nLab adjoint operator

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.)

Related concepts

• self-adjoint operator

References

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

• Saunders MacLane, The Influence of M. H. Stone on the Origins of Category Theory, in Functional Analysis and Related Fields, Springer (1970) [doi:10.1007/978-3-642-48272-4_12]

• Saunders Mac Lane: §5 in: Concepts and Categories in Perspective, in: P. Duren, A century of mathematics in America Part 1, AMS (1988) 323-365. [pdf, ISBN:0-8218-0124-4]

Original discussion in print is due to:

• John von Neumann:

  Mathematische Grundlagen der Quantenmechanik, Springer (1932, 1971) [doi:10.1007/978-3-642-96048-2]

  Mathematical Foundations of Quantum Mechanics Princeton University Press (1955) [doi:10.1515/9781400889921, Wikipedia entry]

Lecture notes:

• Bergfinnur Durhuus, Operators on Hilbert Space [pdf, pdf]