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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 2nd 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex[[adjoint operator]] was in a very poor state, so I copied in at least the start of [[self-adjoint operator]].

   adjoint operator was in a very poor state, so I copied in at least the start of self-adjoint operator.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 26th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexBefore the line saying that adjoint operators need not exist in general, I added the sentence that on finite-dimensional Hilbert spaces they do. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+operator/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+operator/4">v4</a>, <a href="https://ncatlab.org/nlab/show/adjoint+operator">current</a>

   Before the line saying that adjoint operators need not exist in general, I added the sentence that on finite-dimensional Hilbert spaces they do.

   diff, v4, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 16th 2023
   • (edited Nov 16th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded ([here](https://ncatlab.org/nlab/show/adjoint+operator#History)) a quote by MacLane on the history of the notion: *** > Two of [[John von Neumann|von Neummann]]'s papers on this topic &lbrack;[[Hilbert spaces]]&rbrack; 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 &lbrack;[1932](#vonNeumann1932)&rbrack;. > This story (told to me by [[Marshall Stone]]) illustrates the important conceptual advance represented by the definition of adjoint operators. &lbrack...&rbrack; I have written elsewhere &lbrack;[1970](#MacLane70)&rbrack; 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](adjoint+functor#InTermsOfHomIsomorphism) > $\;\;\;\;\; hom(F a, b) \;\simeq\; hom(a, G b)$ > between [[hom-sets]] in suitable [[categories]]. *** <a href="https://ncatlab.org/nlab/revision/diff/adjoint+operator/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+operator/5">v5</a>, <a href="https://ncatlab.org/nlab/show/adjoint+operator">current</a>

   added (here) a quote by MacLane on the history of the notion:

   ---

   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.

   ---

   diff, v5, current