nLab adjoint operator




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

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

  • every BB satisfying the above property for A *A^* is a restriction of AA.

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.


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 TT on a Hilbert space could be much more effective if he were to use the notion of an adjoing T astT^ast to the linear transformation TT — one for which the now familiar equation

Ta,b=a,T *b\;\;\;\;\; \langle T a, b \rangle \;=\; \langle a, T^\ast b \rangle

would hold for all suitable aa and bb. 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 GG right adjoint to a functor FF, in terms of a natural isomorphism

hom(Fa,b)hom(a,Gb)\;\;\;\;\; hom(F a, b) \;\simeq\; hom(a, G b)

between hom-sets in suitable categories.

(Cf. discussion at adjoint functor – idea.)


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.