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.
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.)
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:
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:
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