Contents

# Contents

## Definition

Let $A: H\to H$ be an unbounded operator on a Hilbert space $H$. An unbounded operator $A^*$ is its adjoint if

• $(A x|y) = (x|A^*y)$ for all $x\in dom(A)$ and $y\in dom(A^*)$; and

• every $B$ satisfying the above property for $A^*$ is a restriction of $A$.

On finite-dimensional Hilbert spaces, adjoint operators always exists, in matrix-components with respect to any orthonormal linear basis given by passage to the complex conjugate transpose matrix.

On infinite-dimensional Hilbert spaces an adjoint operator does not need to exist, in general.

## History

Recounted by MacLane 1988, p. 330:

Two of von Neummann‘s papers on this topic [Hilbert spaces] had been accepted in the Mathematische Annalen, a journal of Springer Verlag. Marshall Stone had seen the manuscripts, and urged von Neumann to observe that his treatment of linear operators $T$ on a Hilbert space could be much more effective if he were to use the notion of an adjoing $T^ast$ to the linear transformation $T$ — one for which the now familiar equation

$\;\;\;\;\; \langle T a, b \rangle \;=\; \langle a, T^\ast b \rangle$

would hold for all suitable $a$ and $b$. Von Neumann saw the point immediately, as was his wont, and wishes to withdraw the papers before publication. They were already set up in type; Springer finally agreed to cancel them on the condition that von Neumann write for them a book on the subject — which he soon did [1932].

This story (told to me by Marshall Stone) illustrates the important conceptual advance represented by the definition of adjoint operators. &lbrack…] I have written elsewhere [1970] that it is a step toward the subsequent description of a functor $G$ right adjoint to a functor $F$, in terms of a natural isomorphism

$\;\;\;\;\; hom(F a, b) \;\simeq\; hom(a, G b)$

between hom-sets in suitable categories.

(Cf. discussion at adjoint functor – idea.)

## References

The notion of adjoint operators is originally due to Marshall Stone, see also the history section above, as recounted in

Original discussion in print is due to:

Lecture notes:

Last revised on November 16, 2023 at 07:59:08. See the history of this page for a list of all contributions to it.