D(T)}$ ${\displaystyle T}$ ${\displaystyle T\colon D(T)\subset X\rightarrow Y}$ ${\displaystyle S\colon D(S)\subset Y\rightarrow X}$

${\displaystyle \langle y,Tx\rangle _{Y}=\langle Sy,x\rangle _{X}}$

for everyone and applies. Under these conditions is generally not clearly given by. If it is densely defined , there is an excessively maximal, formally adjoint operator . This is called the adjoint operator of . ${\displaystyle x\in D(T)}$ ${\displaystyle y\in D(S)}$ ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle T}$ ${\displaystyle T}$ ${\displaystyle T^{*}}$ ${\displaystyle T}$

Examples

• If one chooses the finite-dimensional unitary vector space as Hilbert space , then a continuous linear operator on this Hilbert space can be represented by a matrix. The operator adjoint for this is then represented by the corresponding adjoint matrix . Therefore the adjoint operator is a generalization of the adjoint matrix. ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle T}$ ${\displaystyle T^{*}}$
• In this example the Hilbert space of square integrable functions is considered. With a corresponding function (for example ) is the integral operator ${\displaystyle L^{2}([0,1])}$ ${\displaystyle k\colon [0,1]\times [0,1]\to \mathbb {R} }$ ${\displaystyle k\in C([0,1]\times [0,1])}$

${\displaystyle Tx(s):=\int _{0}^{1}k(s,t)x(t)\mathrm {d} t}$

steadily on . Its adjoint operator is ${\displaystyle L^{2}([0,1])}$ ${\displaystyle T^{*}}$

${\displaystyle T^{*}y(t):=\int _{0}^{1}{\overline {k(s,t)}}y(s)\mathrm {d} s\,.}$

It is the complex conjugate of . ${\displaystyle {\overline {k(s,t)}}}$ ${\displaystyle k(s,t)}$

properties

Be tightly defined. Then: ${\displaystyle T\colon X\supset D(T)\rightarrow Y}$

• Is tight, so is , that is, and on ${\displaystyle D(T^{*})}$ ${\displaystyle T\subset T^{**}}$ ${\displaystyle D(T)\subset D(T^{**})}$ ${\displaystyle T=T^{**}}$ ${\displaystyle D(T)}$
• ${\displaystyle \operatorname {Ker} (T^{*})=\operatorname {Ran} (T)^{\bot }}$. Ker stands for the core of the operator and Ran (for range) for the image space.
• ${\displaystyle T}$is bounded if and only if is bounded. In this case ${\displaystyle T^{*}}$ ${\displaystyle \|T\|=\|T^{*}\|}$
• Is restricted, the unambiguous continuation of on ${\displaystyle T}$ ${\displaystyle T^{**}}$ ${\displaystyle T}$ ${\displaystyle X}$

Be tightly defined. The operator is defined by for . Is tightly defined, so is . If limited, then even equality holds. ${\displaystyle S\colon X\supset D(S)\rightarrow Y}$ ${\displaystyle T+S}$ ${\displaystyle (T+S)x:=Tx+Sx}$ ${\displaystyle x\in D(T+S):=D(T)\cap D(S)}$ ${\displaystyle T+S}$ ${\displaystyle (T+S)^{*}\supset T^{*}+S^{*}}$ ${\displaystyle T}$

Be a Hilbert dream and . Then the sequential execution or composition of and is defined by for . Is closely defined, the following applies . Is limited, you get . ${\displaystyle Z}$ ${\displaystyle S:Y\supset D(S)\rightarrow Z}$ ${\displaystyle TS}$ ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle TSx:=T(Sx)}$ ${\displaystyle x\in D(TS):=\{x\in D(S):Sx\in D(T)\}}$ ${\displaystyle TS}$ ${\displaystyle (TS)^{*}\supset S^{*}T^{*}}$ ${\displaystyle T}$ ${\displaystyle (TS)^{*}=S^{*}T^{*}}$

Symmetric and self-adjoint operators

A linear operator is called ${\displaystyle T\colon X\supset D(T)\rightarrow X}$

• symmetrically or formally self-adjoint , ifapplies to all. ${\displaystyle \langle Tx,y\rangle =\langle x,Ty\rangle }$ ${\displaystyle x,y\in D(T)}$
• essentially self-adjoint ifsymmetrical, densely defined and its closure is self-adjoint. ${\displaystyle T}$
• self adjoint iftightly defined andholds. ${\displaystyle T}$ ${\displaystyle T=T^{*}}$

There is also the concept of the Hermitian operator . This is mainly used in physics, but not uniformly defined.

Generalization to Banach spaces

Adjoint operators can also be defined more generally on Banach spaces . For a Banach space denotes the topological dual space . In the following, means for and denotes the dual pairing . Let be and Banach spaces and be a continuous, linear operator. The adjoint operator ${\displaystyle X}$ ${\displaystyle X'}$ ${\displaystyle \langle x,x'\rangle :=x'(x)}$ ${\displaystyle x\in X}$ ${\displaystyle x'\in X'}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle T\colon X\rightarrow Y}$

${\displaystyle T'\colon Y'\to X'}$

is defined by

${\displaystyle y'\mapsto (x\mapsto y'(Tx)).}$

In order to distinguish this adjoint operator from the adjoint operators on Hilbert spaces, they are often notated with a instead of a . ${\displaystyle '}$ ${\displaystyle *}$

However, if the operator is not defined continuously but densely, then one defines the adjoint operator ${\displaystyle T:D(T)\subset X\to Y}$

${\displaystyle T'\colon D(T')\subset Y'\to X'}$

by

${\displaystyle {\begin{aligned}D(T'):=&\{y'\in Y':\exists \,x'\in X':\langle Tx,y'\rangle =\langle x,x'\rangle \ \forall \,x\in D(T)\},\\T'y':=&x'\ {\text{für}}\ x\in D(T).\end{aligned}}}$

The operator is always closed , and is possible. Is a reflexive Banach space and , then is dense if and only if is lockable. In particular then applies . ${\displaystyle T'}$ ${\displaystyle D(T')=\{0\}}$ ${\displaystyle X}$ ${\displaystyle Y=X}$ ${\displaystyle T'}$ ${\displaystyle T}$ ${\displaystyle (T')'={\overline {T}}}$

Different conventions

In particular, in the linear complex case is for the dual operator instead also (transposition and used the transition to the complex conjugate) to confusion with the complex conjugate matrix to be avoided. The latter is also described, but this is reserved by physicists for averaging. ${\displaystyle T^{*}}$ ${\displaystyle T^{\dagger }}$ ${\displaystyle T^{*}}$ ${\displaystyle {\overline {T}}}$

literature

• Dirk Werner : Functional Analysis. Springer, Berlin 2007, ISBN 978-3-540-72533-6

Individual evidence

1. ↑ Dirk Werner : Functional Analysis. 6th, corrected edition, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , p. 236.