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adjoint (Definition)

Let $ \mathscr{H}$ be a Hilbert space and let $ A\colon \mathscr{D}(A)\subset \mathscr{H}\to \mathscr{H}$ be a densely defined linear operator. Suppose that for some $ y\in\mathscr{H}$, there exists $ z\in\mathscr{H}$ such that $ (Ax,y) = (x,z)$ for all $ x\in \mathscr{D}(A)$. Then such $ z$ is unique, for if $ z'$ is another element of $ \mathscr{H}$ satisfying that condition, we have $ (x,z-z') = 0$ for all $ x\in \mathscr{D}(A)$, which implies $ z-z'=0$ since $ \mathscr{D}(A)$ is dense. Hence we may define a new operator $ A^*:\mathscr{D}(A^*)\subset\mathscr{H}\to\mathscr{H}$ by

$\displaystyle \mathscr{D}(A^*) =$ $\displaystyle \{y\in \mathscr{H} :$   there is$\displaystyle z\in \mathscr{H}$   such that$\displaystyle (Ax,y) = (x,z)\},$    
$\displaystyle A^*(y) =$ $\displaystyle z.$    

It is easy to see that $ A^*$ is linear, and it is called the adjoint of $ A$.

Remark. The requirement for $ A$ to be densely defined is essential, for otherwise we cannot guarantee $ A^*$ to be well defined.



"adjoint" is owned by Koro.
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See Also: transpose operator

Other names:  adjoint operator

Attachments:
properties of the adjoint operator (Theorem) by Koro
existence of adjoints of bounded operators (Theorem) by asteroid
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Cross-references: well defined, easy to see, operator, implies, linear operator, densely defined, Hilbert space
There are 12 references to this entry.

This is version 7 of adjoint, born on 2003-07-28, modified 2006-06-15.
Object id is 4522, canonical name is Adjoint5.
Accessed 7301 times total.

Classification:
AMS MSC: 47A05 (Operator theory :: General theory of linear operators :: General )
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